Velocity Estimation of Fast Moving Targets Using Undersampled SAR

VELOCITY ESTIMATION OF FAST MOVING TARGETS USING UNDERSAMPLED

SAR RAW-DATA

Paulo A. C. Marques JoseM.B.Dias´

ISEL-DEEC IST-IT R. Conselheiro Em´ıdio Navarro 1, Torre Norte, Piso 10, Av. Rovisco Pais, 1949-014 Lisboa Portugal 1049-001 Lisboa Portugal E-mail: [email protected] E-mail: [email protected]

ABSTRACT PRF needs a non-conventional pulse scheduling. Moreover, The paper presents a new methodology to retrieve slant- non-uniform sampling introduces higher complexity in im- range velocity estimates for moving targets which induce age reconstruction. Using typical SAR mission parameters, a Dopler-shift beyond the Nyquist limit determined by the a single sensor, and uniform pulse scheduling, we readily Pulse Repetition Frequency (PRF). The proposed scheme conclude that the maximum unambiguous range velocity is takes advantage of the fact that the range velocity of a usually very small [7]. moving target induces a Doppler-shift in the azimuth spec- The approach herein proposed to estimate the range ve- tra which depends linearly on the fast-time frequency. We locity of moving targets with velocities above the maximum present results that take real and simulated SAR data. imposed by the PRF is based on the knowledge that the Doppler-shift in the azimuth spectra depends linearly on the 1. INTRODUCTION radar fast-time frequency; i.e., the Doppler-shift varies with

k proportionally to the true target range velocity. In the two It is well known that a moving target induces a Doppler-shift dimensional frequency domain, a moving target return will and a Doppler-spread on the returned signal in the slow-time exhibit a slope which is not subject to PRF limitations. We frequency domain [1]. Most of the techniques proposed in will present a methodology to retrieve the linear dependence recent literature take advantage of this knowledge to retrieve of the Doppler-shift in the azimuth dimension with the fast- the moving target image and velocity parameters [2],[3],[4]. time frequency, thus computing an unaliased estimate of the The azimuth velocity of a moving target is the responsible moving target range velocity. for the spread in the slow-time frequency domain whereas The developed methodology is not intended to achieve the range velocity induces the Doppler-shift. Given a PRF, high accuracy on the range velocity estimation. Rather, it the Doppler-shift is conﬁned to, is designed to retrieve the azimuth spectral support where

the Doppler-shift belongs. This information is crucial to re-

PRF PRF

f ;

D (1) trieve the range velocity with high accuracy (see, e.g., [2]).

2 2

f = 2v = r

where D is azimuth Doppler-shift induced by v a moving target with range velocity r , when the car- 2. UNAMBIGUOUS DOPPLER-SHIFT

rier wavelength is . If the signal is aliased (the induced ESTIMATION 2

Doppler-shift exceeds PRF= ) it has been generally ac-

Ak ;k cepted that the true moving target range velocity cannot In [2] we have shown that the returned echo u from

be uniquely determined using a single antenna and a sin- a moving target takes, in the slow-time frequency domain

k g gle pulse scheduling [5],[6]. The traditional solution to re- u the shape of the antenna radiation pattern in the cross- solve such targets consists in increasing the PRF [5], or range direction,

alternatively, in using a non-uniform PRF as proposed in

[6]. PRF increasing leads to a decrease in the maximum un- 1

Ak ;k / g k 2k ; u

u (2) ambiguous range swath, besides the huge memory require- 2

ments to store the received signal. The use of a non-uniform

k = 2= where is the wavenumber corresponding to the

THIS WORK WAS SUPPORTED BY THE FUNDAC¸ AO˜ g

PARAACIENCIAˆ E TECNOLOGIA, UNDER THE PROJECT fast-time frequency domain f .Theshape becomes shifted

= v =V

POSI/34071/CPS/2000. proportionally to and expanded by . Symbol r

k u Table 1. Mission parameters used in simulation.

ku end Parameter Value ku=2k Carrier frequency 10GHz k Chirp bandwidth 250MHz ustart Altitude 10Km Velocity 637Km/h

k 0 Look angle 20 kmin kmax Antenna radiation pattern Raised Cosine Oversampling factor 1.5 Fig. 1. Returned signal from a moving target with relative

range velocity .

k = k

The signal from a moving target for a ﬁxed 0 ,

S k ;k k = k +k 0 exhibit high correlation with u for .

denotes the moving target relative range velocity with re-

k =2k

The correlation will have a maximum at u

V =1+v =V

spect to the sensor velocity ,and a ,where (shown in appendix).

v k

a is the target cross-range speed. Usually is regarded as

=2=

k From the previous statements, one can intuitively estab-

0 0 a constant and equal to 0 where is the carrier

wavelength. In this work we drop this assumption. If the lish the following methodology to achieve an estimate of : k transmitted pulse has bandwidth B ,then is conﬁned to

Estimate the location of the moving targets using one B

B of the methodologies proposed in [2], [4] or [9].

k = + k k k + = k ;

min 0 max

0 (3)

c c

Process the SAR raw-data as if there were only static

where c is the propagation speed. For a moving target with targets. The moving targets will appear smeared and

relative range velocity , one may expect from equation (2), defocused.

S k ;k 2 that the returned signal u will exhibit a slope of

For each moving target: along the k axis, as illustrated in Fig. 1, in the absence of any other returns. We can readily conclude that – Digital spotlight the moving target image in the

spatial domain and re-synthesize its signature

k =2k k ; k

u max min

u (4)

star t

end

k ;k

back to the u domain as described in [7];

R k

k k SS

– Compute the correlation u between

0 u

where u and are the measured Doppler-shifts at

star t

end

S k ;k S k ;k

k k

u 0 u

0 and for all the transmitted min

fast-time frequencies max and , respectively. In the k

k pulse bandwidth; u

absence of noise, u and could be inferred using

star t end a simple centroid technique. In this situation the relative – Perform a linear regression on the maximum val-

velocity would thus be estimated as

R k k SS

ues of u along axis to estimate and 0

subsequently retrieve the true target range veloc-

b b

k k

u u star t

end ity.

b =

; (5)

2k k

max min and would not be restricted to the maximum value imposed 3. SIMULATION RESULTS

by the PRF. In real situations the returned signal from a moving target In this section we show some results obtained using the is superimposed on the clutter returns, making impossible proposed methodology. Figure 2 shows a real SAR im- the use of a simple spectral centroid estimator. A scheme age where a simulated moving target signature was super-

based on estimation theory is not easy to derive because we imposed. The moving target is composed of four reﬂectors.

2m do not have any information about the signal to use as ref- The total dimension is 2 , and the target is moving erence. However, the following facts apply: with range speed 6.3 times the maximum allowed by the PRF (the mission parameters are presented in Table 1). In

Assuming that the number of static scatterers is large, Fig. 3a), the corresponding magnitude image is presented

k ;k

none is predominant, and that they are uniformly dis- in the spatial frequency domain u . The returns from

tributed within a wavelength, then the correlation of the static ground behave as noise. In the frequency inter-

k ;k k 2 [2; 3]r ad=m u the static ground returns, in the u domain decays val we can see part of the moving tar- very quickly [8]; get signature, which is very weak compared with the static

spatial frequency domain Result of proposed correlation 50 -3 -3

Moving target 100 -2 -2

150 -1 -1 200

k 0 k 0 250 u u

300 1 1 Cross-range [m]

350 2 2

400 3 3 450 208 210 212 208 210 212 a) k b) k 500 20 40 60 80 100 120 Range [m] Fig. 3. On the left: Spatial frequency magnitude image of the SAR region which contains a moving target. On Fig. 2. Reconstructed SAR image which contains a moving the right: Result of the proposed correlation. The maxima target with range velocity 6.3 times the maximum imposed clearly exhibits a slope due to the moving target range ve- by the mission PRF. locity.

ground (SNR=-4.7dB). By taking as reference the signal 4. CONCLUSIONS

S k ;k = 209:4 0 u and performing the correlation proposed in the previous section, we obtain the result illustrated in

Fig. 3b). This ﬁgure clearly displays a maximum whose

k k

u coordinate varies linearly along axis. Performing a

k; k

linear regression on the pairs u corresponding to the

= 0:0463

referred maxima we obtain ^ (the true value is

=0:0472 0:57k m=h ), which corresponds to an error of

In this work we have shown that it is possible to estimate the K m=h (the object is moving with range velocity of 30 ). Doppler-shift induced by a moving target using undersampled SAR raw-data in the azimuth spectra. We exploited the linear dependency of the Doppler-shift on the fast-time Fig. 4 plots Monte Carlo results for an object with ve- frequency. We have shown that although the static ground locity which goes up to 15 times the maximum imposed by returns are incorrelated in the frequency domain, the same

the mission PRF. The number of runs is 50 and the SNR is is not true for a moving target. Using this knowledge we de-

6dB . The achieved results are good for most of the ap- veloped a simple scheme to retrieve the true Doppler-shift plications and enable us to retrieve the frequency interval induced by a moving target. Good results were shown using where the Doppler-shift belongs. More accurate methods a combination of real and simulated SAR data for moving can afterwards be used to retrieve the residual velocity in- objects with range speeds up to 15 times above the maxi- side this interval such as [2]. mum imposed by the used Pulse Repetition Frequency.

Range velocity estimate standard deviation In (7) a correlation between the antenna pattern

Ak ;k; 2k

u and the same function delayed by is eas-

1.4 0

2k k

X j

4k +k

ily recognized, although it is modulated by e .

1.3 0

2k k

X <

If the phase 2 one should expect that

4k +k

1.2 0

k; k +k; k R

S exhibits a maximum which is linearly

m u

1.1 dependent of the fast-time frequency by a factor equal to k 2 . This phase value can be made negligible by consid-

k k

ering the typical values of u and and that we can com-

pensate the dependency of X by using the knowledge of the 0.9 target area approximate range coordinates. Range velocity standard deviation [Km/h] 0.8 2 4 6 8 10 12 14 fd / (2PRF) 5. REFERENCES

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SS m u u u

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