Digital Beamforming for High Resolution Wide Swath SAR Imaging

Nicolas Gebert1, Gerhard Krieger2, Alberto Moreira3

German Aerospace Center (DLR), Microwaves and Radar Institute, P.O. Box 1116, 82234 Wessling, Germany 1Phone: +49 8153/28-3331, Email: nico.gebert@dlr.de, 2Phone: +49 8153/28-3054, Email: gerhard.krieger@dlr.de

3Phone: +49 8153/28-2305, Email: alberto.moreira@dlr.de

Abstract — Synthetic Aperture Radar (SAR) is a well-proved instrument for remote sensing. Multi-aperture SAR-systems allow for digital beamforming on receive that will enable a wide swath SAR with high azimuth resolution. In classic multi-aperture systems, stringent requirements are imposed on sensor velocity and PRF to ensure uniform sampling of the synthetic aperture. This paper shows that an unambiguous reconstruction of the SAR signal is also possible in case of such a non-optimum PRF. For this, an innovative reconstruction algorithm is derived, which en-ables a recovery of the unambiguous Doppler spectrum also in case of a non-uniform sampling of the synthetic aperture. Further, the influence of perturbations on the reconstruc-tion is presented and general system aspects are discussed.

I. INTRODUCTION

A. Synthetic Aperture Radar

Synthetic Aperture Radar (SAR) is an established re-mote sensing technique for high resolution imaging of the Earth surface. It consists of an air- or spaceborne side looking radar on a moving platform which takes advan-tage of the fact that a target contributes to the received signal as long as it is illuminated by the antenna footprint moving over it (cf. Fig. 1). Hence, a synthetic aperture of length Lsa is built by all the pulses which are backscat-tered and received coherently during the illumination time.

The characteristic of the SAR configuration is the time varying distance between sensor and target ( )tr while the platform moves over the synthetic aperture. For a linear flight track with constant velocity v in x-direction (so-called ‘azimuth’) this leads to the following expression for ( )tr in dependency of time t respectively the sensor position ( ) tvtx ⋅= and the minimum slant range distance R0 [1].

( )220

2220)( txRtvRtr +=⋅+= (1)

Considering the received signal sr(t), this leads to a de-lay caused by the 2-way propagation and an azimuth-modulation of the transmitted baseband signal st(t) de-pendent on ( )tr :

( ) ⎟⎠⎞

⎜⎝⎛ ⋅⋅⋅⎟

⎠⎞

⎜⎝⎛ ⋅−= )(4exp)(2 trj

ctrtsts tr λ

π (2)

Where c is the speed of light and λ the carrier wave-length. The phase term can be expressed using quadratic approximation as follows:

( ) ( ) 20

0

20 :

24 t

RtvRt az ⋅+=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⋅⋅

+⋅≈ ωϕλπϕ (3)

yielding the well-known chirp of the azimuth signal:

( ) ( ) ( )( )200 exp tjttsts aztr ⋅+⋅⋅−≈ ωϕ (4)

The bandwidth Baz of this azimuth modulation is de-temined by the geometry defining the maximum recep-tion time tmax, the sensors’ velocity v and antenna length La:

asaazaz

az Lv

vLtB ⋅

=⎟⎠

⎞⎜⎝

⎛⋅=⋅=22

2max π

ωπω (5)

Processing of the received chirp signal is done by autocorrelation with an expected impulse response. Thus the shorter the antenna is, the longer Lsa is and the more pulses are correlated and hence the better the resolution is.

B. High Resolution Wide-Swath Imaging

In conventional SAR the unambiguous swath width and the achievable azimuth resolution pose contradicting requirements on system design. As mentioned above a good azimuth resolution requires a short antenna for a long synthetic aperture which leads to a high azimuth- or Doppler-bandwidth Baz. Hence, a high pulse repetition frequency (PRF) is needed to sample the Doppler-spectrum without aliasing. If a large unambiguous swath is to be imaged, a low PRF and a long antenna in eleva-tion are favourable. This limitation can be expressed by the well-known minimum antenna area constraint deter-mining the lower bound for a full resolution SAR:

Fig. 1: SAR Geometry.

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cvRLWA m

aaa⋅⋅⋅⋅

>⋅=ηλ tan4

(6)

Where Rm is the medium slant range, η the incidence angle and Wa the antenna height [2].

To overcome this fundamental limitation inherent to spaceborne strip-map SAR, several innovative techniques have been suggested which are based on a splitting of the receiving antenna into multiple sub-apertures (Fig. 2, middle, [3]-[8]). This enables the simultaneous reception of the backscattered signal with individual receiver chan-nels. As shown in Fig. 2 on the left, the signal is mixed, digitized and stored by each receiver. Then, a posteriori, digital beamforming on receive is carried out by an ap-propriate combination of the sub-aperture signals. It is then possible to form multiple independent beams and to gather additional information about the direction of the scattered radar echoes.

The transmitter may be either on the same or on a dif-ferent platform. A further extension of this concept is the distribution of the receiver apertures on multiple plat-forms (Fig. 2, right, [9]-[11]) leading to a multistatic SAR where the size of each individual receiver is re-duced. The operation of such a multi-aperture system can be regarded as collecting additional azimuth samples while forming the synthetic aperture. These additional samples will then allow for a reduction of the PRF with-out an increase of azimuth ambiguities, thereby enabling the mapping of a wide image swath. In a classic DPC system, the PRF has to be chosen such that the SAR platform moves just one half of its antenna length be-tween subsequent radar pulses. This imposes a stringent requirement on the relation between PRF, v and the an-tenna length, and any deviation from this relation will result in a non-uniform sampling of the synthetic aperture which leads to additional ambiguities in the signal.

II. RECONSTRUCTION ALGORITHM

A. Theory

In the following it will be shown that an unambiguous reconstruction of the SAR signal is also possible in case of such a non-optimum PRF. For this, an innovative reconstruction algorithm is presented, which enables a

recovery of the unambiguous Doppler spectrum also in case of a non-uniform sampling of the synthetic aperture.

The reconstruction is based on a generalization of the sampling theorem according to which a band-limited signal )(tu is uniquely determined in terms of the sam-ples )(nThi of the responses )(thi of n linear systems with input )(tu , sampled at n1/ of the Nyquist fre-quency [12]. This is independent from the )(nThi as long as the samples do not coincide.

In our case, the functions hi correspond to the bistatic ‘channel’ between the transmitter and each receiver i and consist of the bistatic phase and the joint antenna pattern [14]. The received signals are sampled with PRF and hence the signal bandwidth can be up to PRFN ⋅ . Basi-cally, the )(thi provide the geometrical information of the configuration. A characterization of the whole system is given by the matrix H. It contains the frequency do-main representations )( fH i of all the reference func-tions )(thi shifted by multiples of the PRF.

Then, as shown in [13], the inversion of H yields in its columns a bandpass decomposition of the reconstruction filters Pi (cf. Fig. 3):

( ) ( )( ) ( )

( )( ) ( )( )⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅−+⋅−+

++=

PRFnfHPRFnfH

PRFfHPRFfHfHfH

n

n

n

111

1

1

L

MOM

L

L

H(f) (7)

( ) ( ) ( )( )( ) ( ) ( )( )

( ) ( ) ( )( )⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−++

−++−++

=−

PRFnfPPRFfPfP

PRFnfPPRFfPfPPRFnfPPRFfPfP

nnnn

n

n

1

11

21

22221

11211

L

MOMM

L

L

(f)H 1 (8)

The suppression of ambiguities is achieved by filtering each channel with its appropriate filter Pi and subsequent coherent combination of all receiver channels. Fig. 3 shows the linear system model for the signal reconstruc-tion from a multi-aperture system with N=3 receivers.

The derived algorithm has a great potential for any multi-aperture system, be it a distributed SAR with mul-tiple satellites or a single platform system relying on the displaced phase centre (DPC) technique, like the high resolution wide swath (HRWS) SAR [4] or the dual re-ceive antenna approach with TerraSAR-X [8].

Tx Rx array

v

Tx Rx array

v

Rx TxvRx Txv

Fig. 2: Digital beamforming on receive principle (left). DPCA configuration (middle). Sparse satellite array (right).

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B. Quadratic Approximation

In principle, it is possible to use the multi-channel SAR signal model for a complete reconstruction of the scene reflectivity. However, in order to concentrate on the essential steps, we will only consider the azimuth modulation in the following derivation. The azimuth signal of a point target at azimuth time 0t = and slant range 0R for each individual receiver channel at position

ix∆ relative to the transmitter may be described as

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ∆−+++−=∆ 22

022

02exp);( iii xvtRvtRjxthλπ (9)

when neglecting the envelope of the azimuth signal. The phase of the azimuth signal is proportional to the sum of the transmit and receive paths which are here approximated by the two square roots assuming a straight flight path with velocity v . A further simplification arises if we expand (9) in a Taylor series up to the second order which will lead to the quadratic approximation:

( )( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ∆−−⋅

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⋅∆

−⋅⎟⎠

⎞⎜⎝

⎛−≅∆

0

22

0

20

22exp

2exp4exp);(

Rvxtv

j

RxjRjxth

i

iii

λπ

λπ

λπ

(10)

Here, the first exponential describes a constant phase offset for a given slant range 0R which is equal for all receivers while the second exponential accounts for an additional constant phase offset which is due to the different along-track displacements between each individual receiver and the transmitter. The time-varying azimuth modulation of the bistatic SAR is then given by the third exponential. By comparing (10) with the point target response of a monostatic SAR:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⋅−⋅⎟

⎠

⎞⎜⎝

⎛−≅∆0

220

2exp4exp);(Rtv

jRjxth ii λπ

λπ (11)

it becomes clear that the bistatic azimuth response evolves from its monostatic counterpart by a time delay

v2xt i∆∆ = and a phase shift 02 2 Rxi λπϕ ∆−=∆ .

Therefore, the multiple aperture system can be regarded as a monostatic SAR which is followed by additional time and phase shifts for each receiver channel.

C. Simulation Results

As demonstrated in more detail in [14] the algorithm has been tested with real SAR data acquired by the DLR E-SAR system and lead to accurate reconstruction re-sults. For this, E-SAR data was low-pass filtered to ob-tain data over-sampled by a factor of 10. Then, two am-biguous ‘channels’ were generated, by taking every 20-th sample of the data. The offset between the data taken for two channels was only one sample and hence the result-ing overall sampling was non-uniform. Fig.4 shows on the left the ambiguous image if only one sub-sampled channel is used. Fig.4 on the right presents the result after combining the two channels to the unambiguous image. Further, simulations for a TerraSAR X similar scenario were carried out and lead to promising results [15].

III. SENSITIVITY AGAINST PERTURBATIONS

As mentioned before, the reconstruction in an ideal system is independent from the configuration as long as there are no identical samples. In a real scenario, where perturbations like noise and jitter are taken into account, the configuration which determines the effective sample positions has to be considered. In general, one can say that the sensitivity will increase the higher the actual sampling deviates from the case of uniform sampling.

For the simple case, where only white Gaussian noise is added to the signal in every receiver channel, the sup-pression of the ambiguities is not affected, but the noise floor after the coherent combination rises, the higher the non-uniformity of the samples is. In a first approxima-tion, the noise generated by the amplifiers of the receive modules can be approximated as such white noise. The resulting linear SNR at the output can then be expressed as a function of the N filters Pi and the respective SNRi at the input of the respective filter as follows:

( )fH 1

( )fH 2

( )fH 3

)(tu( )fP1

( )fP3

)( fU

(Bandwidth B)

3BPRF =

f

f

f

( )fP11

( )fP12

( )fP13

-B/2 B/2

( )fP2

( )fH 1

( )fH 2

( )fH 3

)(tu( )fP1

( )fP3

)( fU

(Bandwidth B)

3BPRF =

f

f

f

( )fP11

( )fP12

( )fP13

-B/2 B/2

( )fP2

Fig. 3: Left: Reconstruction for multi-channel subsampling in case of three channels. Right: Each reconstruction filter Pi consists of N bandpass filters Pij.

Fig. 4: Ambiguous image for one channel (left) and reconstruc-tion after non-uniform DPC sampling (right).

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1

1 ,

2 −

= ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⋅⎥⎦⎤

⎢⎣⎡

= ∑N

i iin

iout SNRN

PESNR , (12)

The subsequent plot gives an example for N=2 receiv-ers. It shows the dependency of the SNRout on the relative position of receiver 2 measured in percent of uniform distance. 100% of uniform distance correspond to a uni-form sampling while 0% represent identical receiver positions. The values are normalized to the optimum SNRout achieved for uniform sampling.

The impact of jitter that can be caused by synchroniza-tion errors between the receivers, oscillator phase noise or errors when estimating the reference functions )(nThi , is different. First investigations of the effect when )(nThi are subject to estimation errors were carried out and showed that the resulting error affects the correct recon-struction and additional azimuth ambiguities occur. The height of these ambiguities rises with increasing non-uniformity.

IV. SYSTEM ASPECTS AND HARDWARE REQUIREMENTS

In the case of the DPCA approach using a single plat-form, the receiver hardware becomes different from a conventional SAR system as digital beamforming on receive requires a completely independent operation of all elements in the receiver chain consisting of the an-tenna aperture elements, the low-noise amplifier, the demodulation and the final A/D conversion. Note that the suggested DPCA configurations support a complete sepa-ration between the transmit and receive chains. This offers the possibility to optimise the transmit and receive antennas separately and enables the use of innovative technologies like an integration of highly sensitive re-ceive-only modules directly into the antenna elements. Another challenge is the large amount of data that rises with the number of receiver channels. This will require either broadband data links or the development of effec-tive quantization and compression techniques to reduce the data volume without a significant loss of information. In a sparse array configuration, there is the need for mul-tiple small satellites which have to fly in a close forma-tion. Major challenges in such a configuration are precise orbit control, accurate baseline determination, and oscil-lator synchronisation.

V. SUMMARY & POTENTIALS

We have shown the benefit of multi-aperture SAR sys-tems for high-resolution wide swath imaging. An innova-tive reconstruction algorithm was presented that allows for an unambiguous recovery of the azimuth spectrum even in the case of non-uniformly sampled data. The influence of noise on the reconstruction was demon-strated and an expression was given, how the noise floor rises when the algorithm is applied in the presence of white Gaussian noise. Finally, general system aspects were discussed.

Besides, such systems offer a wide variety of addi-tional opportunities if compared to conventional monostatic SAR systems. One example is the presence of one or even more baselines allowing for interferometry, be it along-track to detect ground moving targets or be it cross-track to measure terrain heights. If more than one baseline is available, multi-baseline techniques can be used to increase the unambiguous range of the measured data. A challenge for the future is the combination of all these linear and nonlinear imaging techniques in a highly reconfigurable remote sensing instrument for a broad range of powerful applications.

REFERENCES

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[6] M. Younis, C. Fischer, W. Wiesbeck, Digital Beamforming in SAR systems, IEEE Trans. Geosci. Remote Sensing 41 (7), pp. 1735-1739, 2003

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[13] Brown, Multi-Channel Sampling of Low-Pass Signals, IEEE Transactions on Circuits and Systems 28 (2) , pp.101-106, 1981

[14] G. Krieger, N. Gebert, A. Moreira, Unambiguous SAR Signal Reconstruction from Non-Uniform Displaced Phase Centre Sampling, IEEE Geoscience and Remote Sensing Letters, Vol. 1, No. 4, October 2004

[15] G. Krieger, N. Gebert, A. Moreira, Unambiguous SAR Signal Reconstruction from Non-Uniform Displaced Phase Center Sampling, IGARSS 2004

Fig. 5: Relative degradation of SNRout for reconstruction in the presence of noise dependent on sample position.

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