Waveform-Diverse Moving-Target Spotlight SAR

Margaret Cheneya and Brett Bordenb

aRensselaer Polytechnic Institute, Troy, NY, USA;bNaval Postgraduate School, Monterey, CA, USA

ABSTRACT

This paper develops the theory for waveform-diverse moving-target synthetic-aperture radar. We assume thatthe targets are moving linearly, but we allow an arbitrary flight path and (almost) arbitrary waveforms. Weconsider the monostatic case, in which a single antenna phase center is used for both transmitting and receiving.This work addresses the use of waveforms whose duration is sufficiently long that the targets and/or platformmove appreciably while the data is being collected.

Keywords: radar imaging, ambiguity function

1. INTRODUCTION

This paper considers waveform-diverse synthetic-aperture radar in the case when there are multiple movingtargets in the scene. We consider a pulsed system traversing a circular flight path, and assume that the targetsare moving linearly. We consider the monostatic case, in which a single antenna phase center is used for bothtransmitting and receiving. The problem is formulated in terms of forming an image in phase space, where theindependent variables include not only position but also the vector velocity.

We include the case of waveforms whose duration is sufficiently long that the targets and/or platform moveappreciably while the data is being collected. Figure 1 shows the regime of validity of the start-stop approximationfor X-band. This figure plots v ≤ λ/T , where v is the relative velocity, λ is the wavelength at the center frequency,and T is the waveform duration. For shorter wavelengths, the curve moves towards the axes, and the regionof validity is smaller. We see that the start-stop approximation is invalid for high-frequency systems or long-duration waveforms or high-velocity targets. For example, a target moving 30 m/sec (67 mph) moves 3 mm in100 μsec and 3 m in .1 sec, distances that could easily be comparable to the system wavelength. The issue alsoarises when low-power, long-duration waveforms are used.

This paper is an extension of earlier work,2 which showed how to combine the temporal, spectral, andspatial attributes of radar data. In particular, the theory developed in this paper shows how to combine fast-time Doppler and range measurements made from different spatial locations. This approach can be used, forexample, for sar and isar imaging when relative velocities are large enough so that target returns at each lookare Doppler-shifted. Alternatively, this theory shows how to include spatial considerations into classical radarambiguity theory. In addition, this approach provides a connection between sar and Moving Target Indicator(mti) radar.

SAR for moving targets has been studied in previous work: Fienup4 analyzed the phase perturbations causedby moving targets and showed how the motion affects the image. Other work7 identifies ambiguities in four-dimensional space that result from attempting to image moving targets from a sensor moving along a straightflight path. A patent6 and other papers8, 10, 11 all use the start-stop approximation to identify moving targetsfrom their phase history. Other work5 uses a fluid model to impose conservation of mass on a distribution ofmoving scatterers, and a Kalman tracker to improve the image adaptively.

In this paper, we outline a derivation for the phase-space point-spread function for a pulsed waveform-diversespotlight SAR system traversing a circular flight path. We show that this point-spread function can be written in

Further author information: (Send correspondence to M.C.)M.C.: E-mail: cheney@rpi.edu, Telephone: 1 518 276 2646B.B.: E-mail: bhborden@nps.edu, Telephone: 1 831 656 2855

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Figure 1. This shows the region of validity for the start-stop approximation for X-band (wavelength 3cm). The horizontalaxis represents the waveform duration, and the vertical axis shows the relative velocity. For shorter wavelengths, the curvemoves towards the axes, and the region of validity is smaller.

terms of the ordinary radar ambiguity function, evaluated at certain arguments. As an application, we computethe error that would be made if the processing were done instead assuming the start-stop approximation. Wefind that for rapidly-moving high-resolution systems, this error can be significant. A more complete expositionof this work will be published elsewhere.3

2. MODEL FOR RADAR DATA

If the target reflectivity in its own reference frame is denoted by Q(x), then the reflectivity of the target movingwith velocity v is Qv(x − vt) = Q(x − vt, v). We model the antenna as an isotropic point source.

2.1. The received field

The field received at the antenna, which is located at x = γ(t), after scattering from a target Q moving withspeed v, is

EscB (t) = −

∫δ(t − t′ − |γ(t) − y|/c)

4π|γ(t) − y|∫

Q(y − vt′,v)∫

δ(t′ − t′′ − |y − γ(t′′)|/c)4π|y − γ(t′′)| f(t′′) dt′′ dt′ d3y. (1)

Here f(t′′) is the transmitted waveform. The model (1) is more easily interpreted after we make the change ofvariables y �→ z = y − vt′, whose inverse is y = z + vt′. This change of variables converts equation (1) into

EscB (t) = −

∫∫∫δ(t − t′ − |γ(t) − (z + vt′)|/c)

4π|γ(t) − (z + vt′)| Q(z,v)δ(t′ − t′′ − |z + vt′ − γ(t′′)|/c)

4π|z + vt′ − γ(t′′)| f(t′′) dt′′ dt′ d3z. (2)

The right side of (2) can be interpreted as follows. The part of the waveform f that is transmitted at time t′′

from location γ(t′′) travels to the target, arriving at time t′. At time t′, the target that started at z is now atlocation z + vt′. The wave scatters with relative strength Q(z), and then propagates to the receiver, arriving attime t. At time t, the receiver is at position γ(t).

For multiple moving targets, we simply integrate over all the possible velocities.

We assume that the system transmits a train of pulses of the form∑m

fm(t − Tm) m = 0, 1, 2, . . . (3)

where the delay between successive pulses is sufficiently large so that successive pulses do not overlap.

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2.2. Spotlight SARWe now assume that the distance from the origin to the sensor position γ is much larger than the distance fromthe origin to the target, and also much larger than the distance travelled by the target or sensor during any ofthe time intervals TT

m+1 − TTm. With these assumptions we can make the following expansions:

|z + vt′ − γ(t)| = |z + vT ′m + v(t′ − T ′

m) − γ(TRm) − γ̇(TR

m)(t − TRm) + · · · |

= |γ(TRm)| − γ̂(TR

m)· [z + vT ′m + v(t′ − T ′

m) − γ̇(TRm)(t − TR

m)]+ · · ·

|z + vt′ − γ(t′′)| = |z + vT ′m + v(t′ − T ′

m) − γ(TTm) − γ̇(TT

m)(t′′ − TTm) + · · · |

= |γ(TTm)| − γ̂(TT

m)· [z + vT ′m + v(t′ − T ′

m) − γ̇(TTm)(t′′ − TT

m)]+ · · · . (4)

We use (4) in (2) and carry out the t′′ and t′ integrations.

For the mth pulse, expression (2) then becomes

Escm(t) = −

∑m

∫fm(φm(t, z,vm)) Q(z)

(4π)2|γTm||γR

m|(1 − γ̂Rm·vm/c)(1 + βT

m)d3z, (5)

where

φm(t,z,vm) ≈ 11 + βT

m

[αv,m

([1 − βR

m]t − RRm(z,vm)/c

) − RTm(z,vm)/c

] − TTm (6)

and

βTm = γ̂T

m·γ̇Tm/c

βRm = γ̂R

m·γ̇Rm/c

αvm,m =1 + γ̂T

m·vm/c

1 − γ̂Rm·vm/c

RRm(z,vm) = |γR

m| − γ̂Rm· (z + Γm − vmT ′

m + γ̇RmTR

m

)RT

m(z,vm) = |γTm| − γ̂T

m· (z + Γm − vmT ′m + γ̇T

mTTm

). (7)

The quantities βT and βR are determined by the squint angle (angle relative to broadside) of the transmitterand receiver, respectively, and α is the Doppler scale factor.

Example: Circular SAR. For the case of circular SAR, where R = |γ(Tm)|, γ̂m = γ(Tm)/R, and R◦m(z, vm) =

R − γ̂m·z, we obtain

E◦m(t) ∝

∫fm(φ◦

m(t,z,v)) Q(z,v)d3zd3vm, (8)

where the phase is

φ◦m(t, z,v) = αv,m (t − R◦

m(z,v)/c) − R◦m(z,v)/c − Tm, (9)

where the Doppler scale factor is

αv,m =1 + γ̂m·v/c

1 − γ̂m·v/c. (10)

3. IMAGE FORMATION

In the circular SAR case, the estimate I(p,u) (image) of the reflectivity Q(p,u) at position p and velocity u iscomputed by weighted matched filtering:

I(p,u) ∝∑m

∫f∗

m (φm(t,p,u))αu,m(1 − γ̂m·v/c)E◦m(t)dt (11)

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4. IMAGE ANALYSIS

Using the data model (8) in the image formation algorithm (11) gives rise to

I(p,u) =∫

K(p,u;z,v)Q(z,v)d3zd3v (12)

where the point-spread function K is

K(u,p;z,v) =∑m

∫f∗

m (φ◦m(t,p,u)) fm(φ◦

m(t,z,v))1 − γ̂m·u/c

1 − γ̂m·v/cαu,mdt. (13)

If we make the change of variables t �→ t′ = φ◦m, we obtain

K(p,u;z,v) =∑m

Am

(αu,m

αv,m,Δτm(p,u;z,v)

) ∣∣∣∣αu,m

αv,m

∣∣∣∣ 1 − γ̂m·u/c

1 − γ̂m·v/c, (14)

whereAm(σ, τ) =

∫f∗

m(σt − τ)fm(t)dt (15)

is the wideband ambiguity function13 and where

Δτm(p,um;z,vm) =αu,m

c

[R◦

m(z,v)αv,m

− R◦m(p,u)αu,m

+ R◦m(z,v) − R◦

m(p,u) +Tm

αv,m− Tm

αu,m

]. (16)

If the waveforms fm have thumbtack ambiguity functions, then approximations to both downrange position anddownrange velocity can be obtained at each look m, and the variation in aspects as m varies (i.e., over thesynthetic aperture) provides cross-range information.

5. APPLICATION: VALIDITY OF THE START-STOP MODEL

Almost all traditional sar signal processing and analysis makes use of the start-stop approximation, also calledthe stop-and-shoot approximation, which is the assumption that neither target nor antenna is moving while eachinteracts with the wave.

5.1. The Start-Stop Model for Spotlight SAR DataUnder the start-stop assumption, (4) is instead

|z + vt′ − γ(t)| ≈ |z + vT ′m − γ(TR

m)| = |γ(TRm)| − γ̂(TR

m)· [z + vT ′m] + · · ·

|z + vt′ − γ(t′′)| ≈ |z + vT ′m − γ(TT

m)| = |γ(TTm)| − γ̂(TT

m)· [z + vT ′m] + · · · . (17)

The start-stop signal model [(2) and (5)] reduces to

Escss(t) = −

∫δ(t − t′ − |γR

m|/c + γ̂Rm· [z + vmT ′

m] /c)4π|γR

m| Q(z)

δ(t′ − t′′ − |γTm|/c + γ̂T

m· [z + vmT ′m] /c)

4π|γTm| fm(t′′ − TT

m) dt′′ dt′ d3z

=∫

fm

(t − (RT,ss

m (z,v) + RR,ssm (z,v))/c − TT

m

)(4π)2|γR

m||γTm| Q(z,vm) d3z d3vm, (18)

where

RT,ssm (z,v) = |γT

m| − γ̂Tm· [z + vT ′

m]

RR,ssm (z,v) = |γR

m| − γ̂Rm· [z + vT ′

m] . (19)

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5.2. Start-Stop Image Formation for Spotlight SAR

The image Iss (estimate of the target reflectivity Q) is formed as

Iss(p,u) =∑m

(4π)2|γRm||γT

m|∫

f∗m

(t − TT

m − (RT,ssm (p,u) + RR,ss

m ((p,u))/c) Esc

ss(t) dt, (20)

which gives rise to a point-spread function of the form

Kss(p,u,z,v) =∑m

∫f∗

m

(t − TT

m − (RT,ssm (p,u) + RR,ss

m ((p,u))/c)

fm

(t − TT

m − (RT,ssm (z,v) + RR,ss

m (z,v))/c)

dt

=∑m

Am(1,Δτ ssm ), (21)

where

Δτ ssm = (RT,ss

m (z,v) + RR,ssm (z,v))/c − (RT,ss

m (p,u) + RR,ssm ((p,u))/c. (22)

A focussed image will be obtained when

0 = (RT,ssm (z,v) + RR,ss

m (z,v)) − (RT,ssm (p,u) + RR,ss

m ((p,u)) = Bm· [(p − z) + (u − v)T ′m] , (23)

where we have writtenBm = γ̂T

m + γ̂Rm. (24)

5.3. Mismatched Processing

If we incorrectly believe the start-stop data model to be accurate, we would form from Esc of (5) an image viathe same processing (20) as for the start-stop model, namely

I×(p,u) =∑m

(4π)2|γRm||γT

m|∫

f∗m

(t − TT

m − RT,ssm (p,u) + RR,ss

m ((p,u)c

)Esc

m(t) dt, (25)

where Escm(t) is given by (5). Substituting (5) into (25), we obtain

I×(p,u) = −∑m

∫f∗

m

(t − TT

m − RT,ssm (p,u) + RR,ss

m ((p,u)c

) ∫fm(φm(t,z,vm)) Q(z,vm)

(1 − γ̂Rm·vm/c)(1 + βT

m)d3z d3vm dt. (26)

The point-spread function for mismatched processing is

K×(p, u;z,v) =∑m

∫f∗

m

(t − TT

m − RT,ssm (p,u) + RR,ss

m ((p,u)c

)fm(φm(t,z,vm))

(1 − γ̂Rm·vm/c)(1 + βT

m)dt.

=∑m

A(

1 + βTm

(1 − βRm)αvm,m

,Δτ×m

)1

(1 + γ̂Tm·vm/c)(1 − βR

m), (27)

where

Δτ×m =

(TT

m

[(1 − βR

m)αvm,m

1 + βTm

− 1]

+RT,ss

m (p,u)(1 − βRm)αvm,m

(1 + βTm)c

− RTm(z,vm)

(1 + βTm)c

+RR,ss

m (p,u)(1 − βRm)αvm,m

(1 + βTm)c

− RRm(z,v)αvm,m

(1 + βTm)c

). (28)

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5.4. Error Analysis for Ideal High-Range-Resolution Waveforms

We expect that the start-stop approximation will be best in the case of a waveform whose ambiguity function isinsensitive to its first argument, i.e., for a high-range-resolution waveform such as the ideal short pulse f(t) = δ(t).For such an ideal high-range-resolution waveform, a focused image is attained when

0 = Δτ×m = TT

m

[(1 − βR

m)αv,m

1 + βTm

− 1]

+(RT,ss

m (p,u) + RR,ssm (p,u)

) (1 − βRm)αv,m

(1 + βTm)c

− (RT

m(z,vm) + RRm(z,v)αvm,m

) 1(1 + βT

m)c. (29)

For the case of constant platform velocity, (29) becomes

0 = Bm· [(z − p) + (v − u)T ′m] /c

− |γTm|Bm·(v − γ̇)/c2 + RRT

m Bm·(v − γ̇)/c2 − Bm· [p + uT ′m]Bm·(v − γ̇)/c2

+ γ̂Tm·γ̇T

m/c

[RRT

m − Bm·(z + γ̇T ′m)

]/c −

(1 − γ̂T

m·γ̇Tm/c

) [γ̂T

m·γ̇m(|γTm|/c) − γ̂R

m·γ̇m(|γRm|/c)

]

−[|γR

m|Bm·v/c2 − γ̂Rm· (z + γ̇R

m(T ′m + |γR

m|/c))Bm·v/c2

]

γ̂Tm·γ̇T

m

[|γR

m|Bm·v/c − γ̂Rm· (z + γ̇R

m(T ′m + |γR

m|/c))Bm·v/c

]/c2, (30)

where we have writtenRRT

m = |γTm| + |γR

m|. (31)

The first line of (30) we recognize as the condition (23) for focusing under the start-stop approximation. Thebottom line of (30) is of order (v/c)2 and is neglected. Thus to leading order in (v/c), the error made in usingthe start-stop approximation is

|γTm|Bm·γ̇/c2 − RRT

m Bm·γ̇/c2 − Bm· [p + uT ′m]Bm·(v − γ̇)/c2

+ γ̂Tm·γ̇T

m/c

[RRT

m − Bm·(z + γ̇T ′m)

]/c −

(1 − γ̂T

m·γ̇Tm/c

) [γT

m·γ̇m/c − γRm·γ̇m/c

]

+[γ̂R

m· (z + γ̇Rm(T ′

m + |γRm|/c)

)Bm·v/c2

]. (32)

The third line of (32) is error due to the sensor having moved between the time of transmission and the time ofreception. When this error is negligible, we have γT

m = γRm and B = 2γ̂, in which case the error (32) reduces to

1c2

(− 4 (γ̂m· [p + uT ′

m]) γ̂m·(v − γ̇) − 2γ̂m·γ̇m

[γ̂m·(z + γ̇T ′

m)]

+ γ̂m· (z + γ̇m(T ′m + |γm|/c)) 2γ̂m·v

). (33)

If, in addition, there is no squint (γ̂·γ̇ = 0 = B·γ̇), the error (33) reduces to(−Bm· [p + uT ′m] + γ̂m·z

c

)(Bm·v/c) =

(−2γ̂m· [p + uT ′m] + γ̂m·z

c

)(2γ̂m·v/c) , (34)

which vanishes for stationary targets. Thus for a stationary scene, we find that misfocusing due to the start-stopapproximation can potentially take place only in a squinted system. Below we compute the magnitude of thiseffect.

Effects on Image. Resolution in the time delay Δτ is related to range resolution by Δτ = 2ΔR/c. Conse-quently, to compute the shift in the image due to the time-delay error (33), we multiply (33) by c/2.

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The image is affected only when the error due to the start-stop approximation is greater than a resolutioncell, because in the backprojection process these contributions will not add correctly. It is well-known? that thesar cross-range resolution δ is determined by the aperture Δθ and center frequency λ0 as

δ =λ0

2Δθ. (35)

The angular aperture Δθ is determined by the antenna path during which data is collected. Most systems aredesigned so that their down-range and cross-range resolutions are roughly equal.

We consider the specific example of a low-earth-orbit satellite whose altitude is 300 km and whose velocityis 8 km/sec, traversing a straight flight path and forming a spotlight sar image of a scene 500 km from thesatellite. We consider only stationary targets because the speed of most moving targets would be insignificantrelative to the platform speed. We use the assumption γT

m = γRm.

We assume data collection starts at pulse m = 0 and ends at pulse m = M . We denote the angle to the scenecenter relative to the antenna velocity vector by θ0 for pulse m = 0, and θM when for pulse m = M . See Figure2.

Figure 2. Spotlight sar geometry

The duration of the data collection interval is denoted by T , which means for this case that the syntheticaperture is of length T |γ̇|, and the system transmits a pulse every T/(M − 1) seconds. To relate this pulserepetition interval, and hence the angular sampling interval, to the system’s cross-range resolution, we apply thelaw of sines to the triangle shown in Fig. 2:

sin Δθ

T |γ̇| =sin θ0

|γM | . (36)

Solving (36) for T , we find that

T =|γM | sin Δθ

|γ̇| sin θ0≈ |γM |λ0

2δ|γ̇| sin θ0, (37)

where we have used the small-Δθ approximation sinΔθ ≈ Δθ, which from (35) can be written in terms of thesystem resolution as λ0/(2δ).

In our simulations, we computed the error (33) for each pulse, plotted as a function of the resolution δ andthe starting angle θ0. In Figure 3, we show only the region in which the error is larger than a resolution cell.

We see from Figure 3 that for a high-resolution system, when the squint angle θ0 is large, the start-stopapproximation could lead to significant image distortion and smearing.

6. CONCLUSION

We see from (14) that the phase-space point-spread function is a weighted sum of ordinary radar ambiguityfunctions, evaluated at arguments that depend on the difference in positions and velocities.

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Figure 3. This shows the error as a function of squint and resolution for a target traveling 100km/hr in the directionopposite to the platform velocity (8 km/sec). We assume 3-cm wavelength (X-band).

ACKNOWLEDGMENTS

We are grateful to Air Force Office of Scientific Research∗ for supporting this work under the agreements FA9550-06-1-0017 and FA9550-09-1-0013.

REFERENCES1. W. C. Carrara, R. G. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Process-

ing Algorithms, Artech House, Boston, 1995.2. M. Cheney and B. Borden, “Imaging moving targets from scattered waves,” Inverse Problems, 24 (2008)

035005.3. M. Cheney and B. Borden, “Waveform-diverse moving-target spotlight synthetic-aperture radar,” preprint

(2010).4. J.R. Fienup, “Detecting moving targets in SAR imagery by focusing”, IEEE Trans. Aero. Electr. Systems

37 (July 2001) 794–809.5. S. Fridman and L.J. Nickisch, “SIFTER: Signal inversion for target extraction and registration”, Radio

Science 39 RS1S34 (2004) doi: 1029/2002RS002827.6. D.A. Garren, “Method and system for developing and using an image reconstruction algorithm for detecting

and imaging moving targets”, U.S Patent 7,456,780 B1, Nov. 25, 2008.7. Holston, Matthew E.; Minardi, Michael J.; Temple, Michael A.; Saville, Michael A., “Characterizing geolo-

cation ambiguity responses in synthetic aperture radar: ground moving target indication”, Algorithms forSynthetic Aperture Radar Imagery XIV. Edited by Zelnio, Edmund G.; Garber, Frederick D.. Proceedingsof the SPIE, Volume 6568, pp. 656809 (2007).

8. J.K. Jao, “Theory of Synthetic Aperture Radar Imaging of a Moving Target”, IEEE Trans. Geoscience andRemote Sensing 39 (September 2001) 1984 – 1992.

9. G. Kaiser, “Physical wavelets and radar: A Variational Approach to Remote Sensing”, IEEE Antennas andPropagation Magazine , February, 1996.

∗Consequently, the US Government is authorized to reproduce and distribute reprints for governmental purposesnotwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authorsand should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied,of the Air Force Research Laboratory or the US Government.

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10. R.P. Perry, R.C. DiPietro, and R.L. Fante, “SAR imaging of moving targets”, IEEE Trans. Aerospace andElectronic Systems 35 (1999) 188–200.

11. M. Pettersson, “Detection of moving targets in wideband SAR”, IEEE Trans. Aerospace and ElectronicSystems 40 (2004) 780–796.

12. M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms, Wiley, New York,1999.

13. D. Swick, “A Review of Wideband Ambiguity Functions,” Naval Research Laboratory Rep. 6994 (1969)14. S. Tsynkov, “On the use of the start-stop approximation for spaceborne sar”, SIAM J. Imaging Sciences 2

(2009) pp. 646 – 669.

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