AD-A090 838 MOORE SCHOOL OF ELECTRICAL ENGINEERING PHILADELPHIA PA F/B 17/9SUPER-RESOLUTION IMAGERY BY FREQUENCY SWEEPING. U)AUG 80 N H FARHAT, C WERNER AFOSR-77-3256

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UNIVERSITY of PENNSYLVANIA DTICELECTE

1Ue Moore School of Electrical Engineering OCT 2 319808-2 PHILADELPHIA, PENNSYLVANIA 19104

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The Moore School of Electrical Engineering 6102 31 B200 S. 33rd St., Phila., Pa. 19104

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SUPPLEMENTARY NOTES

KEY WORDS (Continue on reverse side it necessary and identify by block number)

Microwave 3-D imaging, frequency diversity, holography, inverse scattering,aperture synthesis, Fourier domain projection, target derived reference,retwork analysis, radar cross-section, projection hologram, tomographic radar.

A9n[TACT (Continue on reverse side It necesa ry and identify by block number)

A longstanding problem in radar and electromagnetic scattering measure-ments is the reconstruction of object shape and detail, i.e. an image, fromfar field scatter data to be used in object identification and classification.During the period covered by this report we have been able to demonstratein a feasibility study the first reconstruction of a 3-D image of a perfectlyreflecting object from its wave-vector diversity (multifrequency and multi-aspect) scatter data makii,, use of a new Wighted Fou'7eA Domain Pojeection

FORM ,. ,... .....

S CLARITY CLASSIFICATIOA OF TH1s, PAGE(Whesn Date In.,.d)

heotern (WFDPT) and establish a relationship between holography with wavelengthdiversity and inverse scattering. Coherent wave vector diversity radar tech-niques are used to access a finite volume of the 3-D Fourier space (also knownas reciprocal space or p-space) of the scattering object. The WFDPT permits the

, use of hybrid (digital/optical) computing that enables the retrieval and displayof 3-D image information in parallel slices or cross-sectional outlines. Themajor attributes of this approach as compared to totally digital computing areits potential for displaying a true 3-D image in real-time. Wave-vector di-versity methods appear suitable for the imaging of two classes of;practical ob-jects namely non-dispersive perfectly reflecting objects of the type oftenencountered in radar (and sonar) and semitransparent weakly scattering objectssuch as certain ultrasound and light scattering objects encountered in biologyand medicine. The method has several unique characteristics. It furnishestrue super-rkM ution, i.e. resolution exceeding the classical Rayliegh Limitof the available recording aperture, in this case a highly thinned (widely dis-persed) broad-band coherent receiver array. True super-resolution is achievedbecause of an inherent aperture synthsis due to frequency diversity (frequencyscanning, stepping or comb illumination) and conversion of spectral degrees offreedom into spatial image detail. Accordingly, when applied to the imaging ofa dispersive object, a tLget ignotwe rather than a geometrical image shouldbe expected. Such a target signature could still be useful in object identifi-

. cation and classification since it contains information pertaining to the*:jw material composition of the object intermixed with geometrical image detail.--. 4 The use of frequency diversity was found to lead to a unipolar impulse response.

This is very useful in suppressing coherent noise (speckle) which is known to be3 the major drawback of coherent imaging.Preliminary work on 3-D image display has yielded encouraging results on

'3-D display from a series of weighted projection holograms of various slices of

f a test object. The projection holograms were viewed in rapid succession using,the virtual Fourier transform.

To identify optimal and practical approaches to wave-vector diversitydata acquisition a unique microwave measurement system has been assembled,

.1installed and tested in our anechoic chamber facility. The system was used alsoin the study of TDR (target derived reference) methods in which a reference forphase measurement can be furnished by the scattering object eliminating thus theneed for costly local oscillator distribution networks and eliminating at the

......same time undesirable range phase ambiguities from the collected data. The-measurement facility is computer aided furnishing thereby semi-automatic control.of object positioning or orientation, frequency stepping, data acquisition andstorage, and final data correction and analysis. A high resolution CRT display

". enables the display of weighted projection hoeo am computed from the p-spacedata making use of the WFDPT. Preliminary results of this measurement system, apabilities included in this report confirm its tremendous versatility, Atthis stage, the program strongly suggests the practical feasibility of a newgeneration of cost effective, real-time, super-resolving 3-D imaging radars thatcan because of their 3-D image slicing characteristic be appropriately referred

• to as Tomogtaphic Radou (Tomoszslice in Greek).

Finally, a study of 3-D imaging using other forms of broadband radiationsuch as impulsive, random noise and particularly thermal emission for passive3-D wavelength diversity imaging has been also initiated.

r1 k

(~I

The findings in this report are thoseof the authors and are not to be in-terpreted as the official position ofthe Air Force Office of ScientificResearch or the U.S. Government.

)B

TABLE OF CONTENTS

Page

1. Introduction I

2. Summnary of Important Results 2

3. Conclusions 7

4. List of Publications 8

5. Appendices 19

1. The Frequency Displaced TargetDerived Reference.

11. Holography, Wavelength Diversityand Inverse Scattering.

III. The Virtual Foijrier Transformand its Application in ThreeDimensional Display.

IV. An Automated Frequency Responseand Radar Cross-Section Measure-ment Facility for MicrowaveImaging.

LLVELUNIVERSITY OF PENNSYLVANIA

PHILADELPHIA, PENNSYLVANIA 19104

FINAL REPORT

SUPER RESOLUTION IMAGERYBY FREQUENCY SWEEPING

AIR FORCE OFFICE OF SCIENTIFIC RESEARCH/NE

BUILDING 410 BOLLING AIR FORCE BASEWASHINGTON, D.C. 20332

AUGUST 15, 1980

GRANT NUMBER R jR?7- 32.i'4

DTICl{!lO ELECTVS_ R%

Acces-qlon For PREPARED BYrTI-S GRA&I- 1) CT 2 -, 1980P MC TAB N.H. FARHAT AND C. WERNER

L BI , ~*IR FORCE o, , 'z Scz',- oF::RE Op SCIE, ,'- NTIFIC RESEARC8 tA73C)

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;t : c J l ui .tr bu t o l lA; AW tFR 2(7 b ).A. D. BLOSLechnjo 0al Information Offi...

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HIGH RESOLUTION FREQUENCY SWEPT IMAGING

1. Introduction

The aim of the research work outlined in this final report was the analysisand investigation of methods by which frequency or wavelength diversity techniquescan be employed to impart to a highly thinned, and therefore cost-effective,longwave (microwave or ultrasound) imaging aperture resolution capabilities betterthan its monochromatic classical (Rayligh) limit achieving thereby super-resolutionby means of frequency synthesized apertures. This approach to longwave imaginggains practical significance when one considers the current highly developed stateof the art of broadband microwave gear suitable for use in a new generation ofcost-effective high resolution microwave imaging radars utilizing frequencydiversity techniques.

It is well known that the development of longwave holographic imaging systemspossessing resolution and image quality approaching those of optical systems ishampered by three factors: (a) prohibitive cost and size of longwave imagingapertures, (b) rapid deterioration of longitudinal resolution with range, (c)inability to view a 3-D image as with optical Fresnel holograms because of awavelength scaling problem and (d) degradion of image quality by speckle orcoherent noise because of the low numerical apertures attainable with presenttechniques. For example, a longwave imaging aperture operating at a wavelengthof 3 cm should be about 3 km in size in order to achieve image resolution comparableto an ordinary photographic camera. In addition to inconvenient size, the costof filling such a large aperture with suitable coherent sensors is clearly prohibitive.Furthermore, recall that in conventional longwave holography when optical imageretrieval is utilized, it is necessary to store the longwave hologram data (fringepattern) in an optical transparency suitable for processing on the optical benchusing laser light. In order to avoid longitudinal distortion* of the reconstructedimage, the size of the optical hologram replica must be m (=Xlong/Xlaser) timessmaller than the longwave recording aperture. For the example cited earlier, thismeans an optical hologram replica of less than a millimeter in size. It iscertainly not possible to view a virtual 3-D image through such a minute hologrameven with optical aids since these tend to introduce their own longitudinaldistortion. As a result, longwave holographers have long learned to forgo 3-Dimagery and settled instead for 2-D imagery obtained by projecting the reconstructedreal image on a screen. This permits lowering of the reduction factor m andconsequently relaxing the resolution requirements of the photographic film whichallows in turn the use of highly convenient Polaroid transparency film forpreparation of the optical hologram replica. Because of the small size (measuredin wavelength) of longwave apertures attainable in practice and the above methodsof viewing the real image, speckle noise is always present leading to degrationIn image quality.

* Longitudinal distortion causes for example the image of a sphere to appear

elongated in the range direction like a very long ellipsoid.

- 1

In this report we summarize the main results of our investigation underthis grant. Our findings show that frequency diversity techniques not onlycircumvent the limitations discussed above but provide a means of viewingtrue 3-D images of distant objects such as satellites and aircraft. It is worth-while to point out that our studies of wave-vector diversity imaging (or frequencyswept imaging) were motivated to some extent by evidence of super-resolved "imaging"capabilities in the dolphin and the bat which are known to use frequency swept(chirp) signals in their "sonar" to discern small objects in their environment.

2. Summary of Important Results

The main findings of the study, details of which are given in the appendices andour publications (see list of publications), are outlined next.

(a) Wave-vector diversity (multifrequency and multiaspect) techniques canbe used to enhance the amount of object information collected by a broadbandcoherent aperture deployed in the far field of the scattering object. Thus thedata collected by a highly thinned array of coherent receivers intercepting thewavefield scattered from a distant 3-D reflecting object, as the frequency of itsillumination (and/or its direction of incidence) are changed (see Fig. 1-a forexample), can be stored as a 3-D data manifold in p-space (Fig. 2) from which animage of the object can be retrieved by means of a 3-D Fourier Transform. Thesize and shape of the 3-D data manifold, and therefore the resolution, depend onthe relative positions of the object,the transmitter (illuminator),and the receivingarray and on the spectral width of the illumination utilized.

(b) The data collected must be corrected for a quadratic phase factor F(caused by the unequal distances between the object and the receiving stationsforming the widely dispersed imaging array) before it is stored in a 3-D manifoldin p-space and an undistorted image of the 3-D reflecting object reconstructedthrough the 3-D Fourier transform operation. A bothersome range-azimuth ambiguityis also avoided through elimination of this quadratic phase term.

The most promising methods for data acquisition and correction is that whichutilizes a target derived reference (TDR) at the synchronous detectors of thevarious receivers to correct for the unequal phase shifts or propagation timedelays from the object to each receiver. In this approach the data furnished bythe various receivers of the recording array is free of the undesired factor F.Therefore no additional processing by a computer will be necessary before filingthe data at the appropriate locations in p-space. The TDR method has severaladvantages which include:

(i) Elimination of the need for a costly and unreliable centrallocal oscillator distribution network.

(ii) Because TDR results in a recording configuration similar tothat of a lensless Fourier Transform hologram, the resolution requirementsfrom the recording device are greatly relaxed*.

*A. Macovski, "Hologram Information Capacity", J. Opt. Soc. Am., Vol. 60,

Jan. 1970, pp. 21-29.

2

A --- MOW

In longwave holography this fact is translated into a significantreduction of the number of receiving elements in the recording aperture.In addition the use of TDR allows us to place all the resolving powerof the recording aperture on the target. This means that highresolution images of distant isolated targets should be feasible witharray apertures consisting of tens of elements. The ability tosynthesize a 2-D receiving aperture with a Wells arrayt consisting oftwo orthogonal linear arrays one of transmitters and the other ofreceivers provides further means of reducing the number of stationsneeded for data acquisition without sacrifice in resolution. Afrequency swept Wells array of 10 transmitters and 10 receivers usinga (2-4) GHz sweep should be able to easily furnish 104 3-D distinguishableresolution cells on the target which is more than sufficient fordiscerning the scattering centers on practical targets.

(iv) Greater immunity to phase fluctuations arising fromturbulance and inhomogenieties in the propagation medium because boththe reference and imaging signals arriving at each receiving elementof the aperture travel roughly over the same path.

(v) TDR eliminates the range azimuth ambiguity and excessivebandwidth problems that arise in fast frequency swept imaging whenthe reference signal for the array aperture is distributed instead fromthe illumination source or a centrally located local oscillator phaselocked to it.

Two TDR-methods have been considered to some extent in ourwork to date. In one method which we term LFTDR (Low FrequencyTwuget Ve'tived Refepence), the object is assumed to be illuminatedsimultaneously with a high frequency imaging signal and a low frequencysignal that is a subharmonic of the illuminating frequency. Thesubharmonic reference frequency w is chosen such that kr <<,l, beingthe maximum linear dimension of the object and k = w /c, c beingthe velocity of light. This places scattering fFom the object inthe Rayliegh region where the object behaves as point scatterer withzero phase contribution. The far field phase of the reference signalat any receiver is therefore entirely due to propagation between areference point formed at the object to the receiver. A method formeasuring this reference signal phase and using it to correct theimaging signal phase due to propagation has been proposed by Porter*and analyzed for a one-dimensional object geometry. The referencesignal phase and the imaging signal phase are measured separatelyat each receiving station with the aid of two receivers whose localoscillators (L.O)'s, one at the reference frequency and one at theimaging frequency, are phase-locked only to each other and not to acentral local oscillator as would be the case were we to use aconventional receiver array. Phase locking of the two LO's can beaccomplished by simply making the imaging L.0 a harmonic of the

tC.N. Nilsen and D.N. Swingler, "Quasi-Real-Time Inertialess MicrowaveHolography", Proc. IEEE (Letters), Vol. 65, March 1975, pp. 491-492.

*R.P. Porter, "A Radar Imaging System Using the Object as Reference",Proc. IEEE (Letters), Vol. 59, Feb. 1971, pp. 307-308.

3

610

reference L.O. This would eliminate the difficulties encountered inthe implementation of large or giant thinned coherent receiving arraysof the type required here, namely the distribution of a central localoscillator signal. A great reduction in cost and effort associatedwith installation of a central L.O. distribution network can thus beachieved. This cost reduction should be compared however with thecost of implementing a LFTDR. Because of the large difference betweenthe high frequency imaging frequencies and the low frequency referencefrequency required for the high resolution imaging of practical objects,the same microwave gear can not be used for both frequencies. Thiscould increase system cost. In addition since the measured referencephase must be multiplied by a factor a equal to the ratio of theimaging to the reference frequency before being used as a referencephase in the imaging signal measurement, any errors in the referencephase measurement will also be amplified by this ratio. The precisionof the reference phase measurement and phase error analysis areimportant and will have therefore to be examined further.

Another TDR methods which we call the Ftequency DLspzaced TaAge~tVeived RefeAence (FDTDR) also shows promise. In this method, theanalytical details of which are outlined in appendix I, the objectis illuminated simultaneously during the sweep with two phase lockedimaging frequencies t and wi= w + Aw, Aw being a small incrementalfrequency. This can le realized ilso by single side band modulationof the swept signal or by phase locking two sweep oscillators.Measurement of the differential phase between the signals scatteredfrom the target at these frequencies yields Aw + b-c PT R )l R being

the distance from the transmitter to the object and R being thedistance from the object to the receiver. Multiplicaion of thisphase by wl/Aw yields the phase factor F at frequency w which wouldbe used to correct the phase measured at w . At firit look thismethod would appear to still require a reference local oscillator.This however is not so since the procedure outlined above need notinvolve explicit phase measurements and multiplications. Forexample by mixing the two received signals at w and w Aw in asquare law detector at each receiver a beat signal at frequency Awis derived whose phase is equal to Aw (R + The phase shift ofc R ) Tepaesito

this signal due to the object is effectively zero because the wavelengthat Aw is much larger than the object extent making it behave effectivelyas a point scatterer. Harmonic mixing of the signal w received ateach receiver with this beat signal should yield the cohrected p-spacedata at w . Because of the small difference Aw between the twofrequencies W and w utilized, the effect of phase errors due to systemand atmospheric propagation could be more completely cancelled in thismethod than in the low frequency TDR methods. The small difference Awmeans also that unlike the LFTDR case the same microwave gear (antennas,transmission lines and other microwave circuit components) can be utilizedin the handling of the reference and ima'iva signals. A variation ofthe TDR technique involving double side-band modulation is also possibleand appears to be more simple to implement than the singleside-band method.

94

(c) Because in addition to being dependent on geometry,the dimensionsof the 3-D data in p-space shown in Fig. 2 are dependent on the spectralrange of the illumination, super-resolution (i.e. resolution beyond theclassical limit of the available physical aperture) is achieved. Thisaperture synthesis by wave-vector or frequency diversity helps cut downarray cost (since a thinned array can be used to frequency synthesize alarge array with higher filling factor).

(d) Fowie Domain Projection Theorems (see appendixes II and IIIfor details) enable the generation of two dimensional holograms fromprojections (or weighted projections) of the corrected p-space 3-D datamanifold of Fig. 2 permittin! thereby optical image retrieval of the 3-Dobject in slices parallel to the projection plane one at a time. Forexample, Fig. 1-b shows the projection hologram for the p-space dataobtained in a computer simulation of the arrangement shown in Fig. 1-a.The central cross-sectional outline of the object (the two 1 m diameterreflecting spheres of Fig. 1-a) retrieved from this projection hologramby means of a 2-D Fourier transform carried out on the optical bench isshown in Fig. 1-c. A similar example is shown in Figs. 3 and 4. Figure3 shows a second test object consisting of 3-D distribution of a set of8 point scatterers with locations and spacings given in the Figure.Figure 4 shows the projection holograms corresponding to the three slicesof the object containing the point scatterers and the image retrievedfrom each. The sweep width in this example, as in the previous example,was (2-4) GHz however the number of receivers in the recording arrayhas been reduced from 50 to 16. These computer simulations demonstratethat a 3-D (lateral and longitudinal) resolution of the order oftwenty centimeters*is easily achieved with a frequency sweep covering only(2-4) GHz using a broad-band array of 16 receivers and one transmitter.Wider-spectral windows should yield better resolution. It is worthwhileto note in this respect that commercial microwave sweepers and synthesizersare available with a spectral coverage of (.1-25) GHz indicating apotential for practical resolutions of the order of possibility fewcentimeters with cosc-effective broad-band apertures consisting of tensof receivers operating with one central illuminator.

(e) The viewing or the display of a true 3-D image of the variousslices or cross-sectional outlines should be possible by reconstructionof the various projection holograms in rapid succession while projectingthe reconstructed real images of the corresponding slices on a rapidlymoving projection screen. The screen would be displaced rapidly (togetherwith the Fourier transforming lens) on the optical bench in the axialdirections by small amounts proportional to the distances between thevarious slices. In another approach we have found that the 2-D virtuatFouieA tAansfow of a projection hologram can be carried out by simplyviewing (with the unaided eye) a transparency containing an array ofreduced replicas of the projection hologram arranged side-by-side with apoint source. The image retrieved in this fashion would lie in theplane of the point source. This approach has the potential for 3-Ddisplay by viewing the virtual images retrieved from a series of projectionholograms corresponding to different slices or cross-sectional outlines

*This means 103 distinguishable 3-D resolution cells in the (2 x 2 x 2)m3

volume of the assumed object.

5

of the object passed in front of the eye in rapid succession whilemoving the reconstruction point source axially back and forth at asuitable rate of incremental axial displacements. A proposed electro-optical scheme that permits carrying out this procedure in real-timeusing a rapidly recyclable spatial light modulator (SLM) operating ina reflection mode is shown in Fig. 5. The computer, the high resolutionCRT and the projection optics are used to project reduced noncoherentimages of the various projection holograms in rapid succession on theSLM while the axial position of the reconstruction point sources isaltered rapidly also under computer control. The point source neednot be derived from a laser in order to yield an image but could alsobe a miniature "grain of wheat" light bulb. Details of this task arefound in Appendix III.

(f) As seen in (e), unlike monochromatic longwave holographicimaging, there is no specific scaling requirements imposed on theprojection holograms in order to avoid longitudinal distortion in theoptical reconstruction circumventing thus the wavelength scalingproblem.

(g) Because of the broad spectral extent of the illuminationused and ability to display the reconstructed image in separate slices,speckle or coherent noise, which is known to plague coherent imagingsystems, is suppressed making the system behave in as far as imagenoise is concerned like a noncoherent imaging system but at the sametime enjoy the superior detection characteristics associated withsynchroneous detection techniques.

(h) The broad-band nature of the imaging process also helpssuppress undesirable image detail that could arise from object resonance.which could seriously degrade image quality in a monochromatic imagingsystem.

(i) The data collected at every receiver, represents aftercorrection, essentially the frequency response of the scattering objectmeasured from a different aspect angle. Assuming the scattering processis linear, this frequency response is related to the impulse response ofthe object by a Fourier transform (see ref. 5 in List of publications).This suggests that impulse illumination can be utilized instead offreguency swept illumination. When this is done,the 3-D data manifoldin p-space may be generated by Fourier transforming the impulse responseat each receiver, correcting the data for the Factor F mentioned in (b),and storing the result in the apprupriate p-space locations for eachreceiver. The resulting p-space volume accessed in this fashion can thenbe employed as described earlier to yield 3-D image information. Impulseillumination is desirable in certain instances of rapid target motionbut may be more difficult to implement than frequency swept illumination.Since the impulse reponse of a time invariant linear system can also bededuced from white noise excitation and corellation of the outputresponse with the input as described elsewhere in more detail (see 5in list of publications), it follows that the techniques described inthis report for coherent broadband radiation should be equally applicablewith minor signal processing modification to noise-like broadband

.6

radiation including passive black-body radiation.

(j) Experimental verification for both the principle offrequency diversity imaging and the TDR concept were obtained with theaid of a semi-automated network analyzer configured and installed in arecently refurbished anechoic chamber within the scope of this program(seeFigs. 6 and 7). This versatile system is capable of vector (amplitudeand phase) measurements of wavefields scattered from test objectssituated in the anechoic chamber over any frequency range lying inthe (.1-18)GHz range for a variety of polarizations. A test objectconsistinn of two parallel cylinders 25 cm apart each 5 cm in diameterand 50 cm long was mounted on a rotating styrofoam pedestal that is

under computer control and illuminated as shown in Fig. 8. Thedistance from the center of the object to the illuminating parabolicantenna to the left and the receiving horn feeding the networkanalyzer was 2.5 m. The complex frequency response of this objectwas measured in the (5-14)GHz range and the data stored for 128 objectorientation covering 3600. The stored data was corrected for range-phase with a synthetic TDR generated in the computer and the correcteddata displayed and photographed yielding the frequency swept hologramshown in Fig. 9 (c). The image retrieved from this hologram via an

optical Fourier transform carried out on the optical bench is shown inFig. 9 (d). For reasons of comparison a computer simulation of thisexperiment assuring a (2-18)GHz sweep was performed. The resultantrange-phase corrected hologram and the image retrieved from it opticallyare shown in Fiv'. 9 (a) and (b). Further detail on this phase of theprogram were reported in an MSc. thesis made part of this report inAppendix IV. This part of the program is being continued with the aimof further enhancinq measurement accuracy and demonstrating imaging ofa nonsimple 3-D test object such as a model aircraft utilizing polarizationdiversity to further enhance image quality.

3. Conclusions.

The primarily analytical and numerical study of frequency diversityimaging performed under this grant demonstrates conclusively the feasibilityof a new generation of coherent broadband imaging radars capable offurnishing 3-D image detail of distant target with cost effective giantapertures and efficient digital/optical signal processing.

Future work in this area will focus more on the analysis and identificationof optimal methods for data acquisition, processing and 3-D display. Theultimate aim is the generation of design criteria for a prototype systemand its assessment in the 3-D imaging of low flying aircraft passing withinrange of our facilities on route for landing at the Philadelphia Airport.

7

List of Publications

1. N.H. Farhat, "Frequency Synthesized Imaging Apertures", Proc. 1976,International Optical Computing Conference, IEEE Cat. #76 CH 1100-7C,pp. 19-24.

2. N.H. Farhat, M.S. Chang, I.D. Blackwell and C.K. Chan, "FrequencySwept Imaging of a Strip", Proc. 1976, Ultrasonics Symposium,IEEE Cat. #76 CH 1120-5SU.

3. J.D. Blackwell and N.H. Farhat, "Image Enhancement in LongwaveHolography by Electronic Differentiation", Optics Communications,Vol. 20, Jan. 1977, pp. 76-80.

4. C.K. Chan, N.H. Farhat, M.S. Chang and J.D. Blackwell, "New Resultsin Computer Simulated Frequency Swept Imaging", Proc. IEEE (Letters),Vol. 65, pp. 1214-1215, Aug. 1977.

5. N.H. Farhat, "Principles of Broad-Band Coherent Imaging", J. Opt. Soc.Am., Vol. 67, pp. 1015-1020, Aug. 1977.

6. N.H. Farhat, "Comment on Computer Simulation of Frequency Swept Imaging",Proc. IEEE, Vol. 65, pp. 1223-1226, Aug. 1977.

7. N.H. Farhat, "Comment on a New Imaging Principle", Proc. IEEE (Letters),Vol. 66, pp. 609-700, May 1978.

8. N.H. Farhat, "Microwave Holographic Imaging - Prospects For a Real-TimeCamera", SPIE, Vol. 180, Reat-T~me Signal Proceing I, (1979).

9. N.H. Farhat and C.K. Chan, "Three-Dimensional Imaging by Wave-VectorDiversity", Acousticat Imaging, Vol. 8, A. Metherell (ed.), PlenumPress, New York (1980), pp. 499-515.

10. C.K. Chan and N.H. Farhat, "Frequency Swept Imaging of Three DimensionalPerfectly Reflecting Objects", IEEE Trans. on Antennas and Propagation -Special Issue on Inverse Scattering. (Accepted for publication.)

11. C.K. Chan, "Analytical and Numberical Studies of Frequency SweptImaging", University of Pennsylvania, Ph.D. Dissertation (1978).

12. N.H. Farhat, "Microwave Holography and Coherent Tomography", (Invitedpaper). Presented at 1980 IEEE/MTT's International Microwave Symposium,Electromagnetic Dosimetric Imaging. (To be published in specialconference proceedings).

mom

8

Related Publications

I. N.H. Farhat, "New Imaging Principle", Proc. IEEE (Letters), Vol. 64,pp. 379-380, March 1976.

2. N.H. Farhat, T. Dzekov and E. Ledet, "Computer Simulation of FrequencySwept Imaging", Proc. IEEE (Letters), Vol. 64, pp. 1453-1454, Jan. 1977.

3. G. Tricoles and N.H. Farhat, "Microwave Holography: Applications andTechniques", Invited paper, Proc. IEEE, Vol. 65, pp. 108-121, Jan. 1977.

4. M.A. Kujoory and N.H. Farhat, "Microwave Holographic Substraction forImaging of Buried Objects", Proc. IEEE (Letters), Vol. 66, pp. 94-96, Jan. 1978.

5. M.A. Kujoory and N.H. Farhat, "Format Generation For Double CircularScanners For Use in Longwave Holography", Acoustical Imaging andHolography, Vol. 1, No. 2, pp. 133-141 (1979).

6. N.H. Farhat and J. Bordogna, "An Electro-Optics and Microwave-OpticsProgram In Electrical Engineering", IEEE Trans. on Education - SpecialIssue on Optics Education (accepted for publication).

7. N.H. Farhat, "Holographically Steered Millimeter Wave Antennas",IEEE Trans. on Antennas and Propagation, Vol. AP-28, July 1980,pp. 476-480.

9

Reflecting Spheres

x 15cm (2GHz)

T X (t x 2=7.5cm (tGHz)

oz =O Reeier(a)

Trans

(C)

Fig. 1.COMrnput r .Lult o of Wave-vec:tor Diversity Imaging,

(a) GeuoneLry, (b) 1'rojcL ion Ho-logr-am, (c) Retrieved

Cet'rral (:rosS-SL(!~ ionat Imnage.

10

/'/I ' ~__ .3-D perfect conduictin object

-3-D rec duV voke inS\ \ -space syritsiz by.

\ frequncy sw&&Pin

2-D array of

trnmtter" °•

Fig.2 Three dimensionzl -s c data generated by

frequency sweeping and collected by a 2-D

array of receivers.

1]

JI

-_,09 7 0

(7(xoib ,-f0)

R'yX

y Fez 2 zo plane

- K0 O

-- yo

I' Rl z0 plane

Fig. 3. 3-D object consisting of a set of eight pointscatterers shown in isometric and R'-RI planeviews at R-z =O100 xyz OOcmT

12

(a)

(b)

Fig. 4. Projection holograms and their opticalreconstructions for the set of point scatterersin Fig.7.lO at different RI planes. (a) Hologramand reconstructed image of Z scatterers at R'=-zplane. (b) Hologram and image at R'=O piaRe. 0(c) Hologram and image at R plane

13

A LU

cr- z(1)LL 0

- c3 WO

zz07 U 'U

uw wa-

0i

uLiU- 0

z (.nI0L 0~ 4-'pu Lu5

I-. cr

0-0U)(n

zG

00-o

0J

LLIn

CO 0

0 1- - --rw 0 '

C'C

LLIL

x

0 CD

Z C)

IZ W

00

> Z

01 O ~0

(LI 4 0

-2~

cr 1 clLZ - 4$

z.E Ir O7:C 04zI 3t .Hz1 .410

L ) ri L L

0 P-- L >

Dj~ C) c~

Z N

w L) 4)

c~ "N

Cr OD w -

LU C wy4 4 4z %aw-

15 4 0

zoJ --

- u _ _ _ _

z r~~~- t Gin u L

11 41

$4

44.0

4.r4

(a)

ir

-4 1 9

(b)

Fig. 8. Two views of dual-cylinder test object in Anechoic chamber.(a) View showing illuminator to the left and the receivinghorn on the right separated by absorbing barrier. (b) Viewshowing test object mounted on rotating styrofoam pedastel.Cylinders are.5 cm in diameter, 50 cm long, 25 cm apart.

17

(a) (c)

(b) (d)

Fig. 9. Frequency swept holograms and retrieved images for a dual-cylindertest object. (a) Computed frequency swept hologram for a (2-18)GHzSweep and (b) retrieved image; (c) measured frequency swept hologramfor a (5-,14)GHz sweep and (d) retrieved image.

Iv . . .. .... ........ 8

APPENDICES

19

APPENDIX I

The Frequency Displaced Target Derived Reference

A second TDR method. which we refer to as a Frequency DzspacedTagtgeut VeAied RefeAenco_ (FDTDR) method also shows promise. Thismethod involves simultaneous illumination of the object with two phaselocked imaging frequencies w and w2 = W1 + Aw that differ by a smallfrequency increment Aw. Referring to eq. (9) of ref.10 (see list of publications) we can write for the far field at a givenreceiver location RR,

Jk1 -jkl (RT +RR) -j P1.r1pI(kIRR) = 2--R e fu( ) e 1 dr (1)

J(kl+Ak) -JkI(RT+RR) -JAk(RT+RR)1p2(k2,RR) - 2TIRR e e

_-j5 (1+ N2 )xf U(i)e 11 6 2)dr (2)

where k1, 2 = w1,2/c and Ak = Aw/c.

By making Aw/w <<1 the integral in (2) will approach that in (1). Theonly difference between the far fields i and * at the receivers is thenthe phase term Ak(R + RR). Measur~meni of th~s phase difference yields(R + RR) since Ak is knOwn. This information can be used to correct

thl phase of either the or 2 signals to obtain the required p-spaceinformation.

r0 (r) e j r 6

(31

d1

. . .. . . .. (.)

Appendix II

HOLOGRAPHY, WAVE-LENGTH DIVERSITY

AND INVERSE SCATTERING

ABSTRACT

The use of wavelength diversity to enhance the performance ofthinned coherent imaging aperatures is discussed. It is shown thatwavelength diversity lensless Fourier transform recording arrangementsthat utilize a reference point source in the vicinity of the objectcan be used to access the three-dimensional Fourier space of non-dispersive perfectly reflecting or weakly scattering objects. Hybrid(opto-digital) computing applied to the acquired 3-D Fourier spacedata is shown to yield tomographic reconstruction of 3-D image detaileither in parallel or meridonal (central) slices., Because of aninherent ability of converting spectral degrees of freedom into spatial3-D image detail true super-resolution is achieved together withsuppression of coherent noise. The similarity of the keyequations derived to those of inverse scattering theory is pointedout and the feasibility of using other forms of broadband radiationsuch as impulsive, noise and thermal is discussed. Finally, thepotential of utilizing wavelength diversity imaging in microscopy andtelescopy are discussed.

INTRODUCTION

A frequently encountered question in the science of image formationis how to make an available aperature collect more information aboutthe scene or object being imaged in order to enhance its resolvingpower beyond the classical Rayleigh limit. This process is known assuper-resolution and is relevant to all imaging systems whether holo-graphic or conventional. There are five known methods for achievingsuper-resolution. These include: weighting or apodization of the

aperature data" 2 ; analytic continuation of the wavefield measured over

the aperature 3,; use of evanescent wave illumination 5; maximum entropy

method6 ; and use of the time channel7 . Weighting and analyticalcontinuation techniques are known to become rapidly ineffective as thesignal to noise ratio of the data collected decreases. Maximumentropy techniques are known to be more robust as far as noise isconcerned but involve ususally extensive computation. Illuminationwith evanescent waves is practical in limited situations where fullcontrol of the recording arrangement exists as in microscopy for example.

ALi

2

This leaves the time channel approach in which one can collect intime more information about the object through the available recordingaperature by altering the object aspect relative to the aperature by

means of roation or linear motion8'9 or by altering the parameters ofthe illumation such as directions of incidence, wavelength and/orpolarization. These later operations are known to increase the degreesof freedom of the wavefields impinging on the recording aperatureenhancing thereby their ability to convey information about the natureof the scattering object. Sophisticated imaging systems endevour toconvert the nonspatial degrees of freedom of the wavefield, e.g.,angular, spectral and polarization to spatialimage detail enhancingthereby the resolution capability beyond the classical Rayleigh limitof the available physical aperature. Obviously such procedures involvemore signal processing than that performed by conventional imagingwith lens systems or holography.

In this paper we consider generalizing the holographic concept toinclude wavelength diversity as a means of enhancing resolution. Aquick examination of the basic equations of holography reveals thatthe lensless Fourier transform hologram recording arrangement isamenable to this generalization. This conclusion is used then as astarting point for a Fourier optics formulation of wavelength diversityimaging of 3-D (three dimensional) nondispersive objects. The resultsshow that measurement of the multiaspect or multistatic frequency (orwavelength) response of the 3-D object permits accessing its 3-DFourier space. The resulting formulas are identical to those obtainedfrom a multistatic generalization of inverse scattering10,11,12

establishing thus a clear connection between holography and the inversescattering imaging problem. The inclusion of wavelength diversity inholography is shown to have several important features: (a) theavailability of the 3-D Fourier space data permits 3-D image retrievaltomographically in parallel or meridonal (central) slices or cross-sectional outlines by the application of Fourier domain projectiontheorems, (b) suppression of coherent noise and speckle in the retrievedimage, (c) removal of several longstanding constraints on longwave(microwave and acoustical) holography such as the impractically highcost of the aperatures needed, the inability to view a true 3-D imageas in optical holography because of a wavelength scaling problem, andminimization of the effects of resonances on the object.

WAVELENGTH DIVERSITY

We start by inquiring into the conditions under which the datafrom N holograms of the same nondispersive object recorded over thesame aperture, each at a different wavelength, can be combined toyield a single image superior in quality to the image retrievedfrom any of the indiyidual holograms.

3

One approach to answering the question posed above would be to

determine the conditions under which the well known formulas 3 for thefocusing condition, magnification and image location in holography canbe made independent of wavelength. This quickly leads to the conclusionthat wavelength independence can be met if a reference point sourcecentered on the object is used and proper scaling of the individualholograms by the ratio of recording to the reconstruction wavelength

is performed before super-positionl52. The former condition is thatfor recording a lensless Fourier transform hologram 14where thepresence of the reference point source in the object plane leads tothe recording of a Fraunhofer diffraction pattern of the object ratherthan its Fresnel diffraction pattern because of the elimination of aquadratic phase term in the object wavefield in the recorded hologram.This is known to result in a highly desirable reduction in theresolution required from the hologram recording medium and is therefore of-practical importance especially in nonoptical holography. More

detail of the processing involved in combining the data in multi15

wavelength hologram can be found elsewhere

Additional insight into the process of attaining super-resolutionby wavelength diversity is obtained by considering the concept of

wavelength or frequency synthesized aperature .62 The synthesisof a one dimensional aperature by wavelength diversity is based onthe simple fact that the Fraunhofer or far field diffraction patternof a nondispersive planar object changes its scale, i.e. it "breathes"1,but does not change its shape (functional dependence), as the wave-length is changed. A stationary array of broadband sensors capableof measuring the complex field variations deployed in this breathingdiffraction pattern at suitably chosen locations would sense differentparts of the diffraction pattern as the wavelength is alteredcollecting thereby mor~e information on the nature of the diffractionpattern and therefore on the object that gave rise to it than ifthe wavelength was fixed (stationary diffraction pattern). Eachstationary sensor in the array is thus able to collect as the wave-length is changed, and the breathing diffraction pattern sweeps overit, the same set of data or information collected by a movablesensor mechanically scanned over the appropriate part of the diffractionpattern when it is kept stationary by fixing the wavelength. Hencethe term wavelength or frequency synthesized aperture.

The orientation and location of the wavelength synthesized aperturefor any planar distribution of sensors deployed in the Fraunhoferdiffraction pattern of a planar object and the retrieval of an image

from the data collected has been treated earlier'6". It was clear,however, that extension of the wavelength diversity concept to thecase of 3-D objects is necessary before its generality and practicaluse could be established.

4

For this purpose we considered20 as shown in Fig. 1(a) an isolatedplanar object of finite extent with reflectivity D(Po), where Pois a two dimensional position vector in the object plane (xo,yo). The

object is illuminated by a coherent plane wave of unit-amplitude andof wave vector ki k 1 produced for example by a distant source

located at R The wavefield scattered by the object is monitored ata receiving oint designated by position vector RR belonging to a

recording aperture lying in the far field region of the object. Thereceiving point will henceforth be referred to as the receiver and thesource point at the transmitter. The position vectors o , RT and R

are measured from the origin of a cartesean coordinate system (xo, Yo'

zo) centered in the object. The object is assumed to be nondispersive

i.e., D is independent of k. However, when the object is dispersivesuch that D(Po,k) = D1(pl)D2(k) and D2(k) is known, the analysis

presented here can easily be modified to account for such objectdispersion by correcting the data collected for D2(k) as k is changed.

Referring to Figure 1(a) and ignoring polarization effects, thefield amplitude at RR caused by the object scattered wavefield may be

expressed as,

ik _~jki. r T _j r*, fD((o)e- -jk rR d(= -, ) =e dpo I

where dpo is an abbreviation for dx0dyo and the integration is carried

out over the extent of the object. Noting that rT RV RT -RTIkiand using the usual approximations valid here: rR RR+p/2RR -R"Po

for the exponential in (1) and rR . RR for the denominator in (1) where

= R/R and I . /k are unit vectors in the RR and k directionsR RR 1k = R i

respectively, one* can write eq. (1) as,

,R -k -jk(RT + RR) -j Po (

2RRR e fD(p o ) e dpo, (2)

where we have used the fact that the observation point is in the farfield of the object so that exp (-jkp?/2RR) under the integral sign can

be replaced by unity. In eq. (2), p = k (Ik. - R P Ix + Py ly +

pz Iz is a three dimensional vector whose length and orientation depend

Transmitter

D (T) RT -rRReceiver

YO (a)

n-th Meridonal

C ~ Yon3- ---,Shadow Region

Ob* jc t c, /(if perfectly. .. . .Reflecting 3-0

ki Object)

nn-th slice

Fig. I. Geometries for wavelength diversity imaging. (a) Two dimensionalobject, (b) Three dimensional object with the n-th meridonal(central) slice and cross sectional outline c shown.

V .- --

6

on the wavenumber k and the angular positions of the transmitter andthe receiver. For each receiver and/or transmitter present, P indicatesthe position vector for data storage. An array of receivers forexample would yield therefore as k is changed ( frequency diversity)or as k (=klk.) is charged (wave-vector diversity) a 3-D data manifold.

The projectioA of this 3-D data manifold on the object plane yields*(k,RT) because p . o = t 0 = p x0 + p9 where Px = k(Ik - IR)x

and p k(ik - i R) are the cartesian components of the projectiony k R y

Pt of p on thi object plane. Accordingly eq. (2) can be expressed as,

e-Jk(RT + RR) "(x ) j(PxXo + PyYo)

*(k, RR) 2rRR dXdY (3)

Because of the finite extent of the object, the limits on the integralcan be extended to infinity without altering the result. The integralin (3) is recognized then as the two dimensional Fourier transformD(p xp y) of D(xooy ). It is seen to'be dependent on the object reflec-

tivity function, the angular positions of the transmitter and thereceiver and on the values assumed by the wavenumber k but is entirelyindependent of range. Information about D can thus be collected byvarying these parameters. Note that the range information iscontained solely in the factor F = jk exp [-jk(RT + RR)J/ 2 RR preceeding

the integral. The field observed at 9R has thus been separated into

two terms one of which, the integral b, contains the lateral objectinformation and the other F, contains the range infonration. Thepresence of F in eq. (3) hinders the imaging process since it complicatesdata acquisition and if not removed, gives rise to image distortionbecause R is generally not the same for all receivers. To retrievean image Of the object via a 2-D Fourier transform of eq. (3), thefactor F must first be eliminated. Holographic recording of thecomplex field amplitude given in (3) using a reference point sourcelocated at the center of the object will result in the elimination ofthe factor F and the recording of a Fourier transform hologram. This

operation yields D over a two dimensional region in the pxpy plane.

The size of this region, which determines the resolution of the retrievedimage depends on the angular positions of the transmitter and thereceiver and on the values assumed by k, i.e. the extent of the spectralwindow used. The later dependence on k implies super-resolution imagingcapability because of the frequency synthesized dimension of the 2-Ddata manifold generated. Because of the dependence of resolution onthe relative positions of the object, the transmitter, and receivingaperture, the impulse response is clearly spatially variant. In facta receiver point situated at RR for which p is normal to the object

7

plane can not collect any lateral object information because for thiscondition (p . Po = 0) the integrals in (2) and (3) yield a constant.

Such receiving point is located in the direction of specular reflectionfrom the object where the diffractionpattern is stationary i.e. doesnot change with k. In this case the observed field is solely propor-tional to F containing thus range information only. Obviously thiscase can easily be avoided through the use of more than one receiver

which is required anyway when 2-D or 3-D object resolution is sought20'21.

The analysis presentedabove can be extended to three dimensionalobjects by viewing a 3-D object as a collection of thin merdional orcentral slices as depicted in Fig. 1(b) each of which representinga two dimensional object of the type analyzed above. With the n-thslice we associate a cartesean coordinate system xo9yo, ,z0 that

n ndiffer from other slices by rotation about the common x0 axis. Since

the vectors p, RT and AR are the same in all n-coordinate systems,

eq. (3) holds. ' n (k,RR) is then obtained from projection of the three

dimensional data manifold collected for the 3-D object on the xoYon

plane associated with the n-th slice. An image for each slice canthen be obtained as described before. An inherent assumption in thisargument is that all slices are illuminated by the same plane wave.This isareasonable approximation when the 3-D object is weaklyscattering and the Born approximation is applicable or when the 3-Dobject is perfectly reflecting and does not give rise to multiplereflections between its parts. In the later case the two dimensionalmeridonal slices D n(P ) deteriorate into contours, such as C in

Fig. 1(b) defined by tie intersection of the meridonal planes withthe illuminated portion of the surface of the object. Accordinglywe can write for the n-th meridonal slice or contour,

*n(k,RR) = F f Dn(Pon ) e P.n dPon (4)

We can regard D( ° ) as the n-th meridonal slice or contour of a

three dimensional oBject of reflectivity U(r) where is a threedimensional 2osition vector in object space. This means thatDn(p0n U(r) 6(z0 n) where 6 is the Dirac delta "function".

Consequently eq. (4) becomes,

8

4n(kRR) F f U(r) 6 (z0 n') e Pn d-on

(5)=F f U(r) 6 (zon e d;

where dr designated an element of volume in object space and wherethe last equation is obtained by virture of the sifting propertyof the delta function.

Summing up the data from all slices or contours of the objectwe obtain,

-j .F£ F $r U(F) e-j~ dr-= $()(6)

n n

because

U(F) 6 (z) =u(F).n 0 n

Assuming that the Factor F in eq. (6) is eliminated as before,equation (6) reduces to

(5) = fU (i) e d6 (7)

which is the 3-D Fourier transform of the object reflectivity U(r).Wavelength diversity permits therefore accessing the 3-D Fourier spaceof a nondispersive object providing thereby the basis for 3-D LenslessFourier transform holography. An alternate formulation to that givenabove of super-resolved wave-vector diversity imaging of 3-D perfectly

conducting objects is possible 22 by extending the formulation of the

inverse scattering imaging problem 10 ,11 to the multistatic case, along

lines that are similar but somewhat different than those given by Raz 12.

The resulting scalarized formulas are identical to (7) establishingthus the connection between the holographic and the inverse scatteringapproaches to the imaging problem.

THREE DIMENSIONAL IMAGE RETRIEVAL

The above considerations of multiwavelength holography have leadus to determining a means by which the 3-D Fourier space of the objectcan be accessed employing synchroneous detection. It is clear thatonce the 3-D Fourier space data is available, 3-D image detail can beretrieved by means of an inverse 3-D Fourier transform .which can becarried out digitally. Alternately, holographic techniques

9

____ ___ _ -(-X 0 b, ZO)

(~COO0, OY7 0 )

to (CO,VO, 20)

__ (oo o)-Z R

-X0 __

do~Z kln

-4---xoO

Fig. 2. 3-0 object consisting of a set of eight point scatterers shownin isometric and R'-R' plane views at R ,- z0 Z. x =yo z0=100 cm.X Z

10

(a)

(b)

Fig. 3. Projection hologramis and their optical reconstructions forthe set of point scatterers in Fig. 2 at different R; planes.(a) Hologram and reconstructed image of scatterers at '-

plane. (b) Hologram and image at R;=0 plane. (c) Hologram

and image at R;-z 0 Plane. xo=y0-z0100o cm.

R'x

R',

zoPLANEWAVE R sy

3-DSCA-tTERING

OBJECT

Z RECEIVERARRAY

CENTRALTRANSMiTTER

yR9

Rg/za 0.5

Fig. 4. Arrangement used in computer simulation of wavelength diversityimaging.

12

can be invoked again. Fourier domain projection theorems23 that are

dual to the spatial domain projection theorem25'26 can be applied tothe Fourier space data to produce a series of projection hologramsfrom which 2-D images of meridonal or parallel slices of the object

can be retrieved on the optical bench20 . This procedure does notinvolve any specific scaling of the size of the optical hologramtransparency relative to the size of the original recording apertureby the ratio of the recording to the reconstruction wavelengths asin longwave holography where the scaling necessary for viewing a 3-Dimage free of longitudinal distortion ususally leads to an impracticallyminute equivalent hologram transparancy that cannot be readily viewedby an observer. The lateral and longitudinal resolutions in theretrieved image depend now on the dimensions of the volume in Fourierspace accessed by wavelength diversity. This volume depends on thewavelength range and on the recording geometry. Thus the longitudinalresolution does not deteriorate now as rapidly with range as inconventional monochromatic imaging systems.

1

An example of computer simulations of frequency diversity holo-graphic imaging of a 3-D object consisting of eight point scatterersdistributed as shown in Fig. 2 is given in Fig. 3. Shown in Fig. 3 arethree weighted Fourier domain projection holograms and the correspondingoptically retrieved images for three equally spaced parallel slicesof the object containing distinguishable 2-D distributions of scatterers.The simulated recording arrangement shown in Fig. 4 consisted of anarray of 16 receivers equally distributed on an arc extending from * =

400 to 0 = 77.50 surrounding a central transmitter capable of providingplane wave illumination of the object. The results shown were obtainedwith microwave imaging in mind assuming a frequency.sweep of (2-4)GHz.They clearly indicate a leteral and longitudinal resolution capabilityof the order of 25 cm. Wider sweep widths yield better resolution.For examp',e a (1-18)GHz sweep would yield a 3-D resolution of the orderof 1.5 cm.

DISCUSSION AND CONCLUSIONS

Seeking means by which the information content in a hologram canbe increased for example by wavelength diversity we have arrived at aformulation of 3-D Lensless Fourier transform holography capable offurnishing 3-D image detail tomographically. This ability ofproducing 3-D images in slices from coherently detected wavefieldsenable us to regard the method also as coherent tomography. The Fourierspace accessed in the above fashion by wavelength diversity can beviewed as a generalized 3-D hologram in which one dimension has beensynthesized by wavelength diversity. Such a generalized hologramcontains not only spatial amplitude and phase data as in conventionalholography but also spectral information and hence can yield better

13

resolution than the classical Rayleigh limit of the available apertureoperating at the shortest wavelength of the spectral window used. Thissuper-resolving property is further enhanced through an inherent sup-pression of the effects of object resonances and coherent noise in theretrieved image, the latter being so because frequency diversity tendsto make the impulse response of the system unipolar resembling that ofa non-coherent imaging system that is free of speckle and coherent noise

artifacts 15. Futher enhancement of information content and resolutioncan be achieved by polarization diversity where the p space can bemultiply accessed for different nonredundant polarizations of theillumination and the receivers and the resulting polarization diversityimages added either coherently or non-coherently in order to achieve a

15degree of noise averaging as discussed elsewhere

The removal of several longstanding constraints on conventionallongwave (microwave and acoustic) holography attained through the useof wavelength diversity as described here leads to a new class of imaging systemscapable of converting spectral degrees of freedom into 3-D spatial imagedetail furnishing thereby true super-resolution. Wavelength diversityis applicable to the imaging of two classes of objects: perfectlyreflecting objects of the type encountered in radar and sonar and weaklyscattering objects of low or known dispersion of the type encoLnteredin biology and medicine. The practical application of the conceptspresented here to optical wavefield is presently under consideration.The availability of tunable dye lasers and electronic imaging devicessuggest interesting possibilities of three dimensional wavelengthdiversity microscopy. Here one can conceive of an arrangement inwhich a minute semitransparent object with homogeneous or known dispersionis transilluminated by a collimated coherent light beam from a tunabledye laser which can also be made to provide a coherent reference pointsource in the immediate vicinity of the object. The resulting referenceand the object scattered wavefields are intercepted by the photocathodeof an electronic imaging device of known spectral response such as avidicon. Because of the minute size of the object, the photocathodecan easily be situated in the far field of the object. Thus nearly alensless Fourier transform hologram recording arrangement results. Thespatial frequency content in the resulting hologram is therefore expectedto be sufficiently low to be resolved by a high resolution electronicimaging device. By recording and digitally storing the resulting detectedhologram fringe pattern as a function of dye laser wavelength one cangain access to the 3-D Fourier space of the object since 1k and 1R

for the recording geometry are precisely known.

A similar recording arrangement can be envisioned in the activecoherent imaging of a distant reflecting object (active telescopy) wherethe object can be made to furnish a reference point source situated onits surface like.a wavelength independent stationary glint point oran intentionally placed retroreflector. Because in such an arrangementthe reference and the object wavefields travel over the same path,atmospheric effects are expected to be minimized. The generation of an

14

otject derived reference geometry in longwave (microwave and acoustic)20,27

wavelength diversity imaging has been described elsewhere

Finally it is worthwhile to note that since the scattering processis linear the multiaspect or multistatic frequency or wavelengthresponse measurements referred to in this paper can be obtained alsoby measuring the multiaspect impulse response followed by Fourier

transformation of the individual impulse responses measured19 . Thismeans that impulsive illumination can also be utilized. Because theimpulse response of a linear system can be measured by using random

19noise excitation and cross-correlating the output with the input , apossibility of using random noise (white light) illumination andcross-correlation detection techniques as a means for accessing the3-D Fourier space of the object also emerges.

15

REFERENCES

1. Schelkunoff, S.A., Bell Syst. Tech. J., 22, 80, (1943).

2. Anderson, A.P. and J.C. Bennet, Proc. IEE (letters), 64, 376, (1976).

3. DiFrancia, G.T., J. Opt. Soc. Am., 45, 497, (1955).

4. DiFrancia, G.T., J. Opt. Soc. Am., 59, 799, (1969).

5. Nassenstein, H., Optics Communications, 2, 231, (1970).

6. Wernecke, S.J. and L. D'Addario, IEEE Trans. on Computers, C-26,

351, (1977).

7. Lukosz, W., J. Opt. Soc. Am., 56, 1463, (1966).

8. Leith, E.N., Advances in Holography, 2, N. Farhat (Ed,), (M. Decker,New York. 1976).

9. Lukosz, W., J. Opt. Soc. Am., 57, 932, (1967).

10. Bojarski, N.N., Final Report, contract BOOO-19-73-C-0316,Naval Air Syst. Command, (1974).

11. Lewis, R.M., IEEE Trans. on Ant. and Prop., AP-17',308, (1969).

12. Raz, S.R., IEEE Trans. on Ant. and Prop., AP-24, 66, (1976).

13. Meier, R.W., J. Opt. Soc. Am., 55, 987, (1965).

14. Smith, H.M., Principles of Holography, (Wiley-Interscience, New York,(1969).

15. Farhat, N.H. and C.K. Chan, in Optica Hoy Y Manana, J. Bescos et. al.(eds.), (Sociedad Espanola De.Optica, Madrid, 1978), 399.

16. Farhat, N.H., Ultrasonics Symposium Proceedings, IEEE Cat. No. 75CHO 944-4SU, (1975).

17. Farhat, N.H., Proc. International Optical Computing Conference,IEEE Cat. No. 76-CH 1100-7C, (1976).

18. Farhat, N.H., Proc. IEEE (letters), 64, 379, (1976).

19. Farhat, N.H., J. Opt. Soc. Am., 67, 1015, (1977).

20. Farhat, N.H. and C.K. Chin, in Acoustical Imaging, A. Metherell (ed.),(Plenum, New York. 1980), 499.

16

21. Waters, W.M., Proc. IEEE (letters), 66, 609, (1978).

22. Chan C.K., Analytical and numerical studies of frequency sweptimagery, Ph.D. Dissertation, Univ. of Pennsylvania, Philadelphia,(1978).

23. Stroke, G.W. and M. Halioua, Trans. Amer. Crystallographic Assoc.,12, 27, (1976).

24. Farhat, N.H., Univ. of Pennsylvania Report No. F1 Annual Report,AFOSR Grant No. 77-3256, July (1978).

25. Bracewell, R.N. and S.J. Wernecke, J. Opt. Soc. Am., 65, 1342, (1975).

26. Bracewell, R.N., Australian Journal of Physicas, 9, 198, (1956).

27. Farhat, N.H., C.K. Chan and T.H. Chu, Proc. AP-S/URSI, Symposium,Quebec, Canada (1980).

APPENDIX III

THE VIRTUAL FOURIER TRANSFORM

AND ITS APPLICATION IN THREE DIMENSIONAL DISPLAY

ABSTRACT

In contrast to the well known and widely used instantaneousFourier transforming property of the convergent lens in coherent(laser) light, the "Virtual Fourier Transform" (VFT) capability ofthe divergent lens is less widely known or used despite many advantages.We will review the principle of the VFT and discuss its advantages incertain applications. In particular a method for viewing the virtualFourier transform of a two dimensional function with the naked eyeusing an ordinary point source will be presented. A scheme for three-dimensional image display based on a "Fourier domain projection theorem"utilizing varifocal VFT is described and a discussion of the properties-of the displayed image given.

INTRODUCTION

Several sophisticated three dimensional (3-D) imaging techniques1 2 2such as x-ray tomography , electron microscopy , crystallography

wave-vector diversity imaging and invierse scattering 3, involve meast~re-ments that give access to a finite volume in the 3-D Fourier space ofa 3-D object function. A 3-D image of the original object can thenbe reconstructed by computing the inverse 3-D Fourier transform. Theretrieved iamge normally represents the spatial distribution of arelevant parameter of the object such as absorption, reflectivity,scattering potential, etc.

Obviously, the required inverse transform can be performed digitally.Digital techniques however often preclude real-time operation partic-ularly when the object being imaged is not simple but contains consider-able resolvable intricate detail. More importantly, because of theinherent two dimensionality of CRT computer displays,direct true 3-Dimage display is not possible. Present day computer graphic displaysare capable of displaying 3-D image detail either in seperate cross-sections or slices, or in a computed perspective (isometric) view ofthe object, or in some instances stereoscopically where an illusion ofa 3-D scene is created in the mind of the observer who is required

4,5usually to use special viewing glasses

2

Hybrid (opto-digital) computing techniques offer an alternateapproach to 3-D image retrieval from 3-D Fourier space data. Theyfurnish as shown in this paper the ability to display true 3-D image

detail. The approach is based on "Fourier Domain Projection Theorems"2'3

that are dual to "Spatial or Object Domain Projection Theorems" used in6,7 1

radio-astronomy and tomography . These theorems permit the recon-struction of 3-D image detail tomographically* i.e. in slices from

2-D projections of the 3-D Fourier space data2 '3 . Although the required2-D Fourier transform can be carried out digitally, the emphasis inthis paper is on coherent optical techniques for performing the 2-DFourier transform.with particular attention to implementations thatpermit the execution of the necessary 2-D optical transforms of thevarious projection hologram sequentially in real-time. Specificattention is given to a technique that ittilizes the virtual Fouriertransform which permits the viewingofa virtual 3-D image in real-time.

FOURIER DOMAIN PROJECTION THEOREMS

There are two Fourier domain projection theorems. One leads totomographic object reconstruction in parallel slcies and is called the"weighted Fourier domain projection theorem,' the other leads totomographic object reconstruction in meridonal or central slices andcan therefore be called the "meridonal or central slice Fourier aomainprojection theorem.

We begin by considering a 3-D object function f(i) with r =x1 + yl + z1 being a position vector in object space. Let F(i)x y zbe the 3-D Fourier transform of f(F) defined by,

F(q) = r f(F) e dF (1)

where dF = dx dy dz and =x WIx + Wyly + zlz is a position vector

in the Fourier or spatial frequency domain.

Consider next the projection of F(w) on the wx, wy plane defined by,

Fp (wx'wy) z F(;) dwz (2)

and combining eq. (1) and (2),

F p(wxwy) f(x,y,z)e-( +wy +WzZ)

p t y) w { dxdydzdwz )

From the Greek work Tomos meaning slice.

3

Integrating with respect to wz first and assuming that the volume in

space occupied by F(w) is sufficiently large we obtain,

-i(WxX + w/y)F p(wxWy) = f(x,y,z)6 (z)e dxdydz (4)

= f(x,y,o) ei(wxx + w9)dxdy (5)

The 2-D Fourier domain projection Fp (wx,w,) and the central slice

f(x,y,o) through the object form thus a Fourier transform pair. Thismay be symbolically expressed as,

F (wxwy) +- f(x,y,o) (6)px

Other parallel slices through the object at z = zn , zn being a

constant describing the z coordinate of the n-th parallel slice, canin a similar manner be related to "weighted" Fourier domain projectionsof F() defined by,

jz w1Fp,n (WxWy) = F(w)e nz dwz (7)

Making use of eq. (1) and again performing the integration with respectto wz first we obtain,

Fp,n (wxwy)+- f(xyzn) (8)

which indicates that the weighted projection F (wx,wy) and the n-th

parallel object slice f(x,Y,Zn) form a Fourier transform pair.

Equation (6) is seen to be a special case of eq. (8) when zn = o.

Given the 3-D Fourier space data manifold F(w) one can digitallycompute and display a set of "weighted projection holograms" F p,n(wxwy).

A corresponding set of images of parallel slices or cross-sectionaloutlines of the 3-D object can then be retrieved via 2-D Fouriertransform operations which can most conveniently be carried outoptically from photographic transparency records of the weightedprojection holograms displayed by the computer.

4

Returning to eqs. (1) and (2) one can also show that projectionsof F(w) on arbitrarily oriented planes other than the Wx,W plane

chosen for eq. (2), yields "meridonal projection holograms" that are2-D Fourier transforms of corresponding merdional (central) slicesof the object. This is the "meridonal Fourier domain projectiontheorem. It furnishes the basis for angular multiplexing of theresulting meridonal projection holograms into a single composite

hologram which can be used to form a 3-D iamge of the object in a

manner similar to that in integral holography which is increasingly

being referred to as Cross holography*.

THE VIRTUAL FOURIER TRANSFORM

In contrast to the well known spatial Fourier transforming pro-9perty of theconvergent lens widely used in coherent optical computing,

the complementary virtual Fourier transform capability of a divergent104

lens is less widely known or used despite many attractive features.This is surprising since the power spectrum associated with the VFTis a phenomenon that is frequently observed in daily life when onehappens to look at a distant point source such as a street lightthrough a fine mesh screen or the fine fabric of transparent curtainmaterial. The spectrum of the screen transmittance appears then asa virtual image in the plane of the point source.

The VFT concept of the divergent lens is easily derived from theFourier transform expression of the convergent lens. Figure 1 illustratesthe well known process of forming a real Fourier transform with aconvergent lens. The object transparency, with complex transmittancet(x ,y ), is placed at a distance d in front of a convergent lens offocal length F and illuminated with a normally incidnet collimatedlaser beam. The complex field amplitude of the wavefield in theback focal plane, the transform plane, is given by the well known formulaS -k d x2 2)

-J2 [(1 - fXT(x,y) = e

SF(x x + yy )x ff t(xoy o )e 0 0 dx0dy0 (1)

-Cc

in which the integral is recognized as the two dimensional Fouriertransform of the object transmittance. T(x,y) becomes the exactFourier transform of t(x0 yo ,) when d = F that is when the object

transparency is placed in the Front focal plane of the lens. Thepower spectrum associated with the transform is real and can beprojected on a screen placed in the back focal plane. It is alsowell known that a scaled version of the transform can be obtainedin the back focal plane by placing the object tranparency in the

9converging laser beam to the right of the lens

Named after Lloyd Cross the originator of integral holography.

5

Real Fourier transformObject plane y plane

CollimatedLaser beam z

KVFig. 1. Real Fourier transform formed with a convergent lens

Virtual Fourier Object planetransform plane

&Inverted

transparency

CollimatedLaser beam / .

Observer

F 2 V u i tF ormd -wth.d de Yo

Fig. 2. Virtual Fourier transform formed with a divergent lens

6

Noting that eq. (1) does not change when we replace d by -d,F by -F, x and y0 by -x0 and -y0 respectively, we. can arrive at the

complementary VFT arrangement illustrated in Fig. 2. An invertedtransparency t(x0,Yo ) is placed now in the divergent coherent beam

to the right of the divergent lens (of focal length -F) and a VFTgiven by eq. (1) is observed in the virtual focal plane of the lens.The same VFT can be seen by removing the divergent lens and replacingthe laser beam with a point source placed at the origin of the VFTplane as depicted in Fig. 3. Thus a simple way of viewing the powerspectrum associated with the VFT of a given diffracting screen(which is usually a Fourier transform hologram or a projectionhologram of the type described above) is to hold the screen closeto the eye and look through it at a distant bright point source. Thepoint source used need not be derived from a laser. In fact it ispreferable for safety purposes to use an LED or a spectrally filteredminute white light source such as a "grain-of-wheat" subminiatureincandescent lamp or a miniature Christmas tree decorating lampcovered by a color or interference filter. This has the addedadvantage of furnishing a measure of control over the coherence proper-ties of the wavefield impinging on the screen providing thereby ameans for reducing coherent noise in the observed VFT and also, aswill be discussed below, a means for coherent or noncoherent super-position of VFT's. As the distance of the point source from thediffracting screen is decreased in order to make it compatible withtypical laboratory or optical bench dimensions, the size of theobserved VFT decreased because of the change in the curvature of thewavefield illuminating the diffracting screen. To compensate for thiseffect it is necessary to reduce the size of the diffracting screen ortransparency often to such a scale where viewing the VFT throught thesmall available aperture becomes difficult. To overcome this limitationthe displacement property of the Fourier transform can be utilized.A composite transparency containing an ordered or random array ofreduced replicas of the transmittance function t(x0,yo ) arranged side

by side as illustrated in Fig. 4 is prepared. When such a compositetransparency is viewed with the point source, the VFT's formed by theindividual elements will overlap in the virtual Fourier plane. The VFT'sare identical except for Linear phase dependence on x,y which dependsin each VFT on the central position of each element in the compositetransparency. This leads to a desirable noise averaging effect andthe appearance of fine checkered texture in the image detail. Allthis leads to an enhancement of the quality of the observed powerspectrum. Both coherent and noncoherent superposition of the over-lapping VFT's is possible using this scheme by varying the coherencearea of the wavefield illuminating the composite transparency. Whenthe coherence area is roughly equal to the size of the individualelements of the composite transparency noncoherent superpositionresults, while a coherence area equal or greater than the size of thecomposite transparency would yield coherent superposition.

7

Virtual Fourier Object planetransform plane transparency

Pointsource ObserverInverted

transparency

x

(F +d)Y

Fig. 3. Arrangement for viewing a virtual Fourier transform with apoint source

Fig. 4. A composite screen consisting of an ordered array of identicalFourier' transform projection holograms.

8

THREE DIMENSIONAL DISPLAY

The VFT concept and the "weighted Fourier domain projection theorem"discussed above can be combined in an attractive scheme for the recon-struction and display of a 3-D image from a series of weighted projectionholograms corresponding to different parallel slices through the object.The scheme is based on viewing a series of weighted projectionholograms sequentially in the proper order of the occurance of theircorresponding slices in the original object while displacing the pointsource axially for one hologram to next by an axial increment proportionalto the spacings between adjacent object slices. In this fashion thereconstructed virtual images of the various slices are seen in depthat different VFT planes that are determined by the positions of theaxially incremented point source. Repeated rapid execution of thisprocedure by displacing the point source back and forth leads theobserver to see a virtual 3-D image tomographically in parallel slicesor sections as he looks through the series of projection hologramspassed rapidly, as in a motion picture film, infront of his eyes.

More specifically the scheme is based on preparing a series ofN weighted Fourier domain projection holograms from the 3-D Fourierdomain data F(%_1) of a given object f(i ) as described in the preceedingsections. Each of the projection holograms would correspond to adifferent parallel slice through the object. A composite transparencysimilar to that shown in Fig. 4 is formed for each projection hologram.In fact Fig. 4 is an example of a computer generated composite hologramcontaining an array of identical weighted projection holograms corres-ponding to one slice of the test object shown in Fig. 5. The testobject chosen consisted of eight point scatterers arranged as shown.The 3-D Fourier space of this test object was accessed in a computersimulation of wavelength diversity imaging as described in a companionpaper in this volume*. The resulting computer generated Fourierspace data manifold F(ii) was used to compute three weighted projectionholograms corresponding to the three planes R;=]m,O,1m of Fig.5icontaining the three different distributions of point scatterers. Acomposite array such as that of Fig. 4 was formed and displayed by thecomputer for each of the three projection holograms, each was photo-graphed yielding a set of three projection hologram composite trans-parencies. Copies of these were then mounted on a rotating wheel asshown in Fig. 6 (a) and viewed with an axially scanned point source.Four sets of transparency copies of these three composite projectionholograms were mounted in the order 1,2,3,2,1,2 ... on the peripheryof a rotating wheel as shown in Fig. 6 (a). The wheel is driven bya computer controlled stepper motor.. The axially scanned point sourcewas produced by scanning a focused laser beam back and forth on alength of fine nylon thread with the aid of a deflecting mirror mountedon the shaft of a second computer controlled stepper motor as shown inFig. 6 (b). The laser and optical bench arrangement for forming thescanned focused beam appear in the background of Fjg. 6 (a). Thecomputer controlled steppers enable precise positioning of the secondarypoint source on the scattering thread in synchronism with the hologram

*See paper entitled "Holography, Wavelength Diversity InverseScattering" in this volume.

9

FIZ

o,(-xo, Z)

(01. 0) (e~ - 0 (xo,Y,zo)........ ~\ \ o( o0,

(___-y_-(o)o

01 -%R

yy

---- -7

R1

WX gR' R-o pke-XO

-Yo

A--114--_

R' R: zo plam-xo Xo

' Fig. 5. A three-dimensional test object consisting of a set of eightpoint scatterers shown in isometric and Rx-R' plane views at

Rz=-zo,O,zo. xo=yo=zo= 10 cm.

Z 0

10

Fig. 6. Quasi real-time three-dimensional image reconstruction andtomographic display in successive slices from a series ofprojection holograms mounted on rotating wheel seen in fore-front of (a); Detail of laser scanner used to producelinearly scanned point source is shown in (b).

0 o

.0.E

0

4- 0

--

'40o

4-)

4-'

*.) 0

4-

4-0oci

--a) L

'4-) -r

c- 0.

-~ S.-00

-P4-j

0)0

__-O-

ci>

12

being viewed so that the VFTsare formed in their proper planes. Aviewer looking at the axially displaced point source through eachtransparency mounted on the wheel as it passes infront of his eyewill see a 3-D virtual image. Photographs of the three virtual imagesseen by an observer in this fashion are shown in Fig. 7. An opto-digital scheme for rapid real-time implementation of the procedure realizedabove is shown in Fig. 8. This scheme, presently under study, utilizesa rapid recyclable spatial light modulator (SLM) such as the Itek PROMin order to form VFT's of the projection holograms displayed by thecomputer in real-time.

CONCLUSIONS

We have presented the basic principles of tomographic 3-D imagedisplay based on Fourier domain projection theorems. One possibleimplementation of the principle using the virtual Fourier transformand a series Fourier domain projection holograms has been described.There are several advantages for using the VFT rather than the realFourier transform (RFT), the most important of which is the ease withwhich the position of the VFT plane can be moved axially by simplymoving the position of the reconstruction point source. The VFT approachwas adopted in the present study because it is much easier to move apoint source rapidly than to move the display screen needed in the RFTapproach. Furthermore focusing in the VFT approach is carried out bythe observer while in the RFT approach it must be performed by thesystem. Other attractive features of the VFT are:

(a) Simplicity - enables direct viewing of the power spectrumof a transparency or a hologram with a variety of simple pointsources.

(b) The scale of the observed VFT can be easily altered bychanging the distance between the projection hologram transparencyand the reconstruction point source.

(c) Lower speckle noise and therefore higher reconstructed imagequality can be attained by using nonlaser point sources in thereconstruction such as LED or miniature spectrally filteredincandescent lamps. Further reduction in speckle noise occurswhen an array of the projection hologram rather than a singlehologram is used and when the hologram is slightly vibrated oris in motion because of a noise averaging effect.

(d) Coherent and noncoherent superposition of VFT's is possibleby altering the coherence area of the reconstruction wavelfield.

(e) Because of the Fourier transform nature of the projectionholograms utilized, the resolution requirements from thestorage medium (photographic film or the CRT/SLM system of Fig. 8)are much lower than would be needed in the recording of a Fresnelhologram of the object as a means of 3-D image display. The 3-Dimage detail contained in the single Fresnel hologram is nowdistributed over a series of lower resolution projection hologramswhich are used to form the 3-D image sequentially in time inindividual slices.

13

a- c

U)UI- UJ

w- U -D 1- r

mU), ~ ~ ~ ~ Z - . ______ U1.66-.

L)Z <-iU EUE0

W a

C-CJ

E

4- LL

0 0 j 4-'

_ Q) >J

C)C

W~~~ C. I-=

L)0 -oCL 0

zz

-a--~.0

W C))

U-) C)

0I>

4-' M~

'w

LL-

14

(f) Because 3-D image reconstruction is tomographic (in separateslices) there is no interference between the wavefields formingthe various slices.

(g) Permits other forms of 3-D image display involving spatial orangular multiplexing in a fashion similar to integral holography.

lIL.I

15

REFERENCES

1. H.H. Barret and M.Y. Chiu, Optica Hoy Y Manana, J. Bescos et.al.,(eds.) 136, (Proc. of ICO-11, Sociedad Espaola De Optica, 1978).

2. N.H. Farhat and C.K. Chan, Acoustical Imaging, 8, 499, A. Metherel(ed.), (Plenum Press, New York, 1980).

3. G.W. Stroke and M. Halioua, Trans. Amer. Crystallographic Assoc.,12, 27, (1976).

4. T. Okoshi, Three-Dimensional Imaging Techniques, (Academic Press,New York, 1976).

5. T. Okoshi, Proc. IEEE, 68, 548, (1980).

6. R.N. Bracewell, Australian Journal of Physics, 9, 148, (1956).

7. R.N. Bracewell and S.J. Wernecke, J. Opt. Soc. Am., 65, 1342,(1975).

8. D.L. Vickers, Lawerence Livermore Laboratory Report, No. UCID-17035,(February 1976).

9. J.W. Goodman, Introduction to Fourier Optics, 83, (McGraw Hill,New York, 1968).

10. J. Knapp and M.F. Becker, App. Optics, 17, 1669, (1976).

I

APPENDIX IV

AN AUTOMATED FREQUENCY RESPONSE AND

RADAR CROSS-SECTION MEASUREMENT FACILITY

FOR MICROWAVE IMAGING

UNIVERSITY OF PENNSYLVANIA

THE MOORE SCHOOL OF ELECTRICAL ENGINEERING

AN AUTOMATED FREQUENCY RESPONSE AND RADAR CROSS-SECTIONMEASUREMENT FACILITY FOR MICROWAVE IMAGING

CHARLES L. WERNER

Presented to the faculty of the Moore School of ElectricalEngineering .(Department of Electrical Engineering &Science) in partial fulfillment of the requirements for thedegree of Master of Science in Engineering.

* Philadelphia, Pennsylvania

May 1980

$Thesis Superviy r..

Graduate Group Chairman

UNIVERSITY OF PENNSYLVANIA

THE MOORE SCHOOL OF ELECTRICAL ENGINEERING

AN AUTOMATED FREQUENCY RESPONSE AND RADAR CROSS-SECTION

MEASUREMENT FACILITY FOR MICROWAVE IMAGING

ABSTRACT

This thesis investigates the development of a broadbandmicrowave holographic imaging facility. Different methodsfor the correction of microwave target scatter data arediscussed and implemented. A minicomputer automates allsystem functions including data acquisition, storage,calculation, and graphic display. The effects of rangephase shift on holographic frequency diversity imaging isconsidered and techniques for the removal of this phaseshift in a laboratory environment .The frequency dependentbackscatter of several test targets is derived analyticallyand simulations done of the corresponding holograms. Theseholograms are compared to those measured experimentally.Finally both simulated and experimental holograms areoptically reconstructed to yield target images using opticalFourier transforms and shown to be in excellent agreement.

Degree: Master of Science in Engineering for graduate workin electrical Engineering and Science

Date: May 1980

Author

ACKNOWLEDGEMENTS

I would like to extend my most sincere thanks to Dr.

Nabil H. Farhat for making my studies at the University of

Pennsylvania possible. His patience and wisdom were an

inspiration for my research. I would also like to thank my

parents for their love and understanding these many years.

Finally I want to express appreciation to the Moore School

for giving me the chance to study here.

LL.

TABLE OF CONTENTS

I............ INTRODUCTION 1

II ........... SYSTEM OPERATION AND ERROR REMOVAL

2.1 ..... Microwave sweeper operation 4

2.2.....Network analyzer-computer interface 6

2.3 ..... Implementation of the data aquisition system 9

2.4.....System error correction 13

III .......... RANGE PHASE ANALYSIS

3.1 ..... The effects of range phase on imaging 27

3.2 ..... Practical considerations for range phase 38removal

IV*........... SYSTEM IMPLEMENTATION OF SWEPT FREQUENCYIMAGING

4.1 ..... System repeatability 41

4.2 ..... Computer control of target rotation 43

4.3 ..... Sphere simulation 45

4.4 ..... Simulation of frequency swept imaging 53

4.5 ..... Experimental results 60

V ............ CONCLUSION 72

BIBLIOGRAPHY 74

Appendjx I...Error Correction Programs and Utilities

Appendix II..Microwave imaging programs-acquisition anddisplay

LILL

I INTRODUCTION

Frequency diversity imaging has been under study at the

Electro-Optics and Microwave Optics laboratory of the Moore

School Graduate Research Center.[I ],[2],[3],[,4] This

study has established the theoretic feasibility of imaging

objects by means of their multiaspect frequency response

For the purpose of experimentally studying frequency

diversity imaging, an experimental measurement system has

been assembled and installed in the Moore School anechoic

chamber. In this thesis we will describe the automation of

this measurement system and characterize its peroformance in

the measurement of complex field amplitudes of scattered

fields. A system block diagram fig.l.1, shows the major

system components. The central element is the DEC MINC LSI

11/2 minicomputer. This computer performs several important

functions. The MINC controls laboratory instrumentation via

the IEEE-488 bus protocol standard. This allows the

Hewlett-Packard 8620C microwave sweeper to be precisely

tuned to any frequency in the 2.0 to 18.0 GHz range

(fig.l.2). The computer collects data from the HP 8410B

network analyzer through the four analog input channels

available. These analog values are proportional to the

amplitude in db and phase in the range (-'R) to (+T) radians

relative to the reference signal supplied to the network

analyzer. The computer stores Experimental data on floppy

discs. The available storage capacity is large, over

500,000 measurement pairs of complex field amplitude and

............

CSAO7RWJdCS yr lo6-

t ADEc

'CEC CO#1PCJT

Fig. 12 HP 820C mirowavesweep r dH 40 ewr

analyzer.

phase may be stored on a single disc. This data can be

accessed for both processing and display on a Tektronix 606A

high resolution CRT monitor. The disc system allows the

MINC to operate under a sophisticated software system, DEC

RT-11 V3.OB permitting programing in MINC BASIC,FORTRAN IV,

and MACRO languages.

The processing capability of the system allows the

removal of system response errors due to anechoic chamber

clutter,antenna cross-coupling and receiver channel

characteristics. The data may be processed for target range

calculations and the removal of the phase shift due to the

target range. In addition the collected data may be

filtered to improve imaging and finally displayed on a high

resolution Tektronix 606A X-Y CRT monitor using the MINC

system D/A converter module.

The second part of this thesis will describe the

simulation and actual operation of a holographic radar

system verifying the theory of frequency diversity imaging.

This is done using the system described in the first

section. The scattering of various targets is derived and

holograms using these results are generated for comparison

with experimental data. Finally a system for the

experimental measurement of the scattering for these targets

is outlined and the results from this system compared to

theory.

waf

II SYSTEM OPERATION AND ERROR REMOVAL

This chapter will cover the operation of the microwave

backscatter data aquisition system. This will include

actual interfacing information and an analysis of the types

of errors encountered when making microwave measurements.

The error correction techniques developed are later used in

the experimental verification of the frequency diversity

imaging theory. (,q]

2.1 Microwave swee _er operation

The first task in the development of an automated data

acquisition system is the implementation of a data

communications link between the intelligent controller and

instrumentation. The IEEE-488 bus protocol is utilized in

this application for the transfer of data to the

Hewlett-Packard 8620C microwave sweeper from the MINC LSI

11/2 computer. This bus is a high speed 8 bit wide

bidirectional data path with 5 additional lines dedicated to

control. Data is transferred in ASCII format over the bus.

For example the number 1 is transmitted as the ASCI code for

the character '. Certain sequences of characters make the

sweeper perform different functions or enter different modes

of operation via its IEEE-488 interface.

In order to set the frequency of the sweeper a number

must be sent to the IEEE-488 interface in the range 1-10000.

Each sweeper frequency band has been split into 10000

frequency points. The frequency of operation is controlled

by an internal analog voltage that varies between 0.0 and

10.0 volts. A D/A converter on the 8620C IEEE-488 interface

changes the data transmitted from the MINC into the

frequency controlling voltage. The correspondence between

voltage and output frequency is essentially linear. In

order to obtain the interpolating function for frequency

versus control voltage; a microwave frequency counter was

used to measure the voltage-frequency characteristic

function. Using the MINC-BASIC program CALAB.BAS a least

squares fit for both linear and quadratic functions was made

on the frequency vs voltage data. This type of program is

used for determining the best polynomial fit to the

frequency-voltage characteristic of the sweeper. Sample

output and program listings are in appendix I along with an

explanation of program operation. The results from this

work indicate that the quadratic fit was statistically

superior for all bands on the microwave sweeper. The

FORTRAN subroutine SWEEP was written which utilizes the

quadratic interpolation polynomial for each of the four

bands of the 8620C. This subroutine automatically

calculates the control voltage and band to generate any

frequency in the 2-18 GHz range and transmits the

appropriate commands over the IEEE-488 bus. The variance of

the frequency setting using the quadratic fit for the three

bands are as follows: .6 MHz in the 2.0-6.3 GHz band; 1.2

MHz in the 6.3-12.0 GHZ band; and 1.6 MHZ in the 12.0-18.0

GHz band. For higher accuracy the sweeper may be phaseS

locked to the reference in a locking frequency counter

yielding very high accuracy as precise as the frequency

reference itself. The EIP 371 locking counter may be used

in this application to lock the HP 8620C sweep oscillator to

the correct frequency once it is within 20 MHz of the

desired frequency. The auxiliary output of the HP 8620C

supplies a sample of the signal generated by the fundamental

2.0-6.3 GHz oscillator module within the sweeper to the

locking counter. Sweeper output on higher bands is this

fundamental multiplied by a factor of 2 or 3 for the

6.3-12.0 and 12.0-18.00 GHz bands respectively. For example

the locking frequency for a 10.0 GHz sweeper output on band

2 would be 5.0 GHZ. In operation, subroutine SWEEP will set

the frequency of the sweeper within 2.0 MHz of the desired

frequency and subroutine SLOCK will be called to calculate

the locking frequency; lock the HP 8620C and return to the

main calling routine when lock will have occurred. Lock

time varies from .1 to 3 seconds and resolution is 100 KHZ.

These subroutines are called whenever the sweeper must be

set to a particular frequency or the sweeper must be placed

in or be released from computer control.

2.2 Network analyzer-comueinrfc

The Hewlett-Packard 8410B network analyzer is the focus

of the measurement capabilities of the microwave measurement

system. it can make vector (amplitude and phase)

measurements in the (.l-l8.0)GHz range. The range of

amplitude measurement is 80 db and phase may be measured

modulo (2T). The system reference signal is fed from a 20db

directional coupler to the HP8411A harmonic converter

sampling head of the network analyzer. This reference is

compared to the backscatter from the illuminated target;

both in amplitude and phase. The reference signal amplitude

is kept constant by leveling the sweeper with a feedback

signal derived from its amplified output by means of a

crystal detector. This allows the sweeper-TWT (Traveling

Wave Tube) system to yield nearly constant output in the

(2.0-16.5) GHz range; see fig.2.1. TWT power output is on

the order of 1 watt over these frequencies.

The complex field amplitude measurements are available

as analog voltages from the back panel of the 8410B. The

outputs are proportional to amplitude and phase: 25 MV/db

and 10 MV/DEGREE These values are digitized by the MINC

using its built in analog to digital conversion channels.

The MINC A/D converters digitize voltages lying in the range

of -5.12 to +5.12 volts to the range of 0-4096 yielding 12

bit resolution. If the signal is corrupted by noise; the

user has the option of employing signal averaging to cancel

the effects of noise uncorrelated to the received signal.

A difficulty encountered when measuring phase angle

modulo (27) occurs when the phase is close to (+-n) or (-Y).

At this point a small change in the signal phase may cause

the phase to flip between these two equivalent extremes

rapidly. If a data sample is taken close to (+I) or (-ff) it

may be in transition between them and therefore incorrect.7

Since such points occur infrequently in a typical

measurement they may be ignored. ;their presence does not

seriously hinder any holographic imaging due to the inherent

redundancy and therefore noise immunity of the holographic

reconstruction process.[5] If it is desired that these

points be identified and removed it is necessary to estimate

the mean and variance of the samples taken at each frequency

point. At frequencies where the phase is flipping between

(+Wr) and (-IT ), the variance will be much larger than at

other frequencies. The mean of the samples when this is

occurring will be near zero. Hence to resolve the ambiguity

problem it is necessary to decide if the variance exceeds a

predetermined threshold when the mean in the neighborhood of

zero. Two FORTRAN subroutines PHAMP and PHAMP2 which

implement these algorithms are listed in appendix 1.

Since such points occur infrequently in a typical

measurement they may be ignored. ;their presence does not

seriously hinder any holographic imaging due to the inherent

redundancy and therefore noise immunity of the holographic

reconstruction process.5s] If it is desired that these

points be identified and removed it is necessary to estimate

the mean and variance of the samples taken at each frequency

point. At frequencies where the phase is flipping between

(+r) and (-IT), the variance will be much larger than at

other frequencies. The mean of the samples when this is

occurring will be near zero. Hence to resolve the ambiguity

problem it is necessary to decide if the variance exceeds a

predetermined threshold when the mean in the neighborhood of

zero. Two FORTRAN subroutines PHAMP and PHAMP2 which

implement these algorithms are listed in appendix I.

23Implementation -of the -dat a _aquis it ioSstm

In an experimental environment it is important to be

aware of the various types of errors inherent in the

equipment and the experimental procedure adopted.

Conditions and equipment always vary from theoretical

ideals. A clear understanding of the error removal process

leads to the development of practical implementations of

theoretical concepts and enhancement of measurement accuracy

unattainable otherwise.

Errors in complex field amplitude measurement may be

caused by several factors. These may be grouped into two

categories. Errors caused by the instruments themselves

fall into the first class. Such factors as measurement

variations caused by electronic noise, ,inaccurate A/D and

logarithmic conversions, and inaccuracy and instability in

the microwave source make up this category. The second

group of errors is caused by the test set,antennas , cables,

connectors, amplifier, and room clutter. All these factors

interact with each other in the microwave region and are the

significant cause of error in microwave measurements.

Little can be done about the first class of errors since

they are inherent in the characteristics of the equipment

used. The second group of errors can be removed through the

use of automated measurement of system parameters in the

frequency range of interest. These errors can be removed

form any measurements of scattering objects by digital

processing and the results stored for later recall. This

essentially provides an automated and improved version of

the conventional two antenna radar cross-section measurement

technique ( '1 ;in which a microwave bridge is balanced in

the absence of the target and the degree of imbalance is

measured when the target is introduced into the microwave

field.

Let us look at the first class of errors more closely

since these errors will set the ultimate performance limits

on the system. The characteristics of the signal source are

important in this regard. In this case the signal source is

a Hewlett-Packard 8620C microwave sweeper. Since the

sweeper is not phase locked frequency and stability'problems

exist. The carrier also has significant FM noise which

appears as phase noise in the scattered signal. The phase

shift of the scattered scattered signal as a function of

frequency for small frequency variations is given by:

AG =, K-R U-0

where k=(21/A) and R is the path length. For R greater than

a few tens of meters ,the FM noise on the signal source

causes measurable variations in the phase of the scattered

signal. The stability of a synthesizer is required for the

implementation of a holographic radar system when target

ranges are in terms of kilometers.

Another limit on the ultimate accuracy of the system is

the resolution of the A/D conversions and the accuracy of

the network analyzer. Given the 2.2MV resolution of the

MINC A/D converter and the network analyzer analog output ofI0

50 MV/db, the system can resolve .048 db steps. This

resolution limit restricts the minimum signal to system

error ratio. If the error signal consisting of clutter,

antenna coupling, system directivity and noise exceeds the

scattered target signal the resolution of the target signal

suffers. For example, if the system error signal and target

scattered signal are of equal intensity, then a 1 db change

in the scattered signal causes a .53 db change in the total

received signal vector. When the system error is 13 db

above the scattered signal it becomes impossible to resolve

a 1 db change in the target signal given the resolution

capabilities of the system. This difficulty is further

compounded by errors in the network analyzer. These too are

amplified when the clutter exceeds the target scatter

signal. In fig.2.1 is a plot of the minimum system

resolution in order to detect a 1 db change in the target

return signal versus the system error signal to scattered

signal level in db. In fig.2.2 is plotted the target signal

resolution versus the noise to signal level in db. When the

system error is 20 db below the target signal then the

resolution of the target scatter signal is very close to the

ultimate resolution, .048db. Clutter is the component of

the received signal not scattered by the target but that

signal that is the result of coupling between antennas and

signal scattered by the anechoic chamber walls. As the

clutter/signal ratio increases the target scatter signal

resolution decreases exponentially. Clutter may be reducedII

Fig. 2.1 The required resolution in db for a dataacquisition system to detect a 1 db change in the desiredsignal vs. signal/error -signal ratio.

Fig. 2.2 Minimum change in desired signal level detectableversus signal/error signal level given .048 db system dataacquisiton resolution.

Ia

by improving the isolation between the transmitter and

receiving antennas with the introduction of absorbing foam

panels such as Emerson and Cummings' Ecosorb panels between

the antennas.

2.4 System error correction

Turning next to the second class of errors; those

directly measurable and therefore removable; define the

following quantities which are functions of frequency :

I(f) -- Isolation of reference to test channels

T(f) -- Transfer characteristic of system

A(f) -- Attenuator characteristic

Al(f)-- Antenna system characteristic

S(f) -- Corrected backscatter for target

Cl(f)-- Uncorrected antenna clutter and coupling

C2(f)-- Corrected antenna clutter and cross-coupling

(uncorrected for antenna system response)

C(f) -- Corrected antenna clutter and coupling

Rl(f)-- uncorrected reference target backscatter

R2(f)-- Corrected reference target data

(uncorrected for antenna system response)

R(f) -- Corrected reference target data

Several possible techniques exist for the removal of system

errors. The particular technique is dependent on the

relative signal levels involved, the accuracy desired, ease

of implementation, and computational speed. The first

technique described here is similar to that used by Weir et

al .(71 13

The first step in the correction procedure measures the

transfer function of an attenuator A(f) as a function of

frequency. The equipment setup for this procedure is shown

in fig. 2.3. The two ports of the reflection-transmission

unit connected through a precision HP 1l605A flexible

coaxial arm. The MINC then steps the sweeper to a number of

frequency points and stores the system response (log

amplitude and phase vs. frequency) in memory. When this

completed the attenuator is placed in series with the arm

and another set of measurements is made at the same

frequency points as before, the system response

characteristic is subtracted from the combined attenuator

plus system response measurement made on the second sweep.

An example of this procedure is shown in fig 2.4. Computer

subroutine PAD performs this operation. It is listed in

appendix I along with all other computer program listings

and output pertaining to system response measurement and

removal.

The next step in the calibration process is measurement

of the reference to test channel isolation I(f). This

characteristic is dependent on the directivity of the

network analyzer harmonic converter,and the reflection

transmission unit. For the Hewlett-Packard 8411A harmonic

converter the isolation is greater that 50 db. In this

measurement the ports of the reflection-transmission unit

are terminated in the cables used for the later target

scatter measurement. These coaxial cables are terminated inI "

A#PU resr hpz

Cq#/7 7mA-,7w,. /ec

4Vzoc.%SLAeeer

Fig. 2.3 Equipment setup for attenuator measurement.

i. m -00 . -..0 *9.oE 0 8. 75 .00

5 PHS NDCHF 0

Fig. 2.4 30 db attenuator characteristic for 8.0-8.5 GHz.

15

50 ohm resistive loads as shown in fig.2.5. The results of

a typical run are shown in fig.2.6. Computer subroutine IST

automates this stage in the correction procedure. As can be

seen the coupled signal is well in the noise of the system

and would not affect later scatter data. If the isolation

effect is ignored later calculation would be simplified

greatly; but is included here to be consistent with the

procedure outlined in the literature.[7]

The next stage in generating the data for correction of

scatter data is measurement of the system transfer function

T(f). This consists of the characteristics of the traveling

wave tube amplifier,system cables and connectors. This is

done by connecting the cable from the transmitting antenna

to the receiving antenna cable in fig.2.A and placing the 30

db attenuator characterized previously in the line to avoid

damage to the harmonic converter. The raw measurement MT(f)

is a combination of several factors:

MT(f) =T(f) *A(f) +I (f) (Z.2)

Solving for T(f):

T(f) - (MT (f) -I (f) )/A(f) (2.24

The equipment setup for this procedure is shown in fig 2.7

and typical uncorrected and corrected transfer function data

in fig.2.8 and 2.9.

Measurement of the antenna cross-coupling and room

clutter is the next step in the correction process. In this

procedure shown in fig.2.7b, the target is removed from the

anechoic chamber and the antennas pointed to the targetI'

VL

between refeenc and unknow ports. O". or

Pig.~~15M0 2.5 System cofiuato fo esurig iolaio

SM5 IM P I

Fig. 2.6 Isolation between ports of reflection-transmissionunit; 8.0-8.5 GHz.

17

4Rh-l-Wwee I WnAA 4 #4e

Sam%~4 5%44/ 4At7i

I. PeAetcrn POqt

".. -Au eie3wwpA5L ."- 4.04-

AW$IdIAe Ve'r4q1

25f4. D7 a zld ~&e~~~toS~

A~ ~ 4i~ PeedeAteo

A'jnkn C04 MLL

07VJ43 7 9 a~h, 4f

Fig. 2.7 a) System configuration for transfer characteristicmeasurement. b) Connection to antennas for scattering andclutter measurement. c) Transmitting antenna with spiralAEL antenna and parabolic dish. d) Receiving dualpolarization horn ; EMI A6100 with Norsal 90 hybrid.

19

rI

(C)

Fig. 2.7 (contd.) a) system configuration for transfercharacteristic measurement. b) Connection to antennas forscattering and clutter measurement c) Transmitting antennawith spiral AEL antenna and parabolic dish. d) Receivingdual polariztion horn; EMI A6100 with Norsal 90 hybrid.

'9

response; 8.0-8.5+0 8.350E

20 X -S 114 D GH.. -

-- ' -- --+ .50 E0 8-35E-,Fig. 2.8 UnorceTransfer characteristic corce of sstem;torso;8.0-8.5 GHz.

02O

lZOEO 825O-0 - .370EC!N U DAT . . . . .NA E O T C . . ..

location. A high gain parabolic dish antenna is used for

illumination of the target since the narrow beam pattern of

the antenna places most of the radiated power on the target

area. A smaller dual polarization horn is used for

receiving in order to sample a small area of the scattered

field. The uncorrected clutter Cl(f) is given by:

Cl(f)=C2(f)*T(f)+I(f) (2.3)

Solving for the corrected clutter and coupling:

C2(f)=(Cl(f)-I(f))/T(f) (2.3)

Subroutine ANTEN does the system clutter and coupling

removal. Examples of the uncorrected and corrected clutter

Cl(f) and C2(f) are shown in figs.2.10 and 2.11. The

corrected clutter represents the actual signal reflected

from the anechoic chamber walls and that signal coupled

between the antennas with the system response removed.

These subroutines: PAD,IST,TRANS, and ANTEN, were

combined into a program SYSRES. Data from each of these

subroutines may be stored on disc for later recall or

display. Theoretically if the system is not disturbed then

the system response will remain constant. Then only the

target data need be recorded in any run for a new corrected

backscatter measurement.

The transfer function of the system and the range

clutter- antenna cross coupling data are utilized the the

next step of the error correction process. A reference

object of known constant cross section is measured and the

result stored. This data includes all the errors previously

21

Fig. 2.10 Uncorrected system clutter; 8.0-8.5 GHz.

Fig. 2.11i System clutter corrected for transfer function;8.0-8.5 GHz.

22.

NI

described but also takes into account the antenna system

variations as a function of frequency:

Rl(f)=C2(f)*T(f)+I(f)+R2(f)*T(f) (a.q)

Here C2(f) and R2(f) contain the antenna system response

multiplying the actual values of corrected reference target

and system clutter data.

R2(f)=R(f)*Al(f) (2.5)

C2(f)=C(f)*Al(f) (211}

This response Al(f) takes into account the varying amount of

power received and transmitted as a function of frequency in

the antenna system. If the reference target is chosen to

have a constant cross section and linear phase over the

frequency range of interest then R(f) is of constant

amplitude and linear phase. Solving for R2(f):

R2(f)=(Rl(f)-I(f)-Cl(f))/T(f) (2.7)

This leaves R2(f) proportional to Al(f) shifted by a linear

phase corresponding to the reference target range.

When the actual target is measured; it is corrected for

system errors as was the reference target data and this

result is divided the corrected reference target data to

yield the final target scatter data.

S2(f)=(Sl(f)-I (f)-Cl(f))/T(f) (2.8)

S (f )=S$2 (f )/R2 (f) M. 9)

This technique may be simplified considerably in the

laboratory environment given the signal to noise ratio is

greater than 10 db for the scattered signals. In this case

only multiplicative errors remain and the additive errors

.#~ISe. . . . . . . .

are masked by the high target scatter signal amplitude.

Hence:

I (f) ,Cl (f)->O(2.10)

R1(f)=R2(f) *T(f)

and therefore:

S (f) =Sl (f)/Rl (f) (2.11)

Weir and his group have reported that this technique has

reduced equivalent range clutter to -45 db below I sq.

meter.

There remains one source of error in the scattered

signal measurement that cannot be removed by calculation.

This error comes from multipath scattering from the object.

The target scatters power in all directions. Some of this

signal may be reflected off the walls or floor of the

anechoic chamber. The signals reflected off the walls is

over 48 db down from the incident wave amplitude in the

6.0-12.0 GHz range. However if the target is small in cross

section; on the order of 100 sq. cm.; then it is possible

for the walls (on the order of 100 000 sq. cm. ) to

contribute a significant component to the received signal.

In order to analyze the effect of the multipath

scattering, let the directly received signal be written:

and the indirectly received signal:

Si(t) - 0 Cos(wt 't'e) (z.13)

Then the total received signal is given by:

Sat) (A1 (,2AS cao,8a) Cos (Wt

a V.

Since theta is a function of the indirect path length

differences and frequency, it will lead to a periodic

variation in the amplitude of the scattered signal. As an

example; if the paths differ in length by 1 meter then the

amplitude oscillations will occur every 300 MHz given all

other factors remain constant.

A series of programs was written to test these various

techniques of error removal. They differ in the only in the

error removal technique employed; not in file storage or

display formats nor in range calculation and removal to be

described. These programs used together:

l)Measure and correct the reference target data with

files of transfer function and clutter data

generated by SYSRES.

2)Measure and correct the .target data and finally

take the corrected reference target data and remove

the antenna system response from the target data.

3)Alternately for high SNR; calculate the correct

scattered target signal directly using the reference

target data.

4)Calculate and remove phase shift due to range from

the target signal after error correction

These programs are briefly described and differ in the

mentioned categories:

SPHERE-Reads transfer function and clutter data

files generated by SYSRES; takes the reference or

object data and corrects for errors in the

2S

r

equipment.

SPHER2-Measures clutter and reference target data.

It then subtracts clutter and divides the target

data by the reference target transfer function.

SPHER3-Measures reference target and object signals

ignoring clutter, and divides the object data by the

reference target complex field amplitude data.

In order to implement the complete error correction process

program SPHERE would be run twice; once for the reference

target and then again for the test object. This data would

be stored for later retrieval. Program SPHER3 would read

these files and process then such that the test object

complex field data would be divided by the reference target

data. For high SNR cases; program SPHER3 alone would be

run: first measuring the reference target and then the test

target and finally dividing them yielding the final result.

Listings and a more complete description of program

operation is given in the program appendix I.

2

III RANGE PHASE SHIFT ANALYSIS

This chapter contains an analysis of the effects on the

phase shift due to target range on coherent imaging of a

target. It includes a discussion of some of the available

techniques for the removal of this phase shift and the

required perfor mance of these systems based on bandwidth

and signal to noise ratio.

3.1 The effects of ranephase on imna in9

As previously described by Farhat and Chan ;['] the

scattered field of the scattering target is given by:

(3-1

Where Rq and R, are the vectors from the receiving and

transmitting antennas to the target respectively and k is

the wave vector of the illuminating wave. In this case the

integration is over the extent of the object. The integral

term is independent of the range of the target. While the

term preceeding the integral is target independent and

contains range information. The argument of the complex

exponential is a linear phase function of frequency.

In the single receiver=transmitter pair arrangement;

shown in fig.3.1, the spatial frequency domain (' space)

data is collected in a plane perpendicular to the axis of

rotation. In this case the multiplication of the range

phase factor leads to the convolution of the transforms of

the the range and target functions in the spatial domain.

The real part of the range frequency domain function is aL 27

7-( 1YI7E

(6)

ft-6....

Fig. 2.12 a) Target and antenna placement relative to targetconsidered in analysis. b) Experimental configuration inanechoic chamber; note target on rotating pedestal inforefront.

radially symmetric sinusoidal function:

This transforms to a circular ring of radius (R,+ Pv. This

indicates that each point of the reconstructed image is

convolved with this circular ring pattern; seriously

degrading the target image. In fact any error in the

removal of phase will distort the image in this manner. The

effect of removing the range phase factor is equivalent to

the focusing of the system on the target.

There are several other reasons for the removal of the

factor: ________ (.3)

from the received data. If the data is discrete then the

considerations of aliasing and sufficient data sampling rate

are introduced. When the sampling rate in the frequency

domain is ( f) then the maximum target range before aliasing

will occur is (c/4* f). However if the phase factor is

removed and the target is of smaller dimension L than the

range, then the sampling rate in the frequency domain need

only be sufficient to prevent aliasing over the dimensions

of the object :

qL

This allows the entire resolution capability of the system

to be placed on the target itself greatly reducing the data

volume.

Another reason for the removal of this phase factor

term can be seen in systems involving multiple

receiver-transmitter pairs. For each pair (f,) and (k ) is

different and in order for the data to be coherent all phase

centers must be equal for an image to be formed.

Several methods have been suggested for the removal of

the range phase factor. (4 1 In all cases it is required

that the range removal technique be accurate to within (2/5)

for there not to be serious image degradation. It is

important to investigate the constraints on the ranging

system parameters necessary to attain the required accuracy.

From the scaling theorem in Fourier analysis; a signal

cannot be both of narrow bandwidth and short duration. [$]

We define the duration and the bandwidth of the signal in

the following manner: [ ]

S/ad) feJtI olsit (,4 (Aw)'zj iq f4_6 C~,~(L

where:

is the signal energy. Then if:

ir- fit o (36)

When applyed to signals scattered by a target this may be

translated to range uncertainty:

3fo (37)30

This represents a limit when the durations of the signal

pairs are as previously defined. However it may be possible

to improve the range resolution given that the signal to

noise ratio is greater than 10 db.

Consider a sinusoid in narrow band gaussian noise:

rft) : q c.4 ~ s t (t) (3.9)

where n(t) is the noise signal and ns(t) and nc(t) are the

quadrature components of the noise signal. The amplitude and

phase may be expressed as:[9]

A r,'(rt..): to.J' A )(3% )

Since the noise is gaussian nc(t) and ns(t) are gaussian.

Assuming that the SNR is high the phase may be approximated

by:

Also since the noise is white n , (t) may be related to the

bandwidth of the receiving system.

a, ( N.3.4 )

where (') is the variance of the n,(t) quadrature phase

component. The probability density of the phase may be

written as:A -No 4"e/,,.

This leads to an interesting result; the variance of the

31.

phase is the inverse of the SNR:

I (3.13)

Translating this result to an uncertainty relationship the

phase uncertainty may be expressed:

The uncertainty in phase is the product of the time and

frequency uncertainties. This leads to an expression for

the range resolution as a function of the SNR.

C

Wide band ranging systems measure range by calculating

the propagation delay of the signal. In one such system a

high speed code is transmitted. The received signal is

correlated with the original coded signal ih a delay locked

loop. The value of the control signal in the loop is

proportional to the target range. Obviously the resolution

is only as good as the the period of one of the code

bits(chips). Other wide band systems use other signals for

ranging(chirps,walsh functions) to obtain high range

resolution. (,] , [ll, [,a]

The system implemented here at the Graduate Research

center of the Moore school is a coherent amplitude-phase

measurement facility. Ranging techniques that may be

integrated into this system are therefore of special

interest. The phase of a point scatterer is a linear

function the frequency. This directly corresponds to the

phase factor preceeding the scattering integral in eq ()

If the target consists of multiple scattering centers; then

each of these will be represented as a delta function in the

in the reconstructed image. This suggests a technique using

Fourier analysis to determine range. First place the

reference target in the microwave field and measure the

phase/amplitude response over as wide a frequency range as

possible. Then inverse transform this one dimensional

collection of data. This will transform to essentially a

delta function occurring at the time corresponding to the

propagation delay. This will occur when the target has a

single scattering center such as a sphere. Even when the

object is more complex the inverse transform will be

centered around the transit time. In this Fourier technique

for range determination the resolution (4Ax) is inversely

related to the frequency sweep width.

AX (3 i C

The factor of 1/2 results from the fact that the range is

half the signal propagation path length.

A factor to be considered is that different scattering

centers are visible from different receiver positions. It

is imperitive that all when several receivers are used

simultaneously that all choose the same scattering center A-

the phase center for range removal. If the various ;

centers do not coincide , the fringes of the holoq? ir'

be skewed. The data sets from each of the recei.,'c -

brought into alignment using an adapt ive53

"A-A90 838 MOORE SCHOOL OF ELECTRICAL ENGINEERING PHILADELPHIA PA F/6 17/9SUPER-RESOLUTION IMAGERY BY FREQUENCY SWEEPING,(U)AUG 80 N H FARHAT, C WERNER AFOSR-77-3256

UNCLASSIFIED AFOSR-TR-80-1068 NL

2 200000

0

I EiEElEliEEEEDiE-u.-.--.m.E..hhghhhlhI.hhinhhinhhin.

Information is lost in the hologram if the phase centers for

the scan lines are separated.

When this process is automated the sweep is done by

measuring the response at discrete frequency points. The

amplitude and phase are stored at N frequency points in the

sweep range. The range at which aliasing will occur is

given by:Re

This system exhibits processing gain ; a quality of

all systems which spread a baseband signal into a wide

spectrum. For this system the processing gain is a function

of the number of measurements and the sweep width.

6&-, J6= 10 1-b3. rV

As an example, for 256 measurement points, this would give

24 db of processing gain.

A set of subroutines was written to test this range

removal technique. In one of them , RANGE, the range of the

strongest scattering center is calculated and in the other

,RANCOR, the phase factor is calculated and removed from the

target data. These subroutines are used in programs

SPHERE,SPHER2 ,and, SPHER3.

The time domain equivalent of this technique is fitting

a linear trend to the phase signal from the network

analyzer. Since the phase signal is modulo (2w)hthis means~34

estimating the frequency of a ramp waveform; either using a

least squares approximation or implementing the equivalent

of a phase locked loop. The slope of the ramp waveform is

proportional to the range of the target. The accuracy to

which the range may be determined depends on the sweep

width, the target structure and the noise in the system. If

the sweep is wide, then there will be more data with which

to estimate the slope. Noise in the data will obviously

interfere with the estimation process as will as any phase

shifts due to the target structure.

Another type of system for generating a reference

signal utilizes a Target Derived Reference. This system has

been extensively studied at the Electro-Optics and Microwave

Optics laboratory.[is] A brief review of the ideas developed

to date in this regard are given below. The complex

exponential in the integral term of scattered signal remains

constant when the target is small relative to the

illuminating signal wavelength.4 4

This is true if ( Lfi ) is sufficiently small such

that( f. '4|). In this case the integral value approaches

a constant multiplied by a linear phase; i.e. a point

scatterer. It will only occur when the target dimensions

are less than a tenth wavelength of the illuminating signal,

placing it in the Rayleigh scattering region. The TDR

signal is mixed harmonically with a phase locked scattered

SS.

estimating the frequency of a ramp waveform; either using a

least squares approximation or implementing the equivalent

of a phase locked loop. The slope of the ramp waveform is

proportional to the range of the target. The accuracy to

which the range may be determined depends on the sweep

width, the target structure and the noise in the system. it

the sweep is wide, then there will be more data with which

to estimate the slope. Noise in the data will obviously

interfere with the estimation process as will as any phase

shifts due to the target structure.

Another type of system for generating a reference

signal utilizes a Target Derived Reference. This system has

been extensively studied at the Electro-Optics and Microwave

Optics laboratory.[%%] A brief review of the ideas developed

to. date in this, regard are given below. The complex

exponential in the integral term of scattered signal remains

constant when the target is small relative to the

illuminating signal wavelength.

This is true if CL~,. ) is sufficiently small such

that( f-f 44-). In this case the integral value approaches

a constant multiplied by a linear phase; i.e. a point

scatterer. It will only occur when the target dimensions

are less than a tenth wavelength of the illuminating signal,

placing it in the Rayleigh scattering region. The TDR

- - signal is mixed harmonically with a phase locked scattered35.

signal at the imaging frequency. here the two signal

sources are phase locked by a phase synchronizer. The high

frequency sweeper acts as the slave signal source. These

two signals simultaneously illuminate the target. Harmonic

mixing of the suitably limited TDR and imaging signals will

yield the desired phase corrected data.

Another technique which is useful for ranging is thej

frequency displaced reference method. Here the carrier is

displaced (,af). It is important that the object not

contribute to the phase shift; hence the displacement must

satisfy the following condition:

In this case the target structure will not contribute to the

* net phase shift.' The range to the target phase center is

given by:

Where ()is the change in phase for the carrier and

displace carrier signals respectively. If the frequency

shift is small then the phase shift will not be large. The

resolution of the system then is directly related to the StIR

in the receiver channel since the accuracy to which the

phase may be measured is a function of the channel noise.

If the displacement is large then the phase shift Is

Increased and the StIR requirements on the signal for a

specific resolution is decreased. This relationship may be

expressed: . .

The factor of 2 comes about due to the phase uncertainty

existing in both the carrier and displaced carrier signal.

An alternate method for implementation of the displaced

frequency ranging system is a swept frequency chirp system.

The ranging signal frequency is given by

titf

T- Sweep period

f,- Initial frequency for sweep

f,- final frequency

The scattered signal from the target is given by:

i* A± - f-'ARr(3.

where (df'/dt) is the change in frequency due to the target

structure. The range of the target may then be simply

calculated using a frequency counter and sweep time T and

sweep width (f,- f). This method could be used in an analog

imaging system.

Other Target Derived Reference systems simulate the low

frequency reference carrier by measuring the change in phase

of the imaging frequency over a narrow band. Over this

small band the phase shift is assumed to be linear. in one

system a series measurements is made for each frequency

point. The first displaced down by a small amount, (Af);

the second at fi and the final measurement at (f+&f). Phase

and amplitude are measured at each frequency and processed

to obtain the target range. in a similar system the three

37

signals are transmitted simultaneously by amplitude

modulation of the carrier. These systems are presently

under intense investigation by other workers at the

Electro-Optics and Microwave Optics Laboratory of the Moore

School.

32Practical cosdrtosfor ranqe phase removal

Several factors influence which of these systems would

be of value in a long range imaging radar system versus a

controlled laboratory environment. The distribution of the

reference signal for complex field amplitude measurement

makes implementation of system requiring a central reference

difficult to implement. Techniques are being considered in

the E.O. laboaraty for reference distribution using fiber

optic that might remove this limitation. Reference

distribution is accomplished in the lab readily since the

distances are small.

Typical transmitter-receiver pattern arrangements

might be the Wells array [11)], an orthagonal pattern of

receivers and transmitters, a circular array of receivers

with central transmitter or a random array. Each

combination of receiver and transmitter contributes another

line in the frequency domain 3D data volume. For this

reason it is advantageous that all combinations of the

receivers and transmitters are utilized for data collection.

Reference signal distribution difficulty therefore leaves

the TDR systems as the only practical alternatives for long

range imaging systems. The AM TDR system eliminates30

reference distribution by transmitting the reference signal

along with the imaging signal and automatically corrects for

the target range. TDR systems have the additional advantage

that they have immunity to turbulence and inhomogeneities in

the propagation medium since both the reference and imaging

signals follow the same path.

There are several considerations for determining the best

TDR system. Narrow band systems yield only a weighted

average of the range to the phase center of the object while

wide band system can resolve individual scatters on the

target body. A wide band system could adaptively choose one

of the scattering centers for the phase reference of the

system. A narrow band system could not do this, and any

error would introduce image distortion. The narrow band

systems have the advantage of automatically correcting the

target data for the range phase factor.

For the laboratory imaging experiments a TDR system

would not yield the correct phase factor since in this

arrangement the object rotates about anaxis and the correct

phase factor would be a constant, representing the phase

shift to the axis of rotation ,not the target. The TDR

system looks only at the range to the strongest specular

reflector on the target surface.

If movement of the target is utilized for aperture

synthesis then only one receiver-transmitter pair is

required for 3-D imaging and the adaptive system is not

required. 3

Knowledge of the placement of the data in the 3-D

frequency domain volume is necessary for the reconstruction

of the hologram. The azimuth and elevation angle of the

target can be obtained from a conventional radar located at

a central location where the data processing and

reconstruction is taking place.

IV SYSTEM IMPLEMETATION OF SWEPT FREQUENCY IMAGING

This section of the thesis will describe the research

that was done in order to obtain a clear understanding of

system performance. Following this will be a section of the

thesis devoted to the experimental verification of the swept

frequency imaging theory.[ I The theory is applied in the

simulation of the experiments performed.

4.1 System .! e eatability

An important parameter of system performance is the

repeatability of an experimental measurement. The frequency

range over which this possible for the equipment used

indicates the bandwidth for which imaging is possible.

There are several feedback loops in the system which allow

it to track variations in the transmitted signal level. The

traveling wave tube amplifier characteristic is shown in

fig.4.1. This plot is on a logarithmic scale, indicating

that the TWT amplifier gain drops off exponentially below

7.0 GHz and above 15.0 GHZ. This measurement was done by

first measuring the system response(cables,

connectors,attenuator) less the TWT amplifier and then

subtracting this response from the TWT amplifier plus system

data. A sample of the amplifier output is sampled using a

20 db directional coupler and this is fed to the RF input of

the HP 8743A reflection-transmission (R-T) unit. A crystal

detector at the 'unknown' port of the R-T unit rectifies a

portion of this signal. The detector output is brought to

the external signal leveling input of the HP 8620C sweeper.

DOTE )L)'1(c-L3

(C)

Fig. 4.1 a) Traveling Wave Tube amplifier gaincharacteristic in db; 4.0-16.0 GHz. b) Varian VA 618G TWTamplifier (bottom) and HP 8743A reflection-transmission unit(top).

2z

The leveling circuit of the sweeper can level the output

over a 20 db range. In addition the AGC in the HP 8410B

network analyzer can track the reference signal amplitude

over a 40 db dynamic range. System performance may be seen

in fig.4.2 for a cylindrical target 7 meters distant from

the receiving and transmitting antennas. Two consecutive

measurements of the target were made and the results

divided. The ideal response would be 0 db flat amplitude

and 0 degree phase difference over the entire frequency

sweep. With few exceptions due to phase noise at the (+/- )

transition point, the system has the desired iepeatability

in the 5.5 to 16.0 GHz range. Below 4.5 GHZ there are phase

errors due to insufficient reference power whereas above 15

GHz errors come about due to the low amplitude of the

received signal. Noise may be cancelled by taking multiple

measurements and finding the mean. These results indicate

the useful data can be recorded in the 5.5-16.0 GHz range

4.2 Computer control of taret rotation

When implementing a frequency diversity system with

just one pair of receiving and transmitting antennas, the

target must then be rotated in the electromagnetic field and

the scattering measured for different rotation angles. The

target used in the experimental system rotates on a stepper

motor driven pedestal. The column of the pedestal is 1 1/2

meters in length and is made of styrofoam material with

minimal cross section. A stepper motor controls table

rotation precisely. In order for the pedestal to rotate one143

II

Fig. 4.2 System repeatability for two consecutive scatteringmeasurements of a 80 cm long cylinder 7 cm in diameter;4.0-16.0 GHz. S

Fig. 4.2 System repeatabiyfrtocneuiesatrn

revolution the motor must be stepped 10 000 times. The

motor is under direct computer control using the digital

output port the MINC. A FORTRAN subroutine STEP2 was

written for control of the stepper motor. It calculates the

number of steps required for a specified angular rotation

and moves the table clockwise or counter clockwise based on

the direction parameters passed in the subroutine call. The

stepper and pedestal are shown in fig.4.3.

4.3 Sphere Simulation

For the calibration of a radar system a reference

target is required. The most commonly used reference target

is the conducting sphere since its high degree of symmetry

does not favor any particular polarization for the incident

illumination. Both the bistatic and monostatic scattering

of a metallic sphere was simulated.

The general solution for the plane wave

electro-magnetic scattering of the sphere was first done by

Mie in 1908.[161,[i&] In the far field approximation, the

scattered field is given by:

where

and a

nsts

and-a

* * I AL

Fig. 4.3 Stepper motor and rotating pedestal.

S() and S () are called the complex far field

amplitudes for the ( ) and ( ) polarizations respectively.

The quantity in the square brackets of eq. 4.1 is called

the scattering function.

The scattering cross section in any arbitrary polariztion

(C ) for an incident wave polarized in the ( ) direction

may be written:

Where ( u ) is the polariztion of the incident wave and

('r ) is the polariztion vector of the receiving system.

For the perfectly conducting sphere the coefficients A~and B

are: w* ,A,)

-1 0 (

j,(k.a) - Spherical Bessel function

h (k~a) - Spherical Hankel function

P,( x) - Associated Legendre function

ko - Wave number of incident wave

a - Sphere radius

The prime on the expression for B. denotes differentiation

with respect to (ka).

Polynomial approximations exist for the Mie series

exact solution.[15] Different polynomials are used for the

three frequency regions for scattering. These are: low

'I7

frequency or Rayleigh region (ka)<.4 ; the resonance regiun

.4<(ka)<20 ; and finally the high frequency or physical

optics region (ka)>20. Two programs BISCAT and SPSCAT

implement both bistatic and monostatic cases in the three

frequency regions. Fig.4.4 shows the monostatic scattering

of the sphere as calculated. This is exactly the same

answer for the scattering of the sphere as the exact

solution. The horizontal axis is in terms of the

dimensionless quantity (koa). Figure 4.5 a,b,c and d show

the bistatic scattering of the metallic sphere of bistatic

angles of 3d,60r,90e, and 12f degrees. Note that the

approximation are only valid in the range:

where

which leads to the discontinuities for small (k a) at large

bistatic angles. BASIC programs BIDISP and SDISP generate

the graphs for, the sphere simulations. These programs are

in appendix II.

An important result from these simulations is that at

high frequencies the scattered signal is of constant

amplitude and linear phase irrespective of the bistatic

scattering angle. The only exception to this is the forward

scattering case when (a) equals ( r), where the cross

section grows without bound as k increases. This indicates

that the only portion of the sphere that is scattering for

large (ka) is the front face closest to both the receiver

and transmitter; and therefore a ray optics approximation

-- -I ----- --

Fig. 4.4 Monostatic scattering for the perfectly conductingsphere. (k a) varies from .2 to 10.5 which corresponds tothe scattering of a 20 cm. diameter sphere in the frequencyrange of .1 to 5.0 GHz. Log normalized crosssection: log(

49

P

Fig. 4.5 Bistatic scattering of a 20 cm. diameterconducting sphere in the frequency range .1 to 6.0 GHz;.2<(k* a)<12.6; polarization of the receiver equal toscattered wave polarization, a) Geometry for scatteringexpression. b) Bistatic angle 30: c) Bistatic angle60 d) Bistatic angle 89t e) Bistatic angle 1200

so

4 e000: 0(0 0um

4 nK TART). 2043 K teesD~ 12. C

M-, EEM I 7 l = SU C11 I

Fig.4.5 cond.) itti scatein of0( a0 200 cm0iaee

conducting~~~v00 spher inDOO th-reunc0ag01 o60G

Fig. 4.5sontd) Bistatic sagen of0a. c m. Bitaianee

60. d) Bistatic angle 89. e) Bistatic angle 120:

codctn spher in the freuec range .1 to 60,0J KGA, z1;.".2C~~k~a)<12.6; ~ poErCa.o of tereevr.qa0t h

Fig. 4.5sontd.) Bistatic sangen of0a. cm) Bitaianee

60.' d) Bistatic angle 89. e) Bistatic angle 120:

4! may be applied to find the scattered field.

for the bistatic case. In the monostatic case this reduces

to:

For targets with features larger than a few wavelengths in

size resonance effects become minimal.

Another possible reference target is the long cylinder

(1 >> a). The scattering of the cylinder in the high

frequency region when it is oriented vertically yields an

answer similar to that of the sphere. For (k a)>5 where a

is the cylinder radius, resonance effects disappear and the

copolarized scattered field is given by: [ 15 ]

E3 %EL W1 /-7i 4elLP(L .4..CoCs 0

The equation for the scattering of the cylinder is used

for the computer simulations of frequency swept holography.

4.4 Simulation of frequency swept imaging

A series of frequency swept hologram simulations were

done of targets that would later be imaged experimentally.

The basic arrangement consists of separate receiving and

transmitting antennas which measure the scattering of a

target that rotates about an axis. The center of rotation

is chosen as the phase center of the imaging system. For

this configuration the fre uency domain data lies in a plane2$

perpendicular to the axis of rotation. Therefore the

transforms of the holograms will be slices in this plane.

The first object hologram simulated was comprised of two

cylinders equidistant from the rotational axis. This target

is shown in fig.4.6. Approximations for the various

distances were derived:

,,--,- (0A /-e) (',.,,-.) ,-e. ,- s,,( -o) (,,.,ib)

the waves striking cylinders Cl and C2 are given by:

E Ee (N.z)

The scattered waves from the two cylinders including the

cylinder response then follows:

This is further simplified by combining terms :

Finally take the real part of this function for display:

R (Es): e.s (z Aocat) cos (Abe av oivt~ (~~

This was done for a two cylinder target with cylinders 5 cm.S'V

tx

I6

Fig. 4.6 Bistatic scattering of two cylinder target.

SS

in radius ,separated by 25 cm. using program CYLIN. The

results were displayed by program CDISP on a Tektronix 606A

CRT display. CDISP gives the option of varying the gray

scale compression of the hologram either logarithmically or

by constant multiplication. These programs are listed in

appendix II. The resultant hologram and the reconstructions

obtained through Fourier transformation on the optical bench

appear in figs.4.7 a,b The hologram simulated a sweep

from 2.0 to 18.0 GHz in 64 frequency steps. The target in

the simulation rotated 360 degrees in 128 steps.

Another target simulated which did not have the symmetry

of the first target was comprised of two cylinders both

mounted to one side of the rotational axis, as shown in

fig.4.8. Two simulations were done of this target with

varying diameter cylinders. In the first case 7 cm radius

cylinders were used. The hologram for this case and the

Fourier transform reconstructions are shown in figs.4.8

b,c,d. For the second simulation the target was two

cylinders 3.5 cm in radius. In both cases the cylinders

were located 10 cm from the center of rotation and the

simulation was for a 2.0 to 18.0 GHz sweep. The hologram

and the transformed images are shown in figs.4.9 a,b,c. The

two cylinder off axis target was simulated by first

* calculating the copolarized scattered field for a single

cylinder: t L z& At A

5'i

II

Fig. 4.7 a) Simulation Hologram of two cylinder target; 10cm. diameter, 25 cm. apart. Frequency range: 2.0-18.0GHz; 128 lines, 64 points/line. b) Optical Fouriertransform of hologram.

57

L.

7o., Cm.

Axis OF R6TRT16N

(b)()

(d)

Fig. 4.8 a) Geometry of two cylinder off-axis target. b)Hologram simulation two cylinder off-axis target; 2.0-18.0GHz; 64 points/line; 128 lines. b) Transform with zeroorder term. c) Transform with zero order removed.

58

IF'

(0(.)

( b ) ( C )

Fig. 4.9 a) Hologram simulation of off-axis two cylindertarget with cylinders 7 cm. in diameter; 2.0-18.0 GHz b)Optical Fourier transform with zero order. c) Transformwith zero order term removed.

*79

taking the real part:

Z E. cos a .Ai -..- 9 J s)

For the two cylinder off axis target, one cylinder is at

9(e)= and the other at ( e)=90, therefore the scattered

field is given by:

1Z jE4: r- Cas ( A?-k- -V-" 1)* Cbzos t.e-0G~

In general for an arbitrary set of circular scatterers of

radius a and distance 1 from the origin; the scattered field

may be written:

This gives the capability to simulate the scattering of any

target given that it can be decomposed into N spherical

scattering centers.

4.6 Experimental results

An experimental system for the implementation of swept

frequency imaging was setup in the anechoic chamber at the

Graduate Research Center in the Moore School. The frequency

range for these experiments was from 6.3 to 16.0 GHz in 64

discrete steps. The targets were rotated 360 degrees in 128

steps. These holograms were then identical in form to the

simulations previously done.

The system for error correction and range phase shift

removal was that used for high signal to noise ratio

signals. The reference target was a cylinder positioned so~~]

that its front face was located on the axis of rotation as

in fig.4.10. A plot of system response is shown in

fig.4.11. This data represents the combined characteristics

of the antennas, amplifier , cables and clutter. In

addition it contains the linear phase shift that is the

range phase factor. As an example for the two cylinder

target shown in 4.12 the raw data, magnitude and phase is

shown in fig.4.13. Figure 4.14 shows how this data has been

corrected for range phase and system response. This data

was generated using the Fortran program SPHER3.

The experimental properties of the two cylinder target

were studied extensively. Both the scattering as a function

of frequency for a specific orientation of the target and

the scattering as a function of angular rotation at specific

frequencies was obtained. In figs. 4.15 a,b,c are shown

the corrected frequency response of the targe for

orientations of 45,90Wand 13T degrees.

Another computer program ANTPAT was written to obtain

the radiation pattern of an arbitrary target or antenna. In

the two cylinder case the pattern was measured at 5.0,10.0

and 15.0 GHz. Note that when the cylinders are collinear

all that is seen is the front surface specular reflection of

the one cylinder hence the pattern of a point scatterer in

the vicinity of 00degrees. These patterns are shown in

figs.4.16 a,b,c. At high frequencies the lobe spacing is

much closer than at low frequencies, consistent with the

theoretical result for the pattern. To see this examine the

AXIS OfRC T T#Op

Fig. 4.10 Reference target on pedestal; 80 cm. longcylinder, 7 cm. in diameter.

Fig. 4.11 Reference target response including system

response and range phase shift; 6.3-16.0 GHz.

'az

AXIS O4'r

ILLUMMONrOM

Fig. 4.12 Two cylinder target geometry.

Fig. 4.13 Uncorrected scatter data for symmetrical twocylinder target ; (6)=o , 6.3-16.0 GHz.

Fig. 4.14 Corrected two cylinder target data using systemresponse of Fig. 4.11.

WV

(b)

Fig. 4.15 a) Corrected two cylinder symmetrical targetresponse; 6.3-16.0 GHz, e -e)=5% b) (G)=9d. c)(0• )s 133 ° .

'S

2.. low

E-01

tQ.

O(6

Fig 416Scttein ptrnof two cylinder1 tage of10 Fig.O4.12~~~~~~~ atdffrn feunce s afnto agtrttoange, ) 50 Gz. ) 1.0 ~z. C) 5.0G3Z

.12510: J A !f-VE D~t;CY-fi.-,

'Iexpression for the monostatic scattering of the symmetrical

two cylinder target:

' C"

The expression for the scattering of the target at a given

frequency as a function of angular rotation may be written:

As for the swept frequency response; 1 sin(e) remains

constant and the hence the swept frequency response is

sinusoidal with period dependent on

(1,) and 1.

The final test for the system was the generation of

actual holograms. The first target measured was the two

cylinder target shown in fig.4.17a. The cylinders are of

aluminum, 80 cm. in length and 7 cm. in diameter. The

real part of the corrected swept frequency data in the range

6.3 to 16.0 GHz was stored and displayed on the CRT. The

targets were rotated 360 degrees in 128 steps yielding a

total of 8192 points in the hologram (64 points/line * 128

lines). The center of the hologram is at 0 HZ with radial

distance directly proportional to frequency. An example is

shown in fig.4.17b and the reconstructions in figs.4.17c and

d. These Fourier transforms where done optically . [17]

This procedure was followed for other targets not

having the symmetry of the first object used. The target

type was the same as the simulations done previously. The67

I

Fig. 4.17 a) Two cylinder target in anechoic chamber onrotating pedestal. b) Hologram of target measured between5.3 and 16 GHz corrected for range and system response; 128lines, 64 points/line. c) Optical Fourier transform ofhologram. c) Optical Fourier transform without zero orderterm.

68

first of these was a single cylinder mounted off axis as

shown in fig.4.18a. This cylinder was the same as the

others used and the frequency range and angular sweep were

identical to that of the two cylinder target. The hologram

and reconstructions are shown in figs.4.18 b,c,d. The final

target was the two off axis cylinder target pictured in

fig.4.19a. The frequency diversity hologram and transforms

are in figs.4.19 b,c,d.

'69

n7 c

Fig. 4.18 a) Single cylinder off-axis target and position inanechoic chamber. b) Experimental hologram of target;6.3-16 GHz; 128 lines, 64 points/line; corrected for rangeand system response. c) Optical Fourier transform ofhologram. d) Optical Fourier transform without zero orderterm.

07

101

W (C)

Fig. 4.19 a) Two cylinder off-axis target geometry andposition in anechoic chamber. corrected for range andsystem response. c) Optical Fourier transform of hologram.d) Optical Fourier transform without zero order term.

.7'

hhi

V Conclusion

This thesis has described an automated swept frequency

measuring system.This system may be used for radar cross

section measurement, antenna pattern measurement and swept

frequency holography. The system has a useful range of

5.0-17.0 GHz in which amplitude and phase of the scattered

microwaves from targets in the anechoic chamber of the

Graduate Research Center may be recorded and stored. A DEC

MINC LSI-11/2 completely automates the data acquisition

process. A complete error correction algorithm was

implemented using the data storage and processing

capabilities of the minicomputer.

The effect of range phase shift on swept frequency

holograms was investigated and various techniques for its

removal were investigated. It is believed that a TDR system

for range phase removal is required for implementation of a

practical radar system. This system must have extremely

high resolution for coherent imaging. The relationship

between TDR system bandwidth, receiver channel bandwidth and

resolution was derived:

ReaeLougon -- n. C- lzT3. aAf zir 14ir A

where (&f) is the imaging bandwidth, (R) the range

uncertainty, B the receiver channel bandwidth and N. the

noise power spectral density.

Simulations were performed for the scattering of

various radar targets which include the conducting sphere,72

. --

V Conclusion

This thesis has described an automated swept frequency

measuring system.This system may be used for radar cross

section measurement, antenna pattern measurement and swot

frequency holography. The system has a useful range of

5.0-17.0 GHz in which amplitude and phase of the scattered

microwaves from targets in the anechoic chamber of the

Graduate Research Center may be recorded and stored. A DEC

MINC LSI-11/2 completely automates the data acquisition

process. A complete error correction algorithm was

implemented using the data storage and processing

capabilities of the minicomputer.

The effect of range phase shift on swept frequency

holograms was investigated and various techniques for its

removal were investigated. It is believed that a TDR system

for range phase removal is required for implementation of a

practical radar system. This system must have extremely

high resolution for coherent imaging. The relationship

between TDR system bandwidth, receiver channel bandwidth and

resolution was derived:

af zir 't1rR af4

where (&f) is the imaging bandwidth, (R) the range

uncertainty, R the receiver channel bandwidth and N. the

noise power spectral density.

Simulations were performed for the scattering of

various radar targets which include the conducting sphere,72

infinite cylinder and combinations of these. The holograms

recorded from these analytical results reconstruct the

targets extremely well and set a goal for practical system

performance. An expression was derived for the scattering

of N spherical/cylindrical cylinders in a plane passing

through their centers:

where 1 is the distance from the axis of rotation ( ) the

angle relative to some reference for the target angular

position and a the target radius.

Finally experimental swept frequency holograms were

generated using a rotating pedestal under computer control

to scan the target in one dimension. The experiments done

indicate the feasibility of implementing a practical

holographic radar system. The holograms obtained for

various targets agree well with theory even though the error

correction and range phase shift removal techniques used

were robust in nature. It is believed that better images

are possible given that the error correction techniques

previously outlined are implemented.

Further work can be done in testing the TDR techniques

for their suitability for an imaging system. The system may

also be expanded to include scanning in the ( ) direction

to give true 3-D imaging capability. This may be

implemented by adding a stepper motor controlled azimuthal

scanner to the top of the rotating column.73

infinite cylinder and combinations of these. The holograms

recorded from these analytical results reconstruct the

targets extremely well and set a goal for practical system

performance. An expression was derived for the scattering

of N spherical/cylindrical cylinders in a plane passing4

through their centers:

where 1 is the distance from the axis of rotation C ,)the

angle relative to some reference for the target angular

position and a the target radius.

Finally experimental swept frequency holograms were

generated using a rotating pedestal under computer control

to scan the target in one dimension. The experiments done

indicate the feasibility of implementing a practical

holographic radar system. The holograms obtained for

various targets agree well with theory even though the error

correction and range phase shift removal techniques used

were robust in nature. It is believed that better images

are possible given that the error correction techniques

previously outlined are implemented.

Further work can be done in testing the TDR techniques

for their suitability for an imaging system. The system may

also be expanded to include scanning in the ()direction

to give true 3-D imaging capability. This may be

implemented by adding a stepper motor controlled azimuthal

scanner to the top of the rotating column.73

BIBLIOGRAPHY

[(] N.H. Farhat, Frequency Synthesized ImagingApertureso, Proceedings of the 1976 InternationalOptical Computing Conference, IEEE Cat. No. 76 CH1100-7c, 1976

[2) N.H. Farhat,'Principles of Broadband CoherentImaging', J. Opt. Soc of Am., Vol.67, pp.1015-1021, Aug. 1977.

(31 N.H. Farhat,'Microwave Holographic Imaging-Prospectsfor a Real time Camera', Proc. SPIE TechnicalSymposium, Vol. 180, Real-Time Signal Processing II,1979.

[4] N.H. Farhat and C.K. Chan, "Three DimensionalImaging by Wave Vector Diversity", Presented at 8Symposium on Acoustical Imaging; Key Biscayne, Fla.May 29-June 2 1978.

[5] Robert J. Collier, C.B. Burckhardt and L.H. Lin,Optical Holography. New York: Academic Press, 1971.

[6] J.S. Hollis, T.J. Lyon and L. Clayton, MicrowaveAntenna Measurements. Atlanta, Georgia: ScientificAtlanta, 1970

[71 William B. Weir, LLoyd A. Robinson,DonParker,"Broadband Automated Radar Cross SectionMeasurements", IEEE Transactions on Antennas andPropagation. Vol. 12 pp. 780-784, Nov. 1974.

[8] Athanasios Papoulis, Signal Analysis. New York:McGraw-Hill, 1977.

[9] John B. Thomas, An Introduction to StatisticalCommunication Theory. New York: John Wiley & Sons,1969.

[101 Advisory Group for Aerospace Research andDevelopment:" Spread Spectrum Communications", U.S.Department of Commerce. AD-766-914, July 1973.

(11] Henning F. Harmuth, Sequency Theory, Foundations andApplications. New York: Academic Press, 1977.

[12) Athanasios Papoulis, Systems and Transforms withApplications in Optics. New York: McGraw-Hill,1968.

[131 CHi Keung Chan,"Analytical and Numerical Studies ofI - 7,V

BIBLIOGRAPHY

[1] N.H. Farhat, Frequency Synthesized ImagingApertures', Proceedings of the 1976 InternationalOptical Computing Conference, IEEE Cat. No. 76 CH1100-7c, 1976

[2] N.H. Farhat,'Principles of Broadband CoherentImaging', J. Opt. Soc of Am., Vol.67, pp.1015-1021, Aug. 1977.

(3] N.H. Farhat,'Microwave Holographic Imaging-Prospectsfor a Real time Camera', Proc. SPIE TechnicalSymposium, Vol. 180, Real-Time Signal Processing II,1979.

[41 N.H. Farhat and C.K. Chan, "Three DimensionalImaging by Wave Vector Diversity", Presented at 8Symposium on Acoustical Imaging; Key Biscayne, Fla.May 29-June 2 1978.

[5] Robert J. Collier, C.B. Burckhardt and L.H. Lin,Optical Holography. New York: Academic Press, 1971.

16] J.S. Hollis, T.J. Lyon and L. Clayton, MicrowaveAntenna Measurements. Atlanta, Georgia: ScientificAtlanta, 1970

[71 William B. Weir, LLoyd A. Robinson,DonParker,"Broadband Automated Radar Cross SectionMeasurements", IEEE Transactions on Antennas andPropagation. Vol. 12 pp. 780-784, Nov. 1974.

[81 Athanasios Papoulis, Signal Analysis. New York:McGraw-Hill, 1977.

[91 John B. Thomas, An Introduction to StatisticalCommunication Theory. New York: John Wiley & Sons,1969.

[101 Advisory Group for Aerospace Research andDevelopment:" Spread Spectrum Communications", U.S.Department of Commerce. AD-766-914, July 1973.

[i1] Henning F. Harmuth, Sequency Theory, Foundations andApplications. New York: Academic Press, 1977.

[121 Athanasios Papoulis, Systems and Transforms withApplications in Optics. New York: McGraw-Hill,1968.

[13] CHi Keung Chan,"Analytical and Numerical Studies of711

Frequency Swept Imaging", Phd. Thesis. Departmentof Electrical Engineering; University ofPennsylvania, 1978.

[14] W.H. Wells, "Acoustical Imaging with LinearTransducer Arrays", Acoustical Holography, Vol. 2,A.F. Melherall and L.Larmaore, (Eds.) New York:Plennum, 1970.

[15] G.T. Ruck, Radar Cross Section Handbook. New York:Plennum, 1970.

116] G. Mie, "Beitrage zur Optik truber Median speziellKolloidaler Metallosungen", Ann. Phys. 25:377 1908.

[17] J.W. Goodman, Introduction to Fourier Optics. New

York: McGraw-Hill, 1968.

7

APPENDIX I

f4

M- 0K 4

4'. z N N 11,d d~~

- ci 4 . IU 0 N ~z & L d dP

I- H N L -T1, .0. - 14U4- N &; N -O

e4 sr KO dN N

- -. 1- .

44z W 0 P" o.e w Pd N ( - 0

P4 IL I-

41W ) NK*4 0A - a 4 P * N)P0

F. 4i-' 1100 . 200 .0 . d LP)

Z8 cC -(

z A.

A go

*w f I I.f

C.1

2.n

-C W

40 La

C. o !4,.g5 ~ ~Vl if

1 J - -

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