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TARGET DRONE IMAGING WITH COHERENT RADAR

S

Imaging a BQM-74E Target Drone Using Coherent RadarCross Section Measurements

Allen J. Bric

ince the early 1980s and the advent of the modern computer, digital radarimaging has developed into a mature field. In this article, the specific problem ofimaging a rotating target with a stationary radar is reviewed and built upon. Therelative motion between the rotating target and the stationary radar can be used tocreate a circular synthetic aperture for imaging the target. Typically, an image isreconstructed by first reformatting the raw data onto a two-dimensional grid in thespatial frequency domain (commonly referred to as k-space) and then inverting into thespatial domain using a Fourier transform. This article focuses on a less popularreconstruction paradigm, tomographic processing, which can be used to coherentlyreconstruct the image incrementally as data become available. Both techniques sufferfrom sidelobe artifacts. It is shown that one-dimensional adaptive windowing can beapplied during the reconstruction phase to reduce sidelobe energy while preservingmainlobe resolution in the image plane. The algorithms are applied to real andsimulated radar cross section data.(Keywords: BQM-74E, Radar imaging, Spatially variant apodization, SVA, Syntheticaperture radar, Tomography.)

INTRODUCTIONUntil 1979, synthetic aperture radar (SAR) images

were formed using analog techniques, incorporatingoptical lenses and photographic film. In 1979, the firstreconstruction of a SAR image was formed on a digitalcomputer.1 Today, provided the data are available, SARimages can be formed on relatively inexpensive person-al computers, which have replaced expensive opticalprocessors and mainframe computers. This ability has

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 3 (1

led to an explosion of SAR processing techniques overthe past decade. Radar imaging is now an invaluableinstrument in many scientific fields.

A radar image is formed by illuminating a scene andresolving the resulting scattering distribution intorange and cross-range dimensions, where the rangecomponent measures a reflector’s radial distance fromthe radar. This procedure provides a two-dimensional

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map of the spatial scattering distribution. Two compo-nents are essential for providing this spatial resolution:frequency diversity and spatial diversity.

Frequency diversity provides range resolving capa-bilities and is achieved by using a wideband radar toilluminate the target scene. After the wideband returndata are processed, a range profile, or projection, canbe generated that characterizes the scattering distribu-tion of the scene along the range axis. This profile isa one-dimensional projection of the three-dimensionalspatial scattering distribution onto the range axis. Theprofile is resolved into range cells whose resolution isinversely proportional to the bandwidth of the transmitsignal. The energy returned from each cell is the inte-grated contribution of all scatterers inside the antennabeam in that particular range cell. Since the beamwidthof a reasonably sized real antenna at radio frequenciesgenerates a correspondingly large cross-range cell in thefar field, the fine resolution necessary to resolve smalltargets in the cross-range (azimuth) direction cannot beobtained directly.

To further resolve the target scene into the cross-range dimension, the radar system must have either alarge aperture or spatial diversity. This requirement canbe met by using a real array of antennas; at radio fre-quencies, however, a large and expensive array is nec-essary to generate reasonable resolution (i.e., 0.1–2.0 m). To solve this problem, the relative motionbetween the target scene and the radar can be used togenerate the necessary spatial aperture. This is the basisof SAR, which was discovered by Carl Wiley in 1951.1,2

This article concerns only SAR processing where therelative motion between the target and the radar formsa circular antenna aperture. If the target is held station-ary while the radar platform traces out the circularaperture, the processing is called spotlight-mode SAR(Fig. 1a). An equivalent result can be obtained if theradar can be held stationary while the target is rotatedto provide the relative motion (Fig 1b). Whenever theobject’s motion is used to generate the syntheticaperture, the processing is called inverse SAR (ISAR).

(a) (b)

Figure 1. Illustration of relative motion where (a) the radar pro-vides the motion, and (b) the target provides the motion forgenerating a circular aperture.

366 JOH

For the rest of this article, ISAR processing refers to thespecific case of a rotating object.

This article reviews the tomographic process and itsapplication to SAR imaging when the relative motionbetween the object and the radar generates a circularaperture. One-dimensional adaptive sidelobe cancella-tion techniques are introduced into the tomographicreconstruction process. Applying adaptive processingcan greatly reduce the sidelobe artifacts formed in thetomographic image without degrading spatial resolu-tion. The results are applied to a simulated data set andto actual radar cross section (RCS) measurementsperformed on a BQM-74E target drone.3

RADAR CROSS SECTIONMEASUREMENTS

RCS measurements were collected at the Naval AirWarfare Center Weapons Division’s Radar ReflectivityLaboratory (RRL) in Point Mugu, California, to char-acterize the scattering mechanisms on the BQM-74Etarget drone for the Mountain Top Project. Four fre-quency bands (UHF, C, X, and Ka) were used. Al-though both bistatic and monostatic measurementswere made, only the monostatic measurements areconsidered here. All of the monostatic RCS measure-ments made on the BQM-74E were collected in RRL’slarge anechoic chamber, which has a usable rangelength of 72 ft. To provide far-field measurements, aScientific-Atlanta compact range collimating reflectorwith a 16-ft quiet zone was used as a plane-wave trans-ducer. During the data collection, the drone was rotatedthrough 180° in 0.1° angular increments. At each step,it was illuminated with a swept linear-FM radar, and thereturn was sampled and recorded on an HP 9000 com-puter.3 Frequency sampling at X and C band provided15 m of unambiguous range extent.

The BQM-74E (Fig. 2a) is a reusable subsonicturbojet-powered target drone used primarily to simu-late a low-altitude antiship missile. It has a fuselagelength of 12.9 ft, a wing span of 5.8 ft, and a maximumfuselage diameter of 14 in. Since these drones are usedfor a variety of missions, optional payload kits can beinstalled to change their functional configurations. Forthe Mountain Top experiment, the drones weremodified to an LEC-305 configuration, which consistsof seven different antennas and a metal-taped, fiberglassnose cone that reduces RCS by shielding internalcomponents.3 The antenna placements are shown inFig. 2b.

TOMOGRAPHIC PROCESSINGThis section reviews the basic concepts necessary for

understanding the tomographic process and shows howthese concepts are applied to reflection tomography or,

NS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 3 (1997)


more specifically, to radar imaging.4,5 Tomography at-tempts to reconstruct an object (form an image) fromits one-dimensional projections. In X-ray tomography,the projections are density measurements, whereas inradar tomography, they are range profiles (see the boxedinsert, Range Processing). We start by examining thesimple case of inverting, or “imaging,” an object withrotational symmetry and then showing how the Fourierslice theorem is used to extend this process to non–rotationally symmetric objects. Next a simple nondif-fracting, parallel ray, coherent back-projection algo-rithm is considered for reconstructing the images. In allthe following calculations, it is assumed that the targetis illuminated in the antenna’s far field.

Rotational SymmetryTwo-dimensional rotationally symmetric functions

can be “imaged” by considering the Fourier–Abel–Hankel cycle (see the boxed insert for a review of thiscycle). Tomographic imaging is a natural extension ofthis cycle for non–rotationally symmetric functions.The immediate implication of this cycle is that theHankel and Abel transforms can be calculated withouthaving to compute Bessel functions and derivatives,respectively. Another implication is that a two-dimensional rotationally symmetric function can beinverted by computing the Abel transform (finding its

Figure 2. BQM-74E target drone (a) in flight, and (b) with antennaplacements used in the tests for the Mountain Top Project. Dashedlines indicate the position of antennas on the underside of thefuselage.

(b)

(a)


projection), and following with a one-dimensionalFourier transform, instead of requiring a two-dimensionalFourier transform. This result is pleasing but is notpractical in real imaging situations, because the objectto be imaged is rarely circularly symmetric. The resultcan be extended to nonsymmetric functions, however,through the use of the Fourier slice theorem.

RANGE PROCESSINGBefore examining any imaging techniques, it is useful to

review the process of generating a range profile. Rangeprocessing is a basic concept that can be explained in termsof propagation delay and Fourier analysis. A radar canmeasure the range to a target by determining the propaga-tion delay. If the radar is monostatic, the range R is simply

Rc= t

2, (A)

where t is the propagation delay, and c is the speed of light.If the transmitted signal has a finite bandwidth, the rangeresolution can be determined by considering a unit time–bandwidth product. The range resolution equation becomes

DRcB

=2

, (B)

where B is the transmitted bandwidth. It follows that therange resolution is inversely proportional to the transmittedbandwidth.

Generating a range profile requires that the scatteringmechanisms be properly sorted into range cells dictated bythe transmitted bandwidth. If the transmitted signal ismodeled as a sum of complex sinusoids subtending the band-width, the total return from one transmitted pulse will be

G f s t e dtj ft( ) ( ) ,–

= ⌠⌡ ∞

∞2p (C)

where s(t) is the scattering distribution represented as afunction of time t (range). It is clear from Eq. C that theradar return measures the Fourier transform of the scatteringdistribution over the defined bandpass region. An estimateof the scattering distribution may now be obtained by in-verting the Fourier transform:

ˆ( ) ( ) ( ) ,

– /

/

– – /

– /

–s t G f e df G f e df

f B

f B

j ft

f B

f B

j ft= ⌠⌡

+ ⌠⌡

+ +

0

0

0

0

2

2

2

2

2

2p p (D)

where ˆ( )s t is commonly referred to as the range profile andis the estimate of the scattering distribution along the rangeaxis over a finite bandwidth. Note that Eqs. A–D assumethat a large instantaneous bandwidth is transmitted. It iseasily shown that these equations also hold for a synthesizedbandwidth (e.g., a swept linear-FM system).6

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THE FOURIER–ABEL–HANKEL CYCLEThe zero-order Hankel transform is

a one-dimensional transform that canbe used to determine the two-dimen-sional Fourier transform of a rotation-ally symmetric function. This apparentsimplification results from the fact thata two-dimensional rotationally sym-metric function f(x, y), when ex-pressed in polar form, is dependent ononly a single parameter, namely, theradial distance r, where r x y= +2 2 ,and is independent of the spatial azi-muth angle u. Furthermore, the Fouri-er transform F(u, v) also exhibits rota-tional symmetry and is dependent onlyon the radial frequency q, whereq u v= +2 2 , and is independent ofthe frequency azimuth angle f. A straightforward derivationof the Hankel transform is given by Bracewell7 and is sum-marized as follows:

F u v f x y e dx dyc

j ux vy( , ) ( , ) ,– ( )= ⌠⌡

⌠⌡

+2p(I)

F u v F q f r e r dr dj q r( , ) ( ) ( ) ,0

– cos( – )= = ⌠⌡

⌠⌡

∞

0

22

pp uu f (II)

where f(r) and F(q) are the polar representations of therotationally symmetric functions f(x, y) and F(u, v), respec-tively, and c is the speed of light. After some manipulation,Eq. II can be expressed as

F q f r J qr r dr( ) ( ) ( ) ,= ⌠⌡

∞

2 20

0p p (III)

where F(q) is defined as the zero-order Hankel transform off(r), and J0 is the zero-order Bessel function.

The Abel transform is used to describe the one-dimensional “density” projection or line integral of a two-dimensional function. For rotationally symmetric functions,this line integral is identical in all directions; thus, it canbe defined on any axis.7 For convenience, it is defined onthe x axis as

f x f x y dyA( ) .–

= ⌠⌡

+( )∞

∞2 2 (IV)

Using Fig. A, and noting that dr/dy = sinu = r x r2 2– , analternative definition expressed in terms of r is given by

f xf r r

r xdrA

x( )

( )

–.= ⌠

⌡∞

22 2 (V)

The inverse Abel transform, derived by Bracewell,8 is pre-sented here as

f rf x

x rdxA( ) –

( )

–.= ′⌠

⌡∞

12 2p r

(VI)

Figure A. Geometry for the Abel transform.

Inverting the Abel transform numerically using Eq. VIrequires determining a derivative. Since numerical errors canoccur during this operation and are exacerbated by theintegration, a better method is to use the Fourier–Abel–Hankel cycle for inversion. This cycle is valid for circularsymmetric functions whose cross section is by definition aneven function. Given that the Fourier transform of an evenfunction is its own inverse (i.e. F = F–1), Bracewell7 provedthat

HFA = I AHF = I FAH = I

FHA–1 = I HA–1F = I A–1FH = I ,(VII)

where I is the identity transform, H is the zero-order Hankeltransform, and A is the Abel transform. Using the identitiesHH = I and FF = I with the preceding identities, the follow-ing equations are easily derived:

H = FA A = FH F = AH

H = A–1F A–1 = HF F = HA–1 .(VIII)

These equations form the basis of the Fourier–Abel–Hankelcycle shown in Fig B.

Figure B. Fourier–Abel–Hankel cycle (based on Bracewell,7

Fig. 14-4).

One-dimensionalFourier

One-dimensionalFourier

y

x

Abel AbelHankel

fA(x)

FA(kx)

kx

kxx

ky

y

drdy

r

x

u

u

x

fA(x)

368 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 3 (1997)


Fourier Slice TheoremThe Fourier slice theorem removes the rotational

symmetry requirement from the Fourier–Abel–Hankelcycle (and effectively removes the Hankel transform,since it is valid only for two-dimensional rotationallysymmetric functions). The modified cycle is given inFig. 3.

The Fourier slice theorem simply states that theFourier transform of an object’s projection (Abel trans-form) will yield a slice of the two-dimensional Fouriertransform oriented at the same angle as the projection.For completeness, this result is derived as follows:

Consider the rotated coordinate frame of Fig. 4,where x9 = x cos u + y sin u and y9 = y cos ␣u ␣ – ␣x␣ sin␣ udefine the projection onto the x9 axis as

p x )= s x , y dy .–

u( ( )′ ⌠⌡

′ ′ ′∞

∞

(1)

Taking the Fourier transform of the projection, we have

P f p x e dx

s x ,y e dx dy

s x,y e dx dy

j fx

j fx

–j f(x +y )

u up

p

p u u

( ) ( ) ,

( )

= ( ) ,

–

–

– –

–

– –

cos sin

= ⌠⌡

′ ′

= ⌠⌡

⌠⌡

′ ′ ′ ′

⌠⌡

⌠⌡

∞

∞′

∞

∞

∞

∞′

∞

∞

∞

∞

2

2

2

(2)

where f is frequency. Relating Eq. 2 to the two-dimen-sional Fourier transform, we have

P f S f S f fu u u u( ) ( , ) ( cos , sin ) .= = (3)

Thus, it has been shown that the Fourier transform ofan object’s projection is equivalent to a slice of its two-dimensional Fourier transform oriented at the projec-tion angle.

The Fourier slice theorem provides another meansfor calculating the Fourier transform of a two-dimensional scene. Unfortunately, it provides no com-putational advantage. If the projections are provided,however, then the two-dimensional Fourier transformcan be built rapidly. Typically, this direct method is notused because the Fourier transform of the projection isnot available, but in radar systems, the projection’sFourier transform is measured directly. Therefore, thismethod is computationally efficient in forming thetwo-dimensional Fourier transform of the scene. This


Figure 4. Geometry for derivation of the Fourier slice theorem(u = azimuth angle).

Figure 3. Cycle of transforms implied by the Fourier slice theorem(based on Bracewell,7 Fig. 14-5).

One-dimensional Fourier

One-dimensional Fourier

Two-dimensional Fourier

Abel Abel

y

x

x

sA(x)

SA(kx)

kx

kx

ky

is nice, but the real interest lies in imaging the objectscene, not its Fourier transform. The problem of imag-ing the scene now involves applying a two-dimensionalinverse Fourier transform. For an equivalent result, onecould project the two-dimensional Fourier transform atvarious angles and then take an inverse Fourier trans-form of each resulting projection to recover a slice ofthe object scene along the angle of projection; however,this method is ill-advised because it is so computation-ally inefficient.

u

y '

y

x '

x

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A. J. BRIC

Both of the aforementioned methods for invertingthe two-dimensional Fourier transform require that theentire frequency domain description be available. Sinceprojections are collected serially, it would be beneficialif one could perform the image reconstruction bycombining the projections directly to get the final image;thus, the image could be formed iteratively. This methodof image reconstruction is called back-projection.

Back-ProjectionBack-projection was devised as a technique to recon-

struct straight-ray tomographic images. The basic ideais to shadow each projection (properly oriented) overthe image scene and sum the shadows to form an image.This algorithm is easily derived by rewriting the Fourierslice theorem.5

The inverse two-dimensional Fourier transform inrectangular coordinates is given by

s x,y S u,v e dudv– –

j (ux+vy)( ) ( ) .= ⌠⌡

⌠⌡∞

∞

∞

∞2p (4)

Changing to polar coordinates with u = f cos u andv = f sin u, the transform can be rewritten as

s x,y S f e f df dj f(x +y )( ) ( , ) .cos sin= ⌠⌡

⌠⌡

∞

0

2

0

2p

p u uu u (5)

Expanding the preceding integral into two integralsfrom 0 to p and p to 2p gives

s x,y S f e f df d

S f e f df d

j f(x +y )

j f x y

( ) ( , )

( , ) .

cos sin

[ cos( ) sin( )]

= ⌠⌡

⌠⌡

+ ⌠⌡

⌠⌡

+

∞

∞+ + +

0 0

2

0 0

2

pp u u

pp u p u p

u u

u p u

(6)

Recall that cos(u + p) = –cos u and sin(u + p) = –sin uand S(–f, u) = S(f, u + 180). After substituting and sim-plifying, we have

s x,y S f e f df d–

j f x y( ) ( , ) .( cos sin )= ⌠⌡

⌠⌡ ∞

∞+

0

2p

p u uu u (7)

Substituting Eq. 3 in Eq. 7 and changing coordinateswith respect to Fig. 4 yield

370 JOH

s x,y P f e f df d–

j fx( ) ( ) .= ⌠⌡

⌠⌡ ∞

∞′

0

2p

up u (8)

Note that the inner integral in Eq. 8 is just the pro-jection pu(x9) filtered by a factor Z f Z. Define the filteredprojection as

q x P f f e df–

j fxu u

p( ) ( ) .′ = ⌠⌡ ∞

∞′2 (9)

Now, substituting Eq. 9 in Eq. 8 gives the back-projectionequation:

s x,y q x d

s x,y q x + y d

( ) ( ) ,

( ) ( cos sin ) .

= ⌠⌡

′

= ⌠⌡

0

0

p

u

p

u

u

u u u

(10)

From Eqs. 9 and 10 it is clear that the image isconstructed successively as each projection is acquired.This process enables the image to be constructed seri-ally, eliminating the need to wait for all projections tobecome available for processing to begin. At each step,the resulting image can be viewed, with successiveprojections increasing its fidelity. In some circumstancesthe intermediate images may provide valuable information(e.g., target identification or classification processors).

At a first glance, the high-pass filtering operationimplied by Eq. 9 may not seem unusual, but a closerinspection reveals that the square of the filter functionZ f Z is not integrable, which implies that its Fouriertransform does not exist. This integral can be approx-imated, as is shown by Kak4 and Bracewell.7 If onenumerically performs the integration implied by Eq. 9,the resulting image will have lost its DC componentand will be distorted because of periodic convolutionand interperiod interference.4 An exact solution for theintegral exists if the filter can be band-limited. Thiscondition can be asserted without loss of generalitybecause the projection is a real signal and is thereforeband-limited. For the band-limited case, the impulseresponse of the filter becomes

h x f e df

h xx

x

–B/

B/j fx( ) ,

( )sin( / )

/–

sin( / )/

,

′ = ⌠⌡

′ = ′′

′

2

22

2 2

21

2

2 22 2

1

4

22

p

t

p t

p t t

pt t

pt t

(11)



where B is the transmitted bandwidth and t = 1/B. Inthe preceding case, the impulse response is computedonly over a finite interval and is then inverted, causingits response to depart from the ideal response at lowerfrequencies. Since only a finite number of points in theimpulse response are considered, the frequency responsetails off to a positive number slightly above zero at DC.4

This offset maintains the actual DC level of the imageand also removes the distortion caused by the interpe-riod interference. When this correction is applied tothe projections, the reconstruction process is calledfiltered back-projection. It is important to point outthat this correction is unnecessary in the tomographiccomputation of circular aperture radar images, becausethe Fourier data typically exist only along an annularregion in the spatial frequency domain (k-space) andtherefore naturally act as a “high-pass” filter, similar toEq. 11.6,9,10

Reflective TomographyThe equations derived in the previous section de-

scribe a method used in straight-beam X-ray tomogra-phy where diffraction effects do not occur. (Other der-ivations exist, such as fan-beam tomography.) Since adensity projection is measured by an X-ray, it is assumedthat a rotation of 180° will produce an identical (butrotated) density projection. Note that this is not thecase when electromagnetic waves are used, as diffrac-tion and refraction effects can occur when an object isilluminated. In this case, to completely characterize theobject, a 360° rotation is required. It can be shown,however, that these straight-beam equations are a goodapproximation for reflective tomography if the objectis in the far field of the radiating source or, equivalently,if a plane-wave transducer is used.4 In this article, weuse the straight-beam equations as derived in the pre-vious section (without the filtering), along with a targetrotation of 180°, to reconstruct the images (data wereunavailable for a target rotation of 360°).

Sidelobe ReductionOne problem prevalent in both conventional SAR

and tomographic SAR imagery is range sidelobes.Wherever the target has a sizable RCS, energy can“bleed” into other range and cross-range cells, formingsidelobes. Sidelobes can greatly degrade image qualityand mask details in the computed image. Since theback-projection technique synthesizes an image con-currently as projections become available, it is desirableto window the projections directly, reducing their side-lobes, prior to back-projecting. Clearly, reducing thesidelobes in the range profile (projection) will reducethe sidelobes in the resulting image. This section focus-es on one-dimensional windowing techniques for re-ducing sidelobes in the range profile.


Hanning Window

A simple but effective technique for reducing side-lobes is to window the data with a raised cosine window.This window can be applied to the spatial frequencydata prior to zero padding and inversion. The resultingrange profile has greatly reduced sidelobes; the cost,however, is a reduction in spatial resolution. In somecases, it is unclear whether the reduced sidelobes areworth the trade-off of lower spatial resolution. Whena tomographic radar image is formed with Hanning-windowed range profiles, the resulting image shows anoticeable loss of resolution.

Spatially Variant Apodization

Spatially variant apodization (SVA) is a nonlinearwindowing technique that adaptively calculates theoptimal type of raised cosine window at every point inthe inverse domain. Apodization, a term borrowed fromoptics, refers to the reduction of sidelobe artifacts. Thistechnique, devised by Stankwitz et al.,10 provides con-siderable sidelobe reduction over traditional windowingwhile maintaining maximum resolution. The basics ofthis technique are easily understood by considering theeffect of multi-apodization.

Multi-apodization combines the results of multiple,uniquely windowed transforms on a data segment (in-cluding a rectangular window). After the transforms arecalculated, a “composite” result is formed choosing theminimum magnitude on a point-by-point basis over allof the transformed data sets. Since the transform of therectangular window provides the narrowest mainlobe,the resulting mainlobe for the composite is identical tothat of the rectangular window. Thus, the high-resolution properties of the rectangular window arepreserved while the sidelobes are reduced as dictated bythe properties of the windows chosen in the scheme.This technique is illustrated in Fig. 5.10

Multi-apodization, while simple to implement, canreduce sidelobes only to an extent dictated by thewindows in the scheme. Although adding windows tothis scheme can further reduce sidelobes, the compu-tational burden of doing so ultimately limits theamount of sidelobe reduction possible. If the windowtype is restricted to the family of raised cosine windows,then SVA can be used to provide an effective contin-uum of multi-apodizations based on a raised cosine ofa specific order K – 1. Consider the general form for thefamily of raised cosine windows given by

w tT

ktTK k

k

k=

K( ) (– ) cos ,=

∑11

2

0a

p(12)

where T is the time extent of the window, 0 < t < T(Nuttall11), and a is a weighting factor. From Eq. 12,

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A. J. BRIC

it is easily seen that a (K – 1)th-order raised cosinewindow will contain a constant term and K – 1 cosineterms. For a first-order raised cosine window, only thefirst two terms in Eq. 12 are considered, and afternormalization we have

w(n)=1 – 2a cos(2pn/N) , (13)

where 0 < n < N. If the minimum value of Eq. 13 isconstrained to be greater than zero, a is bound betweenzero and one-half. As a is continuously varied, an entirefamily of first-order raised cosine windows can be gen-erated. With this restricted family of raised cosine win-dows, the SVA algorithm is tasked with determining theoptimal a (window) for each point in the transformeddomain. This is now equivalent to a continuum of multi-apodizations (for the family of first-order raised cosinewindows), resulting in an adaptive window that optimallyreduces sidelobes while maximizing mainlobe resolution.To determine a, we minimize the square magnitude inthe inverse domain of the windowed data. Stankwitz etal.10 give the derivation of the optimal a for a first-orderraised cosine window, and the result is given as

a =+ +

Re( )

[ ( – ) ( )],

g mg m g m1 1 (14)

where 0 < a < 0.5 and g(*) is the transformed data. Theoutput can now be defined in terms of a three-pointconvolution between the transform of Eq. 13 and g(*):

f(m) = g(m) + a[g(m – 1) + g(m + 1)] . (15)

2. If the windowthe compositevalue is a min

CDA uses the adcomponents to dthe Hanning weithat will null tallows an entire dows to be synthe

The generalizsidelobe roll-off pthe expansion. Fothis corresponds is added in the ef 5, but at the constraining the cosine window fhave

w(n) = 1 – a c

where 0 < n < Nthe optimal winmain. Iannuzzell

a =

Re[ ( – )

21

gg m

where 0 < a < 4in terms of a con16 and g(*):

0

–50

–100

–150

0

–50

–100

–150

Mag

nitu

de 2

0 lo

g (

)

0 200 400 600 800 1000Frequency sample number

0

–50

–100

–150

(a)

(c)

(b)

Figure 5. Results of Fourier transform on sinusoid with (a) rectangular window,(b) Hanning window, and (c) dual apodization using rectangular and Hanning windows.

372 JOH

Stankwitz et al.10 explored nu-merous apodization techniques, in-cluding complex dual apodization(CDA), joint in-phase and quadra-ture SVA (described in Eqs. 14 and15), and separate in-phase andquadrature SVA. CDA is a dualapodization technique that usescomplex information in the data toimprove the sidelobe behavior. Itinvolves transforming data win-dowed with Hanning (a = 0.5) andrectangular (a = 0) windows. Con-sidering each in-phase and quadra-ture component independently, acomposite result is formed on apoint-by-point basis in the follow-ing manner:

1. If the windowed components areof the opposite sign, set the compos-ite value to zero.

ed components are of the same sign, set value to the number whose absoluteimum.ditional information in the complex

etermine if there is a window betweenghting and the rectangular weighting

he composite value. This procedurefamily of first-order raised cosine win-sized from the output of two windows.

ed raised cosine window will have aroportional to the number of terms inr the first-order raised cosine window,

to a roll-off of f 3. If an additional termxpansion, the roll-off will increase toexpense of added complexity. Aftergeneral form of a second-order raisedor continuity at the boundaries, we

os(2pn/N) + (a–1)cos(4pn/N) , (16)

. As before, a can be adapted to yielddow at each point in the inverse do-i12 shows that a is given by

+ ++ + + +

( ) – [ ( – ) ( )]( )] – [ ( – ) ( )]

,2 2

1 2 2m g m g mg m g m g m

(17)

/3. The output can again be definedvolution between the transform of Eq.



f m g m g m g m

g m g m

( ) ( ) – ( – ) ( )

–( – ) ( ) .

= + +[ ]+ + +[ ]

a

a

21 1

12

2 2(18)

COHERENT TOMOGRAPHICALGORITHMS

Generating accurate ISAR images using tomograph-ic algorithms requires careful implementation. Twopotential pitfalls include insufficient interpolation ofthe range profile (projection) and inaccurate represen-tation of the raw frequency data. Either dramaticallyhinders image formation.

The back-projection algorithm presented previouslydid not account for discrete data samples. It assumesthat a continuous range profile is available. Since thisis not the case, the data must be interpolated in asufficient amount to minimize errors when projectingonto a discrete image grid. For the images presented inthis article, all of the range profiles were oversampled10 times relative to the grid dimensions. The rangeprofile was interpolated by zero-padding the frequencydata prior to inversion. When back-projecting, thenearest point in the profile was selected as the valuefor the grid point. This method proved to be compu-tationally more efficient than linearly interpolating therange profile during the back-projection process foreach individual point.

Coherent back-projection requires the phase of eachrange profile to be properly represented. Therefore, theraw frequency data must be translated relative to theplace in the spectrum where the data were acquired.This translation amounts to a multiplication in thespatial domain with a complex sinusoid that affectsonly the phase of the range profile. This operation issimilar to the focusing correction in conventionalISAR processing. Care must be taken in the applicationof the phase correction, as it can be aliased during theback-projection process if the range profile was insuf-ficiently interpolated.

With the preceding discussion in mind, two slightlydifferent tomographic algorithms were designed to re-construct the raw ISAR data. Both algorithms share acommon back-projection implementation but differ inthe application of the adaptive spectral window. The“radial” algorithm applies the adaptive window directlyto the raw data prior to interpolation, whereas the“interpolative” algorithm applies it afterwards. Withthe radial algorithm, the adaptive window parameterand convolution are calculated only on real data points,making the radial algorithm computationally more ef-ficient. The three-point SVA formulation described byEq. 15 was used with both the radial and interpolative


algorithms, and the five-point formulation described byEq. 18 was used only in the interpolative algorithm.These algorithms were compared, and results are givenin the next section.

Both the radial and the interpolative algorithms relyon the application of an SVA window to each projec-tion prior to back-projecting. Since both algorithmsreduce sidelobes in the projection to a degree greaterthan or equivalent to the reduction of the Hanningwindow technique, it seems safe to assume that theresulting image will also display the same character.This would certainly be true if the projections were real(as in X-ray tomography); however, this logic breaksdown when one considers a complex projection. Thecomplex projection requires a back-projection algo-rithm to perform complex integrations to form theimage. Since the formation is linear, the phase of theprojection becomes important. SVA, being a nonlineartechnique, affects the phase of each projection. Whenthe image is built using these projections, the sidelobelevel is not necessarily lower than that of the Hanningwindow everywhere, because no constraint was put onthe phase at each point in the projection. Experimen-tally, it has been shown that sidelobe reduction is hin-dered wherever multiple sidelobes interfere. In one ofthe results, an apodization was performed on the five-point interpolative window with the Hanning window.A more formal solution has been devised by the authorand is currently under investigation.

RESULTS OF ALGORITHMEVALUATIONS

Three data sets were used to evaluate the coherenttomographic algorithms discussed in the previous sec-tion. One data set consisted of eight simulated pointreflectors arranged in an “aircraft” configuration. Thesedata were used to debug and evaluate the algorithms.The other two data sets contained C-band and X-bandRCS data collected on the BQM-74E target drone atRRL. The C-band data are of lower resolution than theX-band data since the measurement bandwidths were1 and 4 GHz, respectively. For comparison with theradial and interpolative algorithms, coherent tomo-graphic images were also computed using rectangularand Hanning windows. The images resulting from ap-plying these various coherent tomographic algorithmsto the three data sets are shown in Figs. 6 to 8.

In all of the data sets, it is apparent that the sidelobesof the rectangular window are contaminating the image(Figs. 6a, 7a, and 8a). The application of a Hanningwindow to the data prior to back-projection reduces thesidelobes; however, the resolution loss is quite apparent(Figs. 6b, 7b, 8b). In the target drone images, thisresolution loss produces blurring in the various images,which is very apparent in the target drone reconstructions.

997) 373

A. J. BRIC

–70 –60 –50 –20–40 0–30 –10

(a) (b)

(c) (d)

(e) (f)

Figure 6. Normalized RCS images formedfrom point simulation data using coherentback-projection. (a) Rectangular window,(b) Hanning window, (c) radial three-pointSVA, (d) interpolative three-point SVA, (e)radial five-point SVA, and (f) interpolativefive-point SVA. Each image is scaled torepresent a 4 3 6 m area.

Figure 7. Normalized RCS images formedfrom 1-GHz C-band BQM-74E RCS datausing coherent back-projection. (a) Rect-angular window, (b) Hanning Window, (c)radial three-point SVA, (d) interpolativethree-point SVA, and (e) interpolative five-point SVA. Each image is scaled to repre-sent a 4 3 8 m area.

–100 –80 –60 –20–40 0

(a) (b)

(e)

(c) (d)

374 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 3 (1997)

DRONE IMAGING WITH COHERENT RADAR
TARGET
Figure 8. Normalized RCS images formed from 4-GHz X-band BQM-74E RCS data usingcoherent back-projection. (a) Rectangular window, (b) Hanning Window, (c) interpolativethree-point SVA, (d) interpolative five-point SVA, and (e) composite (Hanning andinterpolative five-point SVA). Each image is scaled to represent a 4 3 8 m area.

–100 –80 –60 –20–40 0

(a) (b)

(e)

(c) (d)

Concentrating on the simulated data set (Fig. 6), wesee that the images resulting from the application of theradial and interpolative algorithms constrain the widthof the mainlobe of each point source to be equivalentto that of the rectangular window. Thus, the maximumpossible resolution is retained. It is apparent, however,that the radial algorithms are limited in their ability toreduce sidelobes in the reconstructed images (Figs. 6cand 6e). When the interpolative algorithms are appliedto the process, the sidelobe levels are suppressed to amuch greater degree, with the five-point algorithm(Fig. 6f) performing better than the three-point algo-rithm (Fig. 6d).

Now turning to the BQM-74E target drone RCSdata collected at RRL, consider the C-band imagesgenerated with a bandwidth of 1 GHz (Fig. 7). Again,heavy sidelobe artifacts are apparent in the reconstruc-tion with the rectangular window (Fig. 7a), and againthe application of a Hanning window prior to back-projection decreases resolution in the reconstructedimage (Fig. 7b); however, there is good sidelobe rejec-tion. When the radial three-point window is applied tothe data (Fig. 7c), the sidelobe reduction is dubious atbest. This can be contrasted with the three-point andfive-point interpolative algorithms (Figs. 7d and 7e),which provide good sidelobe rejection without tradingspatial resolution. In the C-band reconstructions ofFig. 7 it is unclear why the upper half of the drone bodyshows a significantly smaller RCS than the lower half.This result is not an artifact of the reconstruction

horizontal and vertparachute dome oncomposite image shforming a point-by-and five-point intescribed. This technition; the nonlinear are further suppress

CONCLUSIONWhen the relati

radar and a targetaperture can be formapplied to form animage can be formeavailable. In discrimtions, a partial imagcation; thus, tomoresponse time in apOne particular applwarheads and debri

In this article, imulti-apodization trange profiles priorlobes in the two-dication of the one-dithe ordinal directionimage “shares” in point formulation o

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 3 (

process; whether it is a propagationphenomenon or an artifact of themeasurement process is unknown.

The RCS imbalance in theC-band reconstructions of Fig. 7 isnot evident in the X-band imagesof Fig. 8. Since the X-band imageswere acquired at a bandwidth of4 GHz, the resolution is 4 timesgreater than that of the C-bandimages. In these images, sidelobeartifacts are again evident in therectangular windowed data inFig. 8a, and the application of aHanning window shows loss of res-olution as seen in Fig. 8b. Applyingthe three- and five-point interpola-tive algorithms to these data yieldsFigs. 8c and 8d, respectively. Thefive-point formula is particularlywell suited to the high-resolutionX-band image, where small detailsare much more apparent. Featuresthat can be seen easily includeantennas on the front body of thedrone, air intake on the bottom,ical stabilizers, engine nacelle, and the back of the drone. Finally, theown in Fig. 8e is the result of per-

point minimization on the Hanningrpolative images as previously de-que provides greater sidelobe reduc-artifacts due to the SVA algorithmed.

ve motion of a coherent wideband is such that a circular synthetic

ed, tomographic processing can be image. With this processing, thed incrementally as the data become

ination or identification applica-e might be sufficient for a classifi-

graphic processing could improveplications where images are used.ication might be to image rotatings from missile intercepts.t was shown that one-dimensionalechniques can be applied to the to back-projection to reduce side-mensional image. With this appli-mensional windows, dependence ins is removed and each incremental

the sidelobe reduction. The five-f the SVA algorithm was shown to

1997) 375

A. J. BRIC

,

,

,

.

.

-,

atr

perform best in reducing the sidelobe artifacts in thereconstructed image. Particularly since a native two-dimensional implementation of the five-point formula-tion has not been devised, this technique offersa potentially powerful tool in image reconstruction.

REFERENCES1Carrara, W. G., Goodman, R. S., and Majewski, R. M., Spotlight Synthetic

Aperture Radar Signal Processing Algorithms, Artech House, Boston (1995).2Wiley, C. A., “Synthetic Aperture Radars,” IEEE Trans. Aerospace Electron.

Sys. AES-21(3), 440–443 (May 1985).3Radar Reflectivity Laboratory Report, Radar Cross-Section Measurements of the

BQM-74E Drone Target for the Mountain Top Project, Code 4kM500E, NavalSurface Warfare Center, Point Mugu, CA (Sep 1995).

4Kak, A. C., Principles of Computerized Tomographic Imaging, IEEE Press, NewYork (1987).

5Munson, D. C., Jr., O’Brien, J. D., and Jenkins, W. K., “A TomographicFormulation of Spotlight-Mode Synthetic Aperture Radar,” Proc. IEEE71(8), 917–926 (Aug 1983).

376 JOH

6Mensa, D. L., High Resolution Radar Cross-Section Imaging, Artech HouseBoston (1991).

7Bracewell, R. N., Two Dimensional Imaging, Prentice Hall, Englewood CliffsNJ (1995).

8Bracewell, R. N., The Fourier Transform and Its Applications, 2nd Ed.McGraw Hill, New York (1986).

9Walker, J. L., “Range-Doppler Imaging of Rotating Objects,” IEEE TransAerospace Electron. Sys. AES-16(1), 23–52 (Jan 1980).

10Stankwitz, H. C., Dallaire, R. J., and Fienup, J. R., “Nonlinear Apodizationfor Sidelobe Control in SAR Imagery,” IEEE Trans. Aerospace Electron. SysAES-31(1), 267–279 (Jan 1995).

11Nuttall, A. H., “Some Windows with Very Good Sidelobe Behavior,” IEEETrans. Acoustics, Speech, Signal Process. ASSP-29(1), 84–91 (Feb 1981).

12Iannuzzelli, R. J., “Adaptive Windowing of FFT Data for Increased Resolution and Sidelobe Rejection,” Proc. IASTED Int. Conf. Signal Image Process.pp. 246–252 (Nov 1996).

ACKNOWLEDGMENTS: I would like to thank the technicians and staff at RRLin Point Mugu, California, responsible for providing the excellent quality RCS datof the BQM-74E target drone, without which accurate reconstructions would nobe possible. I also thank Alexander S. Hughes and Russell J. Iannuzzelli for theiguidance and help in certain phases of this work.

THE AUTHOR

ALLEN J. BRIC received B.S. and M.S. degrees in electrical engineering fromPurdue University in 1989 and 1991, respectively. During graduate studies, heworked as a research assistant at Argonne National Laboratories, gainingexperience in computed tomography. He joined APL’s Fleet Systems Departmentin 1992 and is currently a member of the Associate Professional Staff in the AirDefense Systems Department. His research interests include image processing,radar and time series analysis, information theory, nonlinear filtering, andfractals. His e-mail address is [email protected].


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