Ultra-Wideband Coherent Processing€™s wideband imaging radars perform ... the resolution of...

• CUOMO, PIOU, AND MAYHANUltra-Wideband Coherent Processing

VOLUME 10, NUMBER 2, 1997 THE LINCOLN LABORATORY JOURNAL 203

L played a major rolein developing wideband radar systems. Thisdevelopment was motivated by the successful

application of high-power instrumentation radars toresearch in ballistic missile defense and satellite sur-veillance. Today’s wideband imaging radars performreal-time discrimination and target identification.Advanced signal processing methods have improvedthe resolution of processed radar return signals, fur-ther improving wideband-radar technology.

Figure 1 illustrates a ballistic missile defense envi-ronment that relies on accurate target identificationand size-shape estimation, two capabilities critical tomany areas of national defense. The primary goal of adefensive radar system is to intercept and destroy athreat target. This objective is complicated by thepresence of many objects in the radar field of view,some purposefully designed to deceive radar discrimi-nation algorithms. Decoys, for example, may have ra-dar cross section (RCS) levels similar to those of thewarhead, which makes robust target selection basedsolely on RCS levels difficult. Narrowband radars

Ultra-Wideband CoherentProcessingKevin M. Cuomo, Jean E. Piou, and Joseph T. Mayhan

■ Lincoln Laboratory has developed an approach for estimating the ultra-wideband radar signature of a target by using sparse-subband measurements.First, we determine the parameters of an appropriate signal model that best fitsthe measured data. Next, the fitted signal model is used to interpolate betweenand extrapolate outside the measurement subbands. Standard pulse-compressionmethods are then applied to provide superresolved range profiles of the target.A superresolution algorithm automatically compensates for lack of mutualcoherence between the radar subbands, providing the potential for ultra-wideband processing of real-world radar data collected by separate widebandradars. Because the processing preserves the phase distribution across themeasured and estimated subbands, extended coherent processing can be appliedto the ultra-wideband compressed radar pulses to generate superresolved radarimages of the target. Applications of this approach to static test range and fielddata show promising results.

usually lack sufficient range resolution to allow a di-rect measurement of target length, although they aregenerally useful for tracking and coarse motion esti-mation. Unlike narrowband radars, wideband radarsallow a much larger suite of target discrimination al-gorithms to be employed for real-time range-Dopplerimaging and phase-derived range estimation.

Figure 2 illustrates a typical narrowband-radar andwideband-radar target response. The narrowband re-sponse indicates the position of the target as a wholewith the peak RCS corresponding to the electromag-netic size of the target. The wideband response pro-vides resolution within the target’s range profile. Indi-vidual scattering centers are isolated into smallrange-resolution cells that provide a more direct mea-surement of the target’s size and shape.

To achieve fine range resolution, wideband fieldradars utilize coded waveforms with large time-band-width products. Wideband chirp waveforms are com-monly used because of their ease of generation andprocessing in the radar receiver. Mixing the radar re-turn signals with a replica of the transmitted signal


204 THE LINCOLN LABORATORY JOURNAL VOLUME 10, NUMBER 2, 1997

FIGURE 1. A typical ballistic missile defense environment, which demands accurate target identification and size-shape estima-tion. Shortly after launch, the warhead and decoy separate from the main body of the missile. Radar discrimination algorithmsattempt to find the threat target by exploiting differences in size, shape, and motion dynamics between the warhead and non-threatening objects in the radar’s field of view.

FIGURE 2. Comparison of target response—radar cross section (RCS) levels versus relative range—for a typicalnarrowband radar (left) and a wideband radar (right). The narrowband response can identify only the position of the tar-get as a whole. The wideband response provides a direct measurement of individual scatterers within the target’slength, permitting a much larger suite of target discrimination algorithms to be employed than with narrowband radars.

Atmosphere

Debris

Decoy wobbleWarhead

Tank tumble

Launch

Reentryslowdown

Heavy, fast warhead

Light, slow decoy

Impact

Warhead spinDecoy

–40

–30

–20

–10

RC

S (d

Bsm

)

Length

Relative range (m)

0 5 10

Relative range (m)

0 25 50



produces a baseband signal with frequency compo-nents that are proportional to the relative range be-tween scattering centers on the target. The basebandsignal is sampled and Fourier-transformed to providea range-resolved profile of the target. This process iscalled pulse compression. Properties of the com-pressed pulse, such as resolution and sidelobe levels,depend on the extent and shape of the window func-tion applied to the baseband signal samples. The Fou-rier-theory relations define resolution to be inverselyproportional to the total signal bandwidth. In accor-dance with this inverse relationship, the resolution ofthe radar improves as radar bandwidth increases.

Many wideband field radars operate on these basicprinciples. Figure 3 shows an aerial view of theKiernan Reentry Measurement System (KREMS) fa-cility located on Kwajalein Atoll in the central PacificOcean. This facility has been the most sophisticatedand important wideband-radar research center in theUnited States for over thirty years [1]. The photo-

graph depicts several wideband field radars, includingthe ALCOR C-band radar developed in 1970 forthe purpose of wideband discrimination research.ALCOR utilizes a wideband chirp waveform with abandwidth of 512 MHz to achieve a range-resolutioncapability of about 53 cm. Kwajalein’s millimeter-wave radar can operate at the Ka-band and W-band,and is capable of a transmission bandwidth of 2000MHz, providing an impressive 14-cm range-resolu-tion capability. The United States also operates high-resolution wideband radars on ship platforms, such asCOBRA JUDY. Figure 4 shows the COBRA JUDYS-band phased-array radar and the X-band dish-antenna radar.

Although the field radars mentioned above providea high degree of range resolution, important targetfeatures are often much smaller than conventionallyprocessed range-resolution cells. To improve therange resolution of a radar, we can increase the radarbandwidth or process the received signals with

FIGURE 3. The Kiernan Reentry Measurement System (KREMS) facility located on KwajaleinAtoll in the central Pacific Ocean. The ALCOR C-band radar is located under the white ra-dome in the lower left of the photograph. The millimeter-wave radar is located under thesmaller white radome near the center.



superresolution algorithms. Cost and design limita-tions are major drawbacks to increasing radar band-width. Because we want to obtain higher-resolutionradar data without incurring significant hardwarecosts, we researched robust superresolution algo-rithms that can be applied to a wide range of real-world data sets.

In 1990, Lincoln Laboratory developed a superres-olution algorithm that can significantly improve therange resolution of processed radar return signals.The algorithm, called bandwidth extrapolation(BWE) [2, 3], increases the effective bandwidth of aradar waveform by predicting the target’s response atfrequencies that lie outside the measurement bands.For radar applications, BWE typically improves therange resolution of compressed radar pulses by a fac-tor of two to three. BWE often provides striking im-provements in the quality of wideband-radar images.As an example, Figure 5(a) shows a radar image of asimulated three-point target without BWE processingapplied. The resolution is insufficient to resolve thetarget points. Figure 5(b) shows the same target withBWE processing applied to the compressed radarpulses, first in the range dimension and then in cross-range. The BWE processed image is better resolved,allowing us to analyze and identify the target.

Although BWE improves resolution, the approach

has the following inherent limitations. The algorithmis based on signal processing models that characterizea complex target as a collection of point scatterers,each having a frequency-independent scattering am-plitude. BWE algorithms are often sufficient for typi-cal wideband signal processing in which the wave-forms have a small fractional bandwidth comparedwith the center frequency. Over ultrawide frequencybands in which the radar bandwidth is comparable tothe radar center frequency, however, the scatteringamplitude of the individual scattering centers canvary significantly with frequency. Spheres, edges, andsurface joins are examples of realistic scattering cen-ters that exhibit significant amplitude variations as afunction of frequency. Ultra-wideband (UWB) signalmodels must be flexible enough to accurately charac-terize these non-pointlike scattering centers.

The ability to measure or estimate a target’s UWBradar signature is useful for many radar-discrimina-tion and target-identification applications. Not onlyis fine range resolution obtained, but the amplitudevariations of isolated scattering centers are useful foridentifying the type of scattering center. Many ca-nonical scattering centers are known to exhibit f α-type scattering behavior; e.g., the RCS of flat plates,singly curved surfaces (cone sections), and doublycurved surfaces (sphere) vary as f 2, f 1, and f 0, re-

FIGURE 4. The COBRA JUDY ship with a clear view of the S-band phased-array andX-band dish-antenna radars.

X-band

S-band



spectively. The RCS of a curved edge varies as f –1,whereas a cone vertex varies as f –2. One goal of UWBprocessing is to detect these frequency-dependentterms in the measured data and to exploit them forscattering-type identification.

Building a field model of a true UWB radar can beexpensive. A more practical approach is to use con-ventional wideband radars to sample the target’s re-sponse over a set of widely spaced subbands, as illus-trated in Figure 6. In this figure, the COBRA JUDYS-band and X-band radars are used to collect coher-ent target measurements over their respective widelyspaced subbands. Coherently processing these sub-bands together makes it possible in principle to accu-rately estimate a target’s UWB radar signature. This

concept increases processing bandwidth and im-proves range-resolution and target-characterizationcapabilities.

To perform UWB processing, as illustrated in Fig-ure 6, we must address a number of technical issues.First, we need to develop a robust signal processingmethod that compensates for the potential lack ofmutual coherence between the various radar sub-bands. We must then fit an appropriate UWB signalmodel to the sparse-subband measurements. The fit-ted signal model must accurately characterize UWBtarget scattering and provide for meaningful interpo-lations or extrapolations outside the measurementsubbands. In this article, we discuss our approach toUWB coherent processing, and then apply our UWBcoherent processing algorithms to static-range data.We summarize the main results of this work and sug-gest some research strategies for the future.

UWB Coherent Processing

Figure 7 illustrates an overview of our approach toUWB coherent processing. An estimate of the target’sUWB radar signature is obtained by coherently com-bining sparse-subband measurements. While the fig-ure illustrates UWB processing for only two sub-

FIGURE 5. Demonstration of bandwidth extrapolation (BWE) processing. (a) The three-point target image without BWE pro-cessing is not resolved well enough to identify the target. (b) The three-point target image with BWE processing allows us toidentify and analyze the target.

FIGURE 6. Ultra-wideband (UWB) processing concept ap-plied to COBRA JUDY S-band and X-band wideband signa-ture data. S-band and X-band measurements are coherentlyprocessed together to provide an interpolated estimate of atarget’s UWB radar signature.

(a) (b)

C-bandS-band X-band

Frequency

(Interpolated)



bands, it is straightforward to apply this concept to anarbitrary number of subbands.

The illustrated process is divided into three steps:1. Process multiband data samples from the in-

phase (I) and quadrature (Q) channels to makethe radar subbands mutually coherent.

2. Optimally fit an UWB all-pole signal model tothe mutually coherent subbands. The fittedmodel is used to interpolate between and ex-trapolate outside the measurement bands.

3. Apply standard pulse-compression methods tothe enlarged band of spectral data to provide asuperresolved range profile of the target.Step 1 is important when applying UWB process-

ing to field data collected by separate wideband ra-dars. Time delays and phase differences between theradars can make them mutually incoherent. To coherethe subbands, we fit an all-pole signal model to thespectral data samples in each subband and adjust themodels until they optimally match. Correspondingcorrections are then applied to the underlying datasamples. This approach is based on the assumptionthat the target can be accurately characterized by asuperposition of discrete scattering elements. This as-sumption is often valid for targets that are large withrespect to radar wavelength [4–6].

In step 2, we fit a global UWB all-pole signalmodel to the mutually coherent subbands. We thenuse the model for interpolation and extrapolationpurposes. All-pole models are well suited for UWBprocessing because they accurately characterize thetarget by a superposition of discrete scattering cen-ters, each with its own frequency-dependent term.While all-pole models match best to signals that growor decay exponentially with frequency, they can alsoaccurately characterize f α scattering behavior overfinite bandwidth intervals.

In step 3, standard Fourier-based pulse-compres-sion methods are used to generate a range-resolvedprofile of the target. Because the UWB process is fullycoherent, superresolved radar images can also be gen-erated by using standard techniques.

Mutual-Coherence Processing

UWB processing requires a consistent set of spectralsignals in each subband; i.e., the all-pole models for

FIGURE 7. UWB process flow to estimate the target’s UWBradar signature. Sparse multiband data samples for the in-phase (I) and quadrature (Q) channels are selected. Mutual-coherence processing allows two or more independentradar subbands to be used in the model fitting step thatfollows. An all-pole signal model is fitted to the sparse-subband data samples and used for interpolation and ex-trapolation outside the measurement bands. Standardpulse-compression methods are then applied to the UWBtarget data.

Mutually coherent subbands

Mutualcoherence

Model fittingand

parameterestimation

UWB all-polesignal model

Multiband data samples (I and Q channels)

Frequency

Frequency

Frequency

Model fit interpolatesmissing data

Range

Target

Other algorithmssuch as imaging

Standardpulse

compression

UWB compressedpulse



each subband must be consistent. This requirement isnot an issue in multiband radar systems specificallydesigned to be mutually coherent. Mutual-coherenceproblems will most likely occur, however, when thesubband measurements are collected by wideband ra-dars operating independently. This section discusses astraightforward signal processing approach that cancompensate for the lack of mutual coherence betweenany number of radar subbands. The technique allowsus to apply UWB processing across a wider range ofradar platforms used in the field.

For illustration purposes, we simulate the radar re-turns for a hypothetical target consisting of two dis-crete scattering centers. The scattering center closer tothe radar has a scattering amplitude that decays withfrequency, whereas the scattering center away fromthe radar has a scattering amplitude that grows withfrequency. The simulated spectral signal samples sn aregiven by

sf

fe

f

fen

n i n n i n=

+

−

41

14

1

13π π/ / .

The frequency-sampled phase terms correspond to ascattering-center separation of 15 cm. White Gauss-ian noise is added to each signal sample and the sig-nal-to-noise ratio is 20 dB.

We assume that only two subbands are availablefor coherent processing of the noisy sn signal samplesillustrated in Figure 8(a). The sn signal samples in thelower subband have been modulated by the functione i n− π / 9 to simulate the effects of mutual incoherence;i.e., the signal poles for the lower subband have beenrotated 20° clockwise relative to the upper-subbandsignal poles. Figure 8(b) shows the correspondingcompressed pulses, which do not line up because thesubbands are not mutually coherent. In effect, mutualcoherence is seen as a consequence of uncertainty in

FIGURE 8. (a) Sparse multiband measurements of a target consisting of two closely spaced scat-tering centers. The amplitude of one scatterer (blue) decays with frequency, while the amplitude ofthe other scatterer (red) grows with frequency. The two subbands illustrated are not mutually co-herent. For clarity, signals from only the I channel are shown. (b) The corresponding compressedpulses do not line up in range because the subbands are mutually incoherent.

10

–10

0

–30

10

0

–10

–20

Relative range (m)(b)

RC

S (d

Bsm

)I c

hann

el (m

)

Frequency (GHz)(a)

Lower subbandUpper subband

–1 0 1 2

Sample number0

–2

Compressed pulses

3 4 5 6 7 8 119 10 12

N1 – 1 N – N2 N – 1

Vacant band

Measurement band Measurement band

5

–5



position and time sequencing of the separate radars.We begin the mutual-cohering process by model-

ing the spectral signals in each subband with a super-position of complex exponential functions. An all-pole signal model of the form

M f a pn kk

P

kn( ) =

=∑

1

is used for this purpose. As illustrated in Figure 8(a),the lower subband contains N1 data samples, whilethe upper subband contains N2 data samples. Thusthe sample index n ranges from n = 0 , … , N1 – 1 forthe lower subband and from n = N – N2 , … , N – 1for the upper subband. The all-pole model param-eters are physically meaningful. The number of scat-tering centers and their complex amplitudes are de-noted by P and ak , respectively. The poles pkcharacterize the relative ranges and frequency decayof the individual scattering centers; the f α frequencydecay model indicated earlier is approximated by anexponential variation over the band of interest. Thesubbands can be mutually cohered by fitting a sepa-rate all-pole model to each subband and adjusting themodels until they are consistent.

Our approach to all-pole modeling utilizes the sin-gular-value decomposition of the forward-predictionmatrix. Specifically, the forward-prediction matrix forthe lower subband is given by

H1

0 1 1

1 2

1 1 1 1 1

=

−

− − + −

s s s

s s s

s s s

L

L

N L N L N

L

L

M M M M

L

,

where L denotes the correlation window length andthe sn denote the frequency-domain radar measure-ments. The special form of matrix H1 is called aHankel matrix, which is associated with the transientresponse of a linear-time-invariant system. Subspacedecomposition methods exploit the eigenstructure ofHankel matrices to estimate the parameters of linear-time-invariant signal models [7]. Using a correlationwindow length L = N1/ 3 generally provides for robustparameter estimates. Larger values of L can provide

better resolution, but the estimates may not be as ro-bust to noise. The forward-prediction matrix H2 forthe upper subband, constructed in a similar way, is

H2

2 2 1 2 1

2 1 2 2 2

1 1

=

− − + − + −

− + − + − +

− − + −

s s s

s s s

s s s

N N N N N N L

N N N N N N L

N L N L N

L

L

M M M M

L

.

To estimate the all-pole model parameters for thelower and upper subbands, we apply the singular-value decomposition to H1 and H2, respectively,which decomposes H1 and H2 into the product ofthree matrices:

H U S V1 1 1 1= ′

and

H U S V2 2 2 2= ′ ,

where the prime symbol denotes the Hermitian op-erator. The S matrices contain the singular values forthe two subbands. The U and V matrices contain thecorresponding eigenvectors. In particular, the col-umns of the V matrices correspond to the eigenvec-tors of the respective subband covariance matrices. Bydecomposing H1 and H2 in this way, we can estimatethe all-pole model parameters for each subband withthe following four-step process:1. The singular-value matrices S1 and S2 are used

to estimate the model orders P1 and P2 for thetwo subbands.

2. P1 and P2 are used to partition V1 and V2 intoorthogonal subspaces: a signal-plus-noise sub-space and a noise subspace. A modified root-MUSIC (multiple signal classification) algo-rithm described below is applied to estimatethe signal poles for each subband.

3. The all-pole model amplitude coefficients ak aredetermined by using a linear least-squares fit tothe measured data.

4. The resulting subband signal models are ad-justed to optimally match.In step 1, the singular values in S are used to esti-

mate appropriate model orders for the two subbands.The relatively large singular values in S correspond to



strong signal components, while the small singularvalues generally correspond to noise. For low noiselevels, there is a sharp transition between the largeand small singular values. The transition point can beused as an estimate of the model order. At highernoise levels the transition from large to small singularvalues is smooth, making accurate model-orderestimation more difficult. The Akaike InformationCriterion (AIC) [8, 9] and Minimum DescriptionLength (MDL) [10, 11] are two model-order estima-tion methods that work well in these cases. Figure 9shows the singular-value spectra for the two-subbanddata set in Figure 8. The AIC and MDL model-orderestimates are both correctly equal to two.

Once the model orders P1 and P2 have been esti-mated, we proceed to step 2, in which the subspacedecomposition properties of V1 and V2 are used to es-timate the dominant signal poles for each subband.The matrices V1 and V2 are partitioned into orthogo-nal signal-plus-noise and noise subspaces,

V V V1 1 1= [ ]sn n

and

V V V2 2 2= [ ]sn n .

The partitioning is performed so that V1sn and V2

sn

have P1 and P2 columns, respectively. The noise sub-space matrices V1

n and V2n have L – P1 and L – P2

columns, respectively. Pole estimates for eachsubband are obtained by employing a modified root-MUSIC algorithm. Matrices A1 and A2 are definedfrom the noise subspace vectors for each subband as

A V V

A V V

1 1 1

2 2 2

= ′

= ′

n n

n n .

We denote a1i as the elements of the first row of A1and b1i as the elements of the first row of A2. Theseelements are used to form the polynomials A1(z) andA2(z) given by

FIGURE 9. Singular-value spectra for the two-subband dataset in Figure 8. The Akaike Information Criterion (AIC) andMinimum Description Length (MDL) model-order estimatesare equal to two. The signal-to-noise ratio is 20 dB.

FIGURE 10. Pole estimates for the two-subband data set illustrated in Figure 8. The dominant signal poles inthe lower and upper subbands are shown in blue and red, respectively.

Model order

Sin

gula

r val

ue

10

20

30

40

50

60

AIC and MDL model-order estimates

Lower subband Upper subband

0

1 2 3 4 5 6 7 8

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

Im

Re

Dominantsignalpoles

–1.5

–1.0

–0.5

0

0.5

1.0

1.5Lower subband

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

–1.5

–1.0

–0.5

0

0.5

1.0

1.5 Im

Re

Dominantsignalpoles

Upper subband



A z a z

A z b z

ii

i

L

ii

i

L

1 11

1

2 11

1

( )

( ) .

( )

( )

=

=

−

=

−

=

∑

∑

The roots of A1(z) and A2(z) correspond to pole esti-mates for bands 1 and 2, respectively.

This approach can be viewed as a variant of the tra-ditional root-MUSIC algorithm described in Refer-ence 12. Our approach has the important advantageof providing high-resolution pole estimates whileeliminating the symmetric pole ambiguities that re-sult from the traditional root-MUSIC approach.

Pole estimates can also be obtained by applying thespectral-estimation techniques described in Refer-ences 13 through 19. In our algorithm, the pole esti-mates are obtained for each subband by applying themodified root-MUSIC algorithm to V1

n and V2n . The

root-MUSIC algorithm finds poles corresponding tothe signal vectors that are most orthogonal to the

noise-subspace vectors. In general, the f α variation ofthe signal model leads to poles that are displaced fromthe unit circle in the complex z-plane. Over eachsubband, however, the variation of f α is small, so thedominant signals correspond to poles that lie close tothe unit circle. After estimating model orders in step1, we use the P1 poles closest to the unit circle in step2 to characterize the dominant lower-subband signalsand the P2 poles closest to the unit circle to character-ize the dominant upper-subband signals.

Figure 10 shows the resulting pole estimates for thetwo-subband data set illustrated in Figure 8. Thepoles shown in blue and red are considered the domi-nant signal poles for the lower and upper subbands,respectively. Notice that a lack of mutual coherenceprevents the signal poles in the lower subband fromlining up with the signal poles in the upper subband.

In step 3, we estimate the all-pole amplitude coef-ficients ak for the lower and upper subbands. An op-timal set of amplitude coefficients can be found bysolving a standard linear least-squares problem. Step 3

FIGURE 11. Mutual-coherence processing applied to the sparse-subband data set illustrated inFigure 8. Lower- and upper-subband signal models are shown before (a) and after (b) mutual-coherence processing. The mutually cohered signal models are consistent over much of the UWBprocessing interval.

10

–10

0

Vacant band

Mutually incoherent subbands

Vacant band

Mutually coherent subbands

I cha

nnel

(m)

(a)

Sample number0 N1 – 1 N – N2 N – 1

10

–10

0

I cha

nnel

(m)

Frequency (GHz)(b)

3 4 5 6 7 8 119 10 12




completes the all-pole modeling process for eachsubband. The lower- and upper-subband signal mod-els are denoted by M1( fn ) and M2( fn ), respectively.

In step 4, the subband signal models M1( fn) andM2( fn ) are adjusted until they optimally match.There are many ways to accomplish the match. Astraightforward method involves modulating andphase-aligning the lower-subband signal model untilit closely matches the upper-subband signal model.For example, the coherence function

C AM f e M fni n

nn

N

= ( ) − ( )=

−

∑ 1 2

2

0

1∆θ

can be minimized with respect to the pole rotationangle ∆θ and complex amplitude coefficient A. An-other approach for matching the subband signalmodels is to find an appropriate rotation matrix thatbest aligns the signal subspace vectors contained in V1and V2 . Whichever method is employed, the subbandmodel-alignment process tends to promote a strongsense of mutual coherence between the two subbands.

In Figure 11(a), we show the mutually incoherentsubband signal models. In Figure 11(b), an optimalpole rotation angle ∆θ* and complex amplitude coef-ficient A* were applied to the lower-subband signalmodel and corresponding data samples; i.e., thelower-subband data samples were replaced by mutu-ally coherent data samples given by

˜ , , , .[ * arg( *)]s s e n Nn ni n A= = −+∆θ 0 11K

Although the two signal models in Figure 11(b)may not entirely agree, it is important to recognizethat they have approximately the same signal poles.The corresponding all-pole model coefficients ak ,however, significantly differ. The lower subband fa-vors the decaying signal component, whereas the up-per subband favors the growing signal component.

UWB Parameter Estimation and Prediction

Once the radar subbands have been mutually co-hered, a global all-pole signal model is optimally fit-ted to the measured data. Our approach determinesthe all-pole model parameters that minimize the costfunction J given by

J q s M fn n nn

= − ( )∑2

.

The index n ranges over all of the available datasamples. The coefficients qn are used to weight themeasurements appropriately. The function J measuresthe total weighted error between the model given by

M f a pn k kn

k

P

( ) ==

∑1

and the mutually coherent data samples in eachsubband.

Minimizing J with respect to the all-pole modelparameters is a difficult nonlinear problem with noclosed-form solution. Brute-force numerical solu-tions are not feasible because of the potentially largenumber of signal parameters that must be estimated.Figure 12 illustrates an alternative approach to solv-ing this dilemma. Initial estimates of the all-polemodel parameters are obtained by using the tech-nique based on singular-value decomposition. Theseinitial estimates are then iteratively optimized by us-ing a standard nonlinear least-squares algorithm, suchas the Newton-Raphson algorithm. (Detailed infor-mation about the Newton-Raphson algorithm can befound in many standard texts on numerical analysis[20].) If the initial parameter estimates are close tooptimal, the standard nonlinear least-squares algo-rithm rapidly converges to the all-pole model param-eters that minimize J.

Many methods will give an initial estimate of theglobal all-pole model parameters. One method is toconstruct the multiband prediction matrix given by

HH

H=

1

2

.

The submatrices H1 and H2 correspond to the for-ward-prediction matrices for the lower and uppersubbands, respectively. We call this approach sub-aperture processing because it combines the datasamples from both subbands, providing the potentialfor robust parameter estimates from noisy data.

It is also possible to obtain multiband parameter



estimates by allowing for cross-correlation betweenthe subbands, i.e., by defining H as

H H H= [ ]1 2 .

We refer to this method as extended-aperture pro-cessing, which provides the potential for true UWBresolution. However, the resulting pole estimates aretypically more sensitive to noise than those fromsubaperture processing. In principle, the two meth-ods—subaperture processing and extended-aperture

processing—can be combined to provide robusthigh-resolution estimates of the dominant signalpoles. In both cases, multiband parameter estimatesare obtained by decomposing H into the product ofthree matrices:

H USV= ′ .

An estimate P̂ of the model order is obtained byapplying the AIC or MDL techniques to the spec-trum of singular values contained in S. For the sparse-subband data set illustrated in Figure 11(b), both theAIC and MDL model-order estimates are correctlyequal to two. The model-order estimate is used topartition V into orthogonal signal-plus-noise andnoise subspaces. Initial pole estimates are obtained byusing the methods of a previous section, “Mutual-Coherence Processing,” or any other superresolutionspectral-estimation technique.

Figure 13 shows a plot of initial pole estimates forthe sparse-subband data set illustrated in Figure11(b). Including both H1 and H2 into the Hankelmatrix correctly identifies both signal poles and asso-ciates them as f +1 and f –1 pole behavior. The twodominant signal poles are used to initialize the New-ton-Raphson algorithm.

This algorithm uses the initial parameter estimates

FIGURE 13. Multiband pole estimates for the mutually coher-ent subbands illustrated in Figure 11(b). The dominant sig-nal pole inside the unit circle corresponds to the f –1 scatter-ing center. The dominant signal pole outside the unit circlecorresponds to the f +1 scattering center.

FIGURE 12. UWB parameter estimation. Initial parameter es-timates are obtained by using a singular-value decomposi-tion technique. These initial parameter estimates are itera-tively optimized with a standard nonlinear least-squaresalgorithm.

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

Im

Re

–1.5

–1.0

–0.5

0

0.5

1.0

1.5Multiband poles Dominant

signalpoles

f –1

f +1

Data matrix ofmutually coherent

subbands

Singular-value

decomposition

MatrixproductUSV′

Estimate modelorder

(e.g., MDL, AIC)

P^

Modifiedroot-MUSIC

algorithm

Initialpole

estimates

Nonlinearleast squares

(iterative)

Global all-polemodel

parameters



to find the global all-pole model parameters ak and pkthat locally minimize the cost function J. The modelorder P remains fixed during this iterative process andthe algorithm typically converges to a local minimumof J in only a few iterations. We test the approach byoptimally fitting a global all-pole signal model to thetwo subbands illustrated in Figure 11(b).

Figure 14(a) shows a comparison between the glo-bal all-pole signal model and the actual signal; the all-pole model agrees with the actual signal over theentire UWB frequency range. The correspondingcompressed pulses are shown in Figure 14(b). Thesparse-subband compressed pulse uses the mutuallycoherent radar measurements within the two sub-bands and the global all-pole model in the vacantband. With this approach, the two target points arewell resolved and the estimated UWB responseclosely matches the actual signal.

This example also demonstrates the potential forusing all-pole signal models to accurately characterizef α-type scattering behavior over ultrawide processing

bandwidths. In fact, the UWB pole estimates can betransformed into equivalent estimates of the α expo-nents for f α-type signal models. We can always findan f α function that best matches the exponential be-havior of an UWB signal pole over a given frequencyrange. We can also derive an approximate analyticalrelationship between the pole magnitudes and thecorresponding α exponents by matching the func-tions f kα and pk

nat the lowest and highest UWB

frequencies. This relationship is given by

αkkN p

Ndf

f

=− ( )+ −

( ) log

log ( )

,1

1 11

(1)

where df and f1 denote the spectral sample spacingand lowest UWB frequency, respectively. The con-stant N denotes the total number of UWB frequencysamples. In the two-scattering-center example dis-cussed previously, the two dominant signal poles p1and p2 are given by

FIGURE 14. (a) Comparison between the fitted UWB signal model (brown curve) and truth (blackcurve). (b) Corresponding compressed pulses. The two scattering centers are well resolved withthe UWB model closely matching truth.

10

–10

0

Noise floor

–30

10

0

–10

–20


RC

S (d

Bsm

)I c

hann

el (m

)

Frequency (GHz)(a)

Actual signalSparse-subband processing

–1 0 1

Sample number


0

–2

UWB compressed pulses

3 4 5 6 7 8 119 10 12

N1 – 1 N – N2 N – 1

f –1 f +1

Vacant band

2

5

–5



p e

p e

i

i

14

23

0 992

1 005

=

=

.

. .

/

/

π

π

By substituting these poles into Equation 1, we ob-tain an accurate estimate of the true α exponents usedin the simulation. Thus the UWB pole locations pro-vide information on scattering type. This informationis useful for analyzing the details of targets with theviewpoint of constructing an accurate measurement-based model.

Static-Range Experiments

In the previous section, we presented the basic con-cepts behind our UWB processing algorithms. In thissection, we utilize static-range data to demonstratethe applied aspects of UWB processing.

Figure 15 shows our target for the UWB process-ing demonstration—a monoconic model of a reentryvehicle with length of 1.6 m. The spherical nose tip ofthe reentry vehicle has a radius of 0.22 cm; the nosesection is made from a solid piece of machined alumi-num with two grooves and one seam. The firstgroove—approximately 3 mm deep and 6 mmwide—is located 22 cm from the reentry-vehicle nosetip. The second groove is approximately 2 mm deepand 4 mm wide, and is located 44 cm from the reen-try-vehicle nose tip. The midbody of the reentry ve-hicle is made from a single sheet of rolled aluminumwith one groove, one slip-on ring, and three seams.The aluminum slip-on ring (not shown in the photo)is approximately 5 mm thick and 10 mm wide, and isplaced 1.4 m from the reentry-vehicle nose tip.

FIGURE 16. Moment-method RCS calculations for the threemajor grooves on the target, which was at a 20° aspect angle.All three grooves exhibit the expected f 3 scattering behaviorat low frequencies, with break points that depend on the sizeof the groove.

FIGURE 15. Test target for UWB processing experiments.This monoconic model of a reentry vehicle is 1.6 m long. Thespherical nose tip has a radius of 0.22 cm. The nose sectionis made from a solid piece of machined aluminum with threegrooves, two near the front of the model and one at midbody.

The reentry vehicle shown in Figure 15 is ideal forUWB processing experiments because it has severalscattering centers that exhibit significant RCS varia-tions as a function of frequency. Figure 16 shows amoment-method RCS calculation for the three majorgrooves on the reentry vehicle. The grooves exhibitthe expected f 3 scattering behavior at the low-fre-quency end of the spectrum, with break points thatdepend on the size of the groove.

The Lincoln Laboratory static-range radar facilitywas used to collect coherent radar measurements overa wide range of frequencies and viewing aspects of thetarget. Measurements were taken from 4.64 to 18GHz in 40-MHz increments. The target viewingangles, relative to nose-on, ranged from –5° to 95° in0.25° increments.

To demonstrate UWB processing, we focused on asegment of data collected in the 12-to-18-GHz re-gion shown in Figure 16. Figure 17(a) shows anuncompressed radar pulse corresponding to an aspectangle of 20°. To test our UWB processing algorithms,we reduced the bandwidth of the uncompressed radarpulses to two 1.0-GHz-wide subbands, as illustratedin Figure 17(b). Figure 17(c) shows the compressedpulses for the two subbands and for the fullband dataset. The bandwidth of the two subbands is insuffi-cient to resolve many scattering centers on the target,while the fullband compressed pulse resolves all thesignificant scattering centers on the target. The pur-

Frequency (GHz)

RC

S (m

2 )

1810

Firstgroove

Midbodygroove

Secondgroove

6 8 12 14 16

Frequencyrange forUWBexperiments

10–4

10–5

10–6

f 3f –1

First groove

Second groove Midbody groove



pose of this experiment was to use UWB processingto obtain a result highly consistent with the fullbandresult. We then demonstrated the ability to coher-ently process the subband measurements so that wecan accurately estimate the target’s UWB response.

Figure 18 shows the UWB pole estimates obtainedby applying the sparse-subband spectral-estimationtechnique discussed earlier in the section entitled“UWB Coherent Processing.” The pole locations areconsistent with the physical scattering centers on thetarget. The pole corresponding to the nose-tip re-sponse is close to the unit circle, indicating it has anearly constant RCS as a function of frequency. Thegrooves and slip-on ring have non-constant RCSs as afunction of frequency; the corresponding poles are ei-

FIGURE 17. (a) Uncompressed radar pulse of the test target shown in Figure 15 withviewing aspect 20° from nose-on. (b) Sparse-subband measurements used to predictthe target’s response over the fullband from 12 to 18 GHz. (c) Compressed pulses forthe sparse subbands and fullband data sets. The fullband compressed pulse (black)resolves all of the significant scattering centers on the target.

FIGURE 18. UWB pole estimates obtained by using thesparse-subband data set shown in Figure 17(b).

–60

–40

–30

–50

–20

RC

S (d

Bsm

)

Relative range (m)(c)

Compressedpulses

Target

Vacantband

Vacantband

Vacantband

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

0

0.2

–0.2I cha

nnel

(m)

Frequency (GHz)(b)

12 13 14 15 16 17 18

0

0.2

–0.2I cha

nnel

(m)

Frequency (GHz)(a)

12 13 14 15 16 17 18

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

Im

Base edgeRing

Groove

Nose tip

Grooves

Re

–1.5

–1.0

–0.5

0

0.5

1.0

1.5Multibandsignal poles



ther inside or outside the unit circle, as predicted bythe moment-method RCS calculations in Figure 16.

Estimating the corresponding α exponents forthese major scattering centers is straightforward—themagnitude of the signal poles is related to the α expo-nents via Equation 1. A more accurate relationshipcan be obtained by solving for the α exponents thatproduce the best match between the functions f kα

and pkn

over the frequency range of interest, i.e.,from 12 to 18 GHz. Using this approach, we esti-mated the α exponents for the first, second, and mid-body grooves, respectively, on the reentry vehicle tobe α1 = –1.1, α2 = 2.3, and α3 = 2.9. These estimatesare consistent with the moment-method RCS calcu-lations shown in Figure 16.

Figures 19(a) and 19(b) show comparisons be-tween the estimated UWB target response and thetrue UWB radar measurements. The model and themeasurements are in excellent agreement.

Because radar measurements were taken over a

FIGURE 19. Comparisons between the estimated UWB target response and the trueUWB radar measurements. (a) Uncompressed radar pulse for the prediction model(brown line) and the actual radar measurements (black line). (b) The correspondingcompressed pulses that resolve the scattering centers on the target.

wide range of viewing aspects, we could generate two-dimensional radar images of the target. Figures 20(a)and 20(b) show the lower- and upper-subband im-ages, respectively. The resolution is insufficient to re-solve many of the scattering centers on the target. Fig-ures 20(c) and 20(d) show the true and estimatedUWB target images, respectively. All four imageswere generated by applying extended coherent pro-cessing [3] to the corresponding compressed pulsesover the full range of available viewing aspects. Weused target symmetry to process the data as if we hadsampled a range of viewing aspects from –95° to 95°.The UWB images provide a clear picture of the targetand show considerable detail. The sparse-subbandimage closely matches the fullband image and pro-vides an accurate estimate of the locations and α ex-ponents of the many realistic scattering centers on thetarget.

These experimental results suggest that UWB pro-cessing of sparse-subband measurements can signifi-

–0.4

0

0.2

–0.2

0.4

I cha

nnel

(m)

Frequency (GHz)(a)

12 13 14 15 16 17 18

–60

–40

–30

–50

–20

RC

S (d

Bsm

)


Compressedpulse

Target

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

Actual signalSparse-subband

processing



cantly improve range resolution and provide accuratecharacterizations of targets over ultrawide band-widths. We are currently investigating fundamentallimitations and practical payoffs of UWB processing.

Summary

This article presents an approach for accurately esti-mating a target’s UWB radar signature from sparse-

subband measurements. To apply this technology tofield data, we developed an algorithm that couldcompensate for the potential lack of mutual coher-ence between the various radar subbands. Robustmutual-coherence processing was performed by opti-mally matching the all-pole signal models for eachsubband. With the radar subbands mutually cohered,a single UWB all-pole signal model was optimally fit-

FIGURE 20. Comparison of two-dimensional radar images. The upper left and right images show the lower-and upper-subband images, respectively. The fullband image in the lower left uses actual radar measure-ments over the full 12-to-18-Ghz frequency range. The sparse-subband image in the lower right uses thesparse-subband measurements with UWB prediction.

Threshold: –50 dBsm

Lower-subband image

0.5

–0.5

–1.0

1.0

0

Rel

ativ

e r

ange

(m)

0.5–i0.5–1.0 1.00

Cross-range (m)(a)

Upper-subband image

0.5

–0.5

–1.0

1.0

0

Rel

ativ

e ra

nge

(m)

0.5–0.5–1.0 1.00

Cross-range (m)(b)

Threshold: –50 dBsm

Fullband image

0.5

–i0.5

–1.0

1.0

0

Rel

ativ

e ra

nge

(m)

0.5–i0.5–1.0 1.00

Cross-range (m)(c)

Sparse-subband image

0.5

–i0.5

–1.0

1.0

0

Rel

ativ

e ra

nge

(m)

0.5–i0.5–1.0 1.00

Cross-range (m)(d)



ted to the available data. The fitted model was used tointerpolate between and extrapolate outside the mea-surement bands. Standard pulse-compression meth-ods were applied to the enlarged band of spectral datato provide a superresolved range profile of the target.

These UWB processing concepts were demon-strated by using simulations and static-range data. Weshowed that it was possible to accurately estimate atarget’s UWB response when the radar measurementsfill only a small fraction of the total processing band-width. The practical payoff of this technology is thatradar measurements need not be taken over the fullUWB processing interval; signal processing can beused to a certain extent to compensate for any missingdata. Another important benefit of UWB processingis that the α exponents of individual scattering centerscan be more accurately estimated. This accuracy helpsus to better identify the scattering centers that makeup a target, which significantly improves our analysisand understanding of the target.

Suggestions for Future Research

Many unresolved issues in UWB processing of sparse-subband measurements remain. The uniqueness of

our nonlinear optimization process, the accuracy ofthe initial pole estimates, and the performance versusband fill ratio (ratio of measured data to total process-ing interval) are important UWB processing con-cerns. While the nonlinear optimization processcross-correlates the subbands, it may be possible toobtain more resolved UWB signal models by betterexploiting the cross-band correlation informationduring the initial pole estimation stage. We are cur-rently investigating these issues and considering somepotential real-time applications of this technology.

Acknowledgments

The authors would like to thank Michael Burrows ofthe Sensor Systems and Measurements group for pro-viding radar-cross-section predictions for the mono-conic reentry vehicle; we also would like to thankMohamed Abouzahra of the Systems Engineeringand Analysis group and Peter Kao of the Sensor Tech-nology and Systems group for providing the static-range data that we used to demonstrate the appliedaspects of ultra-wideband processing. This work wassponsored by the U.S. Space and Missile DefenseCenter.



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Mrstik, “The Kiernan Reentry Measurements System on Kwa-jalein Atoll,” Linc. Lab. J. 2 (2), 1989, pp. 247–276.

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8. H. Akaike, “A New Look at the Statistical Model Identifica-tion,” IEEE Trans. Autom. Control 19 (6), 1974, pp. 716–723.

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Speech Signal Process. 37 (8), 1989, pp. 1190–1196.12. A.J. Barabell, J. Capon, D.F. DeLong, J.R. Johnson, and

K.D. Senne, “Performance Comparison of SuperresolutionArray Processing Algorithms” Project Report TST-72, LincolnLaboratory (9 May 1984, rev. 15 June 1998).

13. S.W. Lang and J.H. McClellan, “Frequency Estimation withMaximum Entropy Spectral Estimators,” IEEE Trans. Acoust.Speech Signal Process. 28 (6), 1980, pp. 716–724.

14. R.O. Schmidt, “A Signal Subspace Approach to MultipleEmitter Location and Spectral Estimation,” Ph.D. thesis,Stanford University, Stanford, Calif., 1981.

15. T.-J. Shan, M. Wax, and T. Kailath, “On Spatial Smoothing forDirection-of-Arrival Estimation of Coherent Signals,” IEEETrans. Acoust. Speech Signal Process. 33 (4), 1985, pp. 806–811.

16. A. Paulraj, R. Roy, and T. Kailath, “Estimation of Signal Pa-rameters via Rotational Invariance Techniques—ESPRIT,”19th Asilomar Conf. on Circuits, Systems & Computers, PacificGrove, Calif., 6–8 Nov. 1986, pp. 83–89.

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20. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery,Numerical Recipes in C: The Art of Scientific Computing,2nd ed. (Cambridge University Press, Cambridgeshire, U.K.,1992).



. is a senior staff member in theSensor Systems and Measure-ments group, where he special-izes in the development ofadvanced signal processingmethods for high-resolutionradar. He has developed andimplemented various radarimaging techniques, includingextended coherent processing(ECP) to generate high-resolu-tion three-dimensional radarimages, and bandwidth ex-trapolation (BWE) to improvethe range resolution of coher-ent radar returns. He hasauthored several technicalpapers and reports in the areasof superresolution data pro-cessing and applications ofchaotic dynamical systems forsecure communications. Re-cent efforts include the devel-opment of ultra-widebandprocessing techniques formutually cohering and coher-ently combining sparse multi-spectral/multisensor radarmeasurements. Before joiningLincoln Laboratory in 1988,he was a member of the re-search staff at CalspanCorporation.

Kevin received B.S. (magnacum laude) and M.S. degreesin electrical engineering fromthe State University of NewYork (SUNY) at Buffalo, anda Ph.D. degree in electricalengineering from MIT in1993. While attending SUNYat Buffalo, he was awardedDepartmental Honors and wasthe recipient of a UniversityFellowship and a HughesFellowship. He is a member ofTau Beta Pi, Eta Kappa Nu,and Sigma Xi.

. is a staff member in the SensorSystems and Measurementsgroup and works on superreso-lution techniques related toradar target identification andimaging. His research interestsinclude estimation theory andapplication, and eigenstructureassignment for multivariablestochastic systems. Beforejoining Lincoln Laboratory in1995, he taught at CityCollege of New York (CCNY)and the State University ofNew York (SUNY) atBinghamton, where he heldvisiting and assistant professor-ships, respectively.

Jean received B.S. degrees inapplied mathematics andelectrical engineering from theUniversité d’Etat d’Haiti, andM.S. and Ph.D. degrees inelectrical engineering from theCity University of New York(CUNY). He has publishedseveral technical papers inthe areas of eigenstructureassignment for flight controlsystems.

. leads the Sensor Systems andMeasurements group. Sincejoining Lincoln Laboratory in1973, he has worked in theareas of satellite communica-tions antennas, adaptive an-tenna design and performanceevaluation, spatial spectralestimation that uses multiple-beam antennas, electromag-netic scattering from activelyloaded targets, radar systemdesign, and radar data analysis.He served two four-year toursat the Laboratory’s Kwajaleinfield site in the MarshallIslands—the first as leader ofthe ALTAIR deep-space track-ing radar and the second assite manager of the LincolnProgram at Kwajalein.

Joseph received a B.S.degree in electrical engineeringfrom Purdue University,Lafayette, Indiana, and M.S.and Ph.D. degrees in electricalengineering from The OhioState University in Columbus.Upon graduation, he workedon reentry systems analysisfor Avco Corporation inWilmington, Massachusetts.In 1969, he joined the facultyof the University of Akron inOhio as an associate professorof electrical engineering.

Joseph has publishedextensively in the areas ofnonlinear interactions ofelectromagnetic waves inplasma media, adaptive an-tenna design, spectral-estima-tion techniques, and electro-magnetic scattering. Hereceived the R.W.P. Kingaward for papers published inthe IEEE Transactions onAntennas & Propagation.

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