study on the processing scheme for space–time waveform encoding sar system based on...

910 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 3, MARCH 2012

Study on the Processing Scheme for Space–TimeWaveform Encoding SAR System Based on

Two-Dimensional Digital BeamformingFan Feng, Shiqiang Li, Weidong Yu, Member, IEEE, and Shuo Wang

Abstract—The combination of space–time waveform encodingand digital beamforming (DBF) has been proposed as a novelconcept to improve the performance of synthetic aperture radar(SAR) systems in the future. In this paper, we research one suchnew operational mode to reduce azimuth ambiguity in high-resolution wide-swath SAR image and present the processingscheme based on 2-D DBF for this mode. The main proceduresand techniques are described in detail, and a complete mathe-matical derivation of the scheme is given; furthermore, a sampleSAR system is provided, from which numerical simulation resultsare obtained to justify our derivations. In addition, an in-depthanalysis of system performance associated with this mode, fromthe perspectives of both azimuth ambiguity-to-signal ratio (ASR)(AASR) and range ASR (RASR), is carried out. It is shown thata meaningful improvement of AASR can be attained at a smallcost of raised RASR level, as compared with the performance ofsingle-input multiple-output SAR.

Index Terms—Ambiguity suppression, digital beamforming(DBF), displaced phase center antenna (DPCA), high-resolution wide-swath (HRWS), single-input multiple-output(SIMO), space–time waveform encoding (STWE), syntheticaperture radar (SAR).

I. INTRODUCTION

IN THE remote sensing application of spaceborne syntheticaperture radar (SAR), wide swath coverage and high geo-

metric resolution are two desirable merits. However, due to thecontradicting requirements on pulse repetition frequency (PRF)[1], [2], these two demands cannot be met simultaneously.Tradeoff has to be made between spatial coverage and azimuthresolution in a conventional SAR system. On the one hand,one can achieve wide-swath imaging with ScanSAR [3], [4]or TOPSAR [5]–[7] mode at the expense of a coarse azimuthresolution. On the other hand, high azimuth resolution canbe obtained by Spotlight mode [8] which steers the beam toilluminate a specific area all the time, but it suffers the drawbackof noncontiguous coverage along the track.

Manuscript received July 15, 2010; revised December 7, 2010, February 27,2011, April 19, 2011, and June 13, 2011; accepted July 3, 2011. Date ofpublication August 18, 2011; date of current version February 24, 2012.

The authors are with the Department of Space Microwave Remote Sens-ing System, Institute of Electronics, Chinese Academy of Science, Beijing100190, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2011.2162097

To overcome this inherent limitation, several innovativemethods, such as displaced phase center antenna [9], high-resolution wide-swath (HRWS) [10], [11], and multiaperturereconstruction (MAR) [12]–[15] have been put forward aspromising candidates to resolve this problem. The basic ideabehind these modes is to collect additional raw data samplesfor each transmitted pulse by mutually displaced receivingapertures in along-track direction. All these proposals employ asmall fixed aperture to emit pulse along a wide beam in azimuthdirection. The scattered radar echoes are recorded by multiplesubapertures, and then, a posteriori processing is carried outto reconstruct the unambiguous Doppler spectrum. Comparedwith traditional monoaperture systems, all of these techniquesallow for an improved azimuth resolution or a reduction of PRFwithout an increase of azimuth ambiguities, thereby enablingthe mapping of wide swath.

In order to further improve the performance of spaceborneSAR systems, a completely novel concept—multidimensionalwaveform encoding, which indicates that the transmit wave-form should be encoded spatiotemporally and hence becomea joint function of space and time variables, had been advo-cated in [16]–[20]. More recently, a similar concept, referredto as beamspace multiple-input multiple-output (MIMO), wasproposed in [21] as a potential method to reduce clutter inthe application of ground moving target indication. The wave-form encoding will afford waveform diversity on transmit, andthus, better system performance can be obtained, provided thatappropriate processing techniques are applied on receive toexploit the waveform diversity effectively. In [20], a variety ofimplementations of this new concept were conceived to achievedifferent goals. Among them, there was one mode that was torealize the azimuth ambiguity reduction in the HRWS SARimaging. A possible idea for handling the echo data of thismode had been also suggested.

In this paper, we develop this idea into a complete process-ing scheme. The scheme primarily consists of two parts: rawdata preprocessing and conventional SAR imaging processing.Specifically, first-order azimuth ambiguous components, whichaccount for the largest part of the overall azimuth ambiguitypower, will be filtered out by 2-D digital beamforming (DBF) inthe preprocessing stage, and then, existing imaging algorithms,such as chirp scaling [22] and extended chirp scaling [23],can be directly performed on the resulting data to obtain theSAR image of high quality. The focus of our work is on theformer part of the scheme, and all mathematical derivations

0196-2892/$26.00 © 2011 IEEE

FENG et al.: PROCESSING SCHEME FOR STWE SAR SYSTEM BASED ON TWO-DIMENSIONAL DBF 911

Fig. 1. Time–frequency structures of waveforms in (a) STWE SAR and(b) conventional SAR.

in this part will be obtained and provided. In addition, wewill deeply analyze both the azimuth ambiguity-to-signal ratio(ASR) (AASR) and range ASR (RASR) of this operationalmode and derive the explicit expressions for them.

This paper is organized as follows. In Section II, we brieflyrecall the system concept of this space–time waveform encod-ing (STWE) SAR mode already introduced in [20] and alsodiscuss some issues about the implementation of waveformencoding on transmit with planar phased-array antenna andgive the whole processing chain for STWE SAR in the end.Section III describes the algorithm of DBF in elevation toensure a reliable separation between echoes from differentsubpulses. Azimuth beamformers for these separated echoeswill be computed and given in Section IV. In Section V, anexemplary SAR system is presented and simulation results aregiven to confirm the effectiveness of the derived 2-D DBFalgorithm. Section VI is dedicated to the investigation of per-formances in AASR and RASR, as well as of the transmittedpower associated with STWE SAR, and makes a comparisonbetween them and their counterparts in SIMO SAR to betterillustrate the obtained benefits. This paper concludes with ashort summary in Section VII.

II. STWE SAR CONCEPT

In contrast to the traditional space–time coding concept inthe application of wireless communications [24], STWE in thispaper is the combination of subpulsing techniques with multipletransmit subbeams. In this section, we will mainly describe thetransmission mode of STWE SAR and formulate the explicitexpression for the waveform function; also, some issues aboutits realization are discussed, and main steps of the processingscheme will be presented with illustration at the end of thissection.

A. Spatiotemporal Structure of Transmit Waveform

In STWE SAR, a complete radar pulse consists of Mtemporally adjacent linear FM subpulses with the same car-rier frequency, FM rate, and bandwidth. Consider an examplewith three subpulses (M = 3). Fig. 1 shows the differencebetween the time–frequency structure of the waveform adoptedin STWE SAR and that used in conventional SAR system.

As shown in Fig. 1(a), each of these subpulses can be consid-ered as a time-shifted version of another one, and the transmitinterval between them equals the duration of the subpulse.

Fig. 2. STWE on transmit with three chirp subpulses.

Therefore, the complex temporal form of the transmit signalis given by

s(τ) = rect

(τ + T

T

)exp(j2πfc(τ + T ) + jπKr(τ + T )2

)

+ rect( τT

)exp(j2πfcτ + jπKrτ

2)+rect

(τ − T

T

)

× exp(j2πfc(τ − T ) + jπKr(τ − T )2

)(1)

where τ is range time variable, T is the subpulse duration, fc isthe carrier frequency, and Kr is the FM rate.

During the transmission phase, three contiguous azimuthbeams are formed and switched in the transmitter from sub-pulse to subpulse. Each subbeam covers a portion of the fullilluminated azimuth footprint. During the first subpulse, allphase shifters are steered toward the forward beam position,and the entire antenna array is used to coherently transmitsubpulse1 in that direction. During the second subpulse, thephase shifters are reset to point toward the middle beam positionand subpulse2 is transmitted. The third subpulse will then beemitted along the third subbeam that points backward.

Fig. 2 shows the waveform encoding mode described pre-viously in both azimuth and elevation. The azimuth antennapattern for each subbeam can be approximated by sinc(x),where the beamwidth θsub corresponds to the overall lengthof the antenna array La. Consequently, one can obtain theapproximate results of θsub

θsub ≈ λ

La(2)

and azimuth squint angles of three transmit subbeams

θ1 ≈ λ

La(3a)

θ2 ≈ 0 (3b)

θ3 ≈ − λ

La(3c)

with λ being the wavelength.In the elevation direction, all these subbeams, however, illu-

minate the same region. The beamwidth φe in this dimension isdictated by the swath width required in the mission, as shownin Fig. 2(b).


Fig. 3. (a) Different weighting phase ramps in azimuth and (b) corresponding subbeams.

Fig. 4. Block diagram of the processing scheme for STWE SAR.

The final expression of the waveform that features thespatiotemporal characteristics of the transmit pulse is thenobtained as

s(τ, θ) = rect

(τ + T

T

)× exp

[j2πfc(τ + T ) + jπKr(τ + T )2

]· sinc

(La

λ(θ − θc)

)+ rect

( τT

)exp(j2πfcτ + jπKrτ

2)

· sinc(La

λθ

)+ rect

(τ − T

T

)× exp

[j2πfc(τ − T ) + jπKr(τ − T )2

]· sinc

(La

λ(θ + θc)

)(4)

where θ denotes the azimuth squint angle variable measuredfrom the boresight in the slant range plane and θc is themagnitude of θ1 and θ3. From (4), we can see that the transmitwaveform is a joint function of time variable τ and anglevariable θ.

B. Waveform Encoding Implementation by ESA

Current electronically scanned array (ESA) technology givesthe system designer almost complete control of the radar an-tenna pattern and provides rapid beam agility. The increasinguse of DBF on transmit can vary the antenna characteristics

flexibly, so as to enhance the system performance. Consider theaforementioned example, the waveform encoding for transmit-ted pulse is achieved by switching between three linear phaseramps in the antenna excitation pattern. As the beam pattern ofeach subbeam is an approximate sinc(x) function, the weightcoefficient on the lth transmit antenna element is given by

wl = exp

[−j2π

dlλ

· sin(θ)]

(5)

with dl being the space between the lth transmit element and thereference element (e.g., center element of the array in azimuth)and θ being the squint angle of the subbeam. Taking a 15-m-long antenna as an example, Fig. 3 shows different phase rampsin azimuth and correspondingly formed subbeams. Note thatthe phase of wl is proportional to dl, where the phase slopechanges from subbeam to subbeam.

C. Main Procedures of the Processing Scheme for STWE SAR

Fig. 4 shows the processing chain for STWE SAR, where thedashed square and the dashed-and-dotted square are utilized todenote preprocessing part and imaging processing part, respec-tively, in the entire processing scheme.

As can be seen, the echoes associated with different sub-pulses, which arrive at the receiver from different elevationangles simultaneously, should be first separated by DBF onelevation during the echo-receiving period. Then, these echoeswill be downlinked to the ground segment, where they would be


Fig. 5. Spatial separation of echoes from different subpulses.

passed through appropriate azimuth beamformers to completelyremove their first-order azimuth ambiguity components. Thelast step in the preprocessing stage is the coherent combinationof azimuth subbands to achieve the full Doppler bandwidth fora high azimuth resolution [20], as it will be better describedin Section IV. Finally, we can obtain the SAR image with aconventional imaging algorithm applied to the acquired echodata after preprocessing.

III. DBF IN ELEVATION

DBF in elevation has been previously suggested in [10] and[26] as a potential technique to compensate for the transmit gainloss at the swath border and to increase the system signal-to-noise ratio (SNR) in wide-swath SAR imaging. In that case, anarrow and high gain receive beam, which follows pulse echoon the ground, would be formed. In the STWE SAR case,however, the objective of DBF in elevation is to extract theecho of each subpulse from the overlapped received signals.In order to fulfill this requirement, the formed beams shouldhave not only the capability of echo tracking for each subpulsebut also the ability to reject interference from other subpulsessimultaneously. In the following text within this section, wewill develop the corresponding processing algorithm for DBFin elevation in STWE SAR.

A. Time-Varying Spatial Filtering in Elevation

Still consider the system model introduced earlier. Sincesubpulses within one complete radar pulse are transmitted insequence, the scattered signals from different subpulses willthen—at each instant of time—arrive from different elevationangles. Consequently, it is possible to separate radar echoesfrom adjacent subpulses spatially by forming three scanningbeams, each of which is related to one subpulse, during theecho-receiving period [20], as shown in Fig. 5.

In order to effectively separate these echoes, the elevationantenna pattern of each formed subbeam in Fig. 5 shouldhave maximum gain in the direction along which echoes ofcorresponding subpulse arrive and simultaneously deep nulls atthe positions where echoes from other two subpulses are from.

Thus, one can formulate this problem by a set of equations asfollows:

v∗mam =1 (6a)

v∗man =0 (6b)

where vm is the desired beamformer weight vector to form themth subbeam in elevation, the symbol (·)∗ denotes conjugatetranspose, and am and an are the receive steering vectorsrelated to the mth and nth subpulses, respectively. They aregiven by

am =

[1 exp

{j2π

λde sin(αm)

}. . .

exp

{j2π

λ(Ne − 1)de sin(αm)

}]T(7a)

an =

[1 exp

{j2π

λde sin(αn)

}. . .

exp

{j2π

λ(Ne − 1)de sin(αn)

}]T. (7b)

In (7), the symbol (·)T denotes transpose, de is the spacebetween two adjacent subapertures in elevation, Ne is thetotal number of subapertures in elevation, αm and αn are off-boresight angles along which echoes from the mth subpulse andthe nth subpulse arrive, respectively. Based upon the geometrymodel of spaceborne SAR [25] and the approximate relation-ship between off-boresight angle and echo time given in [26],these angles can be expressed in terms of echo time τ as

α1(τ)= arccos

(4 ·R2

orbit−4 ·R2E+(τ+ T )2c2

4 ·Rorbit · (τ+T ) · c

)−β (8a)

α2(τ)= arccos

(4 ·R2

orbit−4 ·R2E+τ2c2

4 ·Rorbit · τ · c

)−β (8b)

α3(τ)= arccos

(4 ·R2

orbit−4 ·R2E+(τ−T )2c2

4 ·Rorbit · (τ−T ) · c

)−β (8c)

where Rorbit is the radius of the orbit, RE is the radius of theEarth, β is the off-nadir angle of the antenna boresight, and c isthe speed of light.

Using matrix notation, one can rewrite (6) in an alternativeway as

v∗mA = eTm (9)

where em is the mth column of a 3 × 3 identity matrix andA is the receiving array matrix with columns being receivingsteering vectors as

A = [a1 a2 a3]. (10)

By using the matrix inversion lemma, the solution to (10) isfound to be

vm = A(A∗A)−1em. (11)


For notational convenience, we do not explicitly show thedependence of A and, consequently, vm on the argument τ in(10) and (11). vm is a time-varying beamformer weight vectordue to the variation of α as subpulses run over the groundsurface.

B. Phase Compensation in Range Frequency Domain

In order to obtain the required range and radiometric res-olution, a frequency-modulated transmit signal is typicallyused instead of a short pulse [see (1) and (4)]. Therefore,it is necessary to take into account the temporal extensionof the pulse, particularly for the chirp, in the beamformingmechanism. Since the off-boresight angles in (8) are actuallycalculated with reference to the center of the subpulse whichcorresponds to the central frequency of a chirp, the beam patternformed by (11) will have the maximum gain and/or deep nullsonly at the central position of these subpulse extents. The otherpart within each subpulse extent, however, will not be coveredwith the same receive gain as is the center of the subpulse. Inthe single-pulse case [26], the solution to this problem was toperform the corresponding time delay subsequently to phaseshifting so as to compensate for the loss of receive gain. In thissection, we will reconsider this issue for the case of multipulseson the ground, where not only receive gain compensation butalso interference suppression should be accounted for.

The received signals in STWE SAR are given by

sr(τ) = sr1(τ) + sr2(τ) + sr3(τ) (12)

where sr1(τ), sr2(τ), and sr3(τ) are the echo signals of sub-pulse1, subpulse2, and subpulse3, respectively. As the reflectedenergy at any illumination instant is the convolution of thepulse waveform and the ground reflectivity g(τ) within theilluminated patch, these signals can be expressed in an integralform as

sr1(τ) =

+T/2∫−T/2

exp{jπKrt

2}· g(τ + T + t)dt (13a)

sr2(τ) =

+T/2∫−T/2

exp{jπKrt2} · g(τ + t)dt (13b)

sr3(τ) =

+T/2∫−T/2

exp{jπKrt2} · g(τ − T + t)dt. (13c)

(In (13), we have neglected the carrier frequency termexp(j2πfct), as downconversion will be performed to move theecho signal from the high-frequency band to the baseband onceit has been received.)

For simplicity, we assume a homogeneous reflectivity sceneand g(τ) is assumed to be a complex constant C. Taking intoaccount the impact of beam pattern formed by (11) on thereceived signals, the echo signal from the nth subpulse, afterpassing through the mth beamformer [as shown in (11)], isgiven by

snm(τ) = C ·T/2∫

−T/2

exp{jπKrt2} · {v∗

mA(τ + t)en} dt

(14)

where A(τ + t) · en denotes receive steering vectors related todifferent positions within the extent of the nth subpulse.

Substituting (11) into (14) and with the definition of

qTm = eTm [A(τ)∗A(τ)]−1 (15)

rn =A(τ)∗A(τ + t)en (16)

(14) can be rewritten in a compact form as

snm(τ) = C ·T/2∫

−T/2

exp{jπKrt2} ·[qTmrn

]dt. (17)

At a given time instant τ , qm is a constant vector, whereasrni is a vector function of intrapulse time variable t whichvaries within [−T/2, T/2] (In order to make the analysis moreclearly and understandable, we employ the time variable t todenote the position relative to the center within the extent ofa subpulse). The existence of the term rn can be seen as themathematical explanation for the fact that other parts withinthe extent of a subpulse are not illuminated with the samereceiving gain as is the subpulse center. In order to derive thecompensation factor, we rewrite rn in its vector form as (18),shown at the bottom of the page.

From (18), one can see that if the termexp{j(2π/λ)(k − 1)de[sinαn(τ + t)− sinαn(τ)]} involvingthe variable t can be compensated, all parts within the extentof a subpulse would then obtain the uniform receive gain.Recalling the linear relationship between time and frequencyin a chirp, this compensation can be achieved by performingphase multiplications in frequency domain.

rn =

⎡⎢⎢⎢⎢⎢⎢⎣

Ne∑k=1

exp{j 2π

λ (k − 1)de [sinαn(τ + t)− sinαn(τ)]}exp{j 2π

λ (k − 1)de [sinαn(τ)− sinα1(τ)]}

Ne∑k=1

exp{j 2π



Ne∑k=1

exp{j 2π



⎤⎥⎥⎥⎥⎥⎥⎦

(18)


Fig. 6. Block diagram of DBF on elevation for separating simultaneously arriving echoes generated from three sequential subpulses. The symbol wmk denotesthe element of the matrix A∗, where subscripts signify its position within matrix. Dk represents the time delay for the kth subaperture as shown in (23), and Rn

corresponds to the nth row vector in matrix (A∗A)−1.

By combining the aforementioned phase term and the tempo-ral waveform of the subpulse shown in the integrand, we definea new function of t as

Ik(t) = exp{jπKrt2}

· exp{j2π

λ(k − 1)de [sinαn(τ + t)− sinαn(τ)]

}. (19)

Since the off-boresight angle α can be approximated as alinear function of echo time τ , we recast Ik(t) as

Ik(t)≈exp{jπKrt2} · exp

{j2π

λ(k − 1)de ·

∂α(τ)

∂τ

∣∣∣∣τ=τc

· t}

(20)

where ∂α(τ)/∂τ)|τ=τc is the value of the first-order derivativeof α with respect to τ at the swath center time τc.

Applying the Principle of Stationary Phase to (20), we obtainan approximate expression for the Fourier transform of Ik(t)with respect to its argument t given by

Ik(f) ≈ exp

{−j

π

Krf2

}

· exp{j2π

λKr(k − 1)de

∂α(τ)

∂τ

∣∣∣∣τ=τc

· f}. (21)

Comparing (19) and (21), we note that the termexp{j(2π/λ)(k−1)de[sinαn(τ+t)−sinαn(τ)]} in the timedomain has been transformed into the second term in (21).As a consequence, this unwanted term can be eliminated bymultiplying the spectrum of the received signal from the kthsubaperture in elevation by the linear phase function Hk(f)given by

Hk(f) = exp

{−j

2π

λKr(k − 1)de

∂α(τ)

∂τ

∣∣∣∣τ=τc

· f}. (22)

Fig. 7. (Solid) Multiple broad azimuth beams receiving the echoes of eachsubpulse transmitted by (dashed) a single narrow beam. (a), (b), and (c) corre-spond to the situations of subpulse1, subpulse2, and subpulse3, respectively.

In practice, the modification of the spectrum with a linearphase can be realized with a time delay of the signal adaptedfor each subaperture, and the delay quantity Dk for the kthsubaperture can be derived from (22) as

Dk =(k − 1) · de

λ ·Kr· ∂α(τ)

∂τ

∣∣∣∣τ=τc

. (23)

From the aforementioned analysis and the finally obtainedresults shown by (22) and (23), we can see that in the mul-tipulse model, the same time-delay module, as in the single-pulse model, should be adopted to approximately eliminate thevariation of the receive gain within the extent of a subpulse.The reason for this lies in the fact that these two models sharethe same pulse waveform (i.e., LFM pulse) and the same geom-etry model, namely, spherical earth model. Nevertheless, unlikethe DBF configuration in elevation presented in [26], null-steering modules are added in our DBF configuration to achievethe separation between echoes from different subpulses. InFig. 6, the whole block diagram of DBF on elevation for STWESAR is shown. After initial downconversion and digitization,the received signals are modified by phase shifting accordingto the matrix A∗. Then, the adapted time delay is performed to


Fig. 8. (Solid curved line) Transmit and receive antenna pattern for each subpulse. (a) (Asterisk) Subpulse1. (b) (Rectangle) Subpulse2. (c) (Triangle) Subpulse3.

correct the aforementioned phase shift for a different positionwithin the extent of the subpulse. The remaining steps willbe the same as in conventional null-steering beamforming,which keeps the signal of interest undistorted while suppressinginterferences substantially.

From the block diagram shown earlier, one can see that thetotal output data volume of a digital beamformer in elevation isthe sum of the data volume created by each subpulse scanningover the whole swath. In other words, the ratio of the input datavolume to the output data volume is equal to the ratio of thenumber of subapertures in elevation to the number of subpulses.Since the former is always several times larger than the latter(see Section V-A), the data downlink rate can thus be drasticallylowered.

Since the weighting coefficients shown in Fig. 6 are timevarying during the data take, DBF in elevation for STWESAR can be regarded as a real-time beamforming approach.In the ideal scenario, the sampled data and phase-shift valueshould have one-to-one correspondence between them. In somehigh-resolution applications where the phase shifter may notswitch as fast as the sampling rate of analog-to-digital converter(ADC), one can use one phase-shift value to weigh a setof successive sampled data as a makeshift. This method willunavoidably introduce phase errors, leading to paired echoesin range dimension after range compression. Nevertheless, themagnitude of these artifacts are actually extremely weak (if not

negligible) compared with the magnitude of real images; thus,tradeoff can be made between the system performance and therequirement on the hardware, e.g., the switching frequency ofthe phase shifter (detailed discussion on the issue of switchingfrequency of a phase shifter is given in the Appendix). Simula-tion results for DBF in elevation will be provided in Section Vto demonstrate the effectiveness of our method in separatingechoes from adjacent subpulses.

Next, after these separately stored echo data have beendownlinked to the ground, a posteriori multiaperture azimuthprocessing will then be carried out to suppress their correspond-ing first-order azimuth ambiguity.

IV. DBF IN AZIMUTH

In azimuth dimension, multiple subapertures are also em-ployed to receive pulse echoes. Therefore, for each output ofdigital beamformer in elevation in Fig. 6, it can be processed bya specific azimuth beamformer to steer nulls in the resultingjoint azimuth antenna pattern to the angles corresponding tothe ambiguous Doppler frequencies. Since each subaperturecorresponds to a broad Rx azimuth beam, it can be regarded asa mode of single narrow beam transmitting and multiple broadbeams receiving for each subpulse, as shown in Fig. 7.

Taking into account both transmit and receive patterns, weshow the corresponding Doppler subbands where the useful


Fig. 9. Coherent combination of azimuth spectrum associated with different subpulses after DBF processing.

signal and first-order azimuth ambiguities are located for eachsubpulse in Fig. 8. From this, one can observe that differentazimuth beamformer weight vectors should be applied to echosignals of different subpulses, since their azimuth spectrumcharacteristics are largely different.

In [12] and [27], a joint spatiotemporal azimuth pro-cessing of the recorded subaperture signals is performed tounambiguously recover the original azimuth spectrum of broadbandwidth. With a slight modification, we can adopt similarstrategies in our scheme to process these separated echoesaccording to their respective space–frequency spectrum shownin the upper part of Fig. 9 (Although a similar processingtechnique is employed, there is an important difference betweenazimuth beamforming in our scheme and that in [12]. This willbe discussed in detail in Section VI).

To get a compromise between system performance and com-putational complexity, we assume that the total antenna is splitinto three subapertures in azimuth direction. Thus, two extradegrees of freedom (DOFs) afforded by this azimuth config-uration can be used to suppress first-order azimuth ambiguitythat has been aliased into the [−PRF/2, PRF/2] range, whileretaining the useful signal without distortion (shown in Fig. 9).

For the mth subpulse echo, the corresponding azimuth beam-former weight vector pm, which needs to steer nulls in di-rections where its ambiguity components come from, can beobtained by solving the following matrix equation:

Hmpm = [0 1 0]T (24)

where Hm is the receive array manifold matrix with rowsbeing the steering vectors associated with signal and ambiguitycomponents. It can be expressed as (25), shown at the bottomof the next page. In (25), da denotes the length of azimuthsubaperture, θm−s is the view direction of the useful signal,and θm−amb−l and θm−amb−r represent the view directions offirst-order azimuth ambiguities on the left and right sides of theuseful signal, respectively.

Using the relation between azimuth squint angle θ andDoppler frequency fd as

fd =2v

λsin(θ) (26)

and the knowledge about frequency intervals where ambiguitiesand signals are located for different subpulses’ echoes, as given


TABLE IAZIMUTH CHARACTERISTICS FOR SUBPULSE1’S ECHO

TABLE IIAZIMUTH CHARACTERISTICS FOR SUBPULSE2’S ECHO

TABLE IIIAZIMUTH CHARACTERISTICS FOR SUBPULSE3’S ECHO

in Tables I–III, we replace the azimuth angle variable θ in(25) with the frequency variable fa; thus, azimuth beamformerweight vectors p1, p2, and p3 can be computed and expressedin terms of fa as follows:

p1 =

[exp {−jπ/v · da(fa + fdc)}2 [1− cos(π/v · da · PRF )]− cos(π/v · da · PRF )

1− cos(π/v · da · PRF )

exp {jπ/v · da(fa + fdc)}2 [1− cos(π/v · da · PRF )]

]T(27a)

p2 =

[exp{−jπ/v · da · fa}

2 [1− cos(π/v · da · PRF )]− cos(π/v · da · PRF )


exp{jπ/v · da · fa}2 [1− cos(π/v · da · PRF )]

]T(27b)

p3 =

[exp {−jπ/v · da(fa − fdc)}2 [1− cos(π/v · da · PRF )]− cos(π/v · da · PRF )


exp {jπ/v · da(fa − fdc)}2 [1− cos(π/v · da · PRF )]

]T(27c)

with v being the velocity of the platform and fdc being the mag-nitude of Doppler centroid associated with the squint transmitsubbeams.

Fig. 10. Block diagram of azimuth processing after the separation betweenechoes from different subpulses by DBF on elevation.

The overall procedure of azimuth processing is shown inFig. 10. It can be observed that proper time-shift compensationshould be first made for the echo of each subpulse, account-ing for the different time delays. Then, these echoes will beprocessed by their corresponding azimuth beamformers. Atlast, a coherent combination of these beamformer outputs isperformed in azimuth frequency domain to recover the fullDoppler spectrum for a high resolution. The obtained widespectrum will no longer suffer from the impacts of aliasingcaused by the first-order azimuth ambiguity, as shown by Fig. 9.Since the first-order ambiguities are primary contributors tothe total azimuth ambiguity power, this achieved result has animplication that the AASR will be sufficiently lowered in thefinal SAR image.

Hm =

⎛⎝ exp

{j 2π

λ da sin θm−amb−l

}1 exp

{−j 2π

λ da sin θm−amb−l

}exp{j 2π

λ da sin θm−s

}1 exp

{−j 2π

λ da sin θm−s

}exp{j 2π

λ da sin θm−amb−r

}1 exp

{−j 2π

λ da sin θm−amb−r

}⎞⎠ (25)


V. SYSTEM DESIGN AND SIMULATION RESULTS

In this section, we will present a system design exampleof spaceborne X-band STWE SAR, with swath coverage over100 km and azimuth resolution less than 3 m. Using the systemparameters in this example, numerical simulation results willthen be obtained and given to demonstrate the effectivenessof our preprocessing techniques in both elevation and azimuthdimensions.

A. System Design Example

Consider a spaceborne HRWS SAR system at a mean orbitheight of 576 km, which entails a sensor velocity of 7600 m.To be consistent with the number of subpulses and azimuthsubapertures assumed in the preceding sections, the system isstill configured such that the total antenna array is divided intoNa = 3 subapertures in azimuth. One radar pulse is comprisedof M = 3 adjacent subpulses, each of which has a durationof T = 50 μs and bandwidth of B = 120 MHz to ensure aground range resolution of 2.5 m at the swath center. Regardingthe azimuth geometric resolution, the total processed Dopplerbandwidth BDs should be on the order of ∼2700 Hz to achievethe value of 2.8 m. As the whole azimuth spectrum is acquiredby combining three Doppler subbands corresponding to threetransmit subbeams, the bandwidth of each subbeam shouldbe 900 Hz approximately, indicating that the azimuth lengthof the antenna has a value of about 15 m. Furthermore, theminimum PRF of the system should satisfy the inequalityNa · PRFmin ≥ BD to ensure that the echo signals associatedwith each subbeam can be sampled adequately in azimuthdimension.

In the following, we select the area with the off-nadir angleranging from 20◦ to 29.1◦ for wide-swath imaging, where theslant range distances at the near and far edges of the swath are616.69 and 668.71 km, respectively. Therefore, the upper boundof the system PRF is determined by

PRFmax =

[2(Rf −Rn)

c+M · T

]−1

(28)

with Rf being the far range distance and Rn being the nearrange distance. In the scope of [PRFmin,PRFmax], a PRFranging from 1025 to 1075 Hz can be chosen to ensure theground coverage with a width of 115 km, as shown by thetiming diagram in Fig. 11.

In elevation dimension, the beamwidth of each receive sub-beam, which is inversely proportional to the total height of theantenna h, should be less than the value of the angle extendedby the covered area of a subpulse, thereby ensuring a reliableseparation between echoes from adjacent subpulses. This re-lationship can be mathematically expressed by the followinginequality:

λ

h≤ c · T

2 · tan(θinc,f ) ·Rf(29)

where θinc,f is the incident angle at the far edge of the swath.In this system example, the height of an optimally orientedantenna should be larger than ∼2.3 m. As the scan angle of

Fig. 11. Timing diagram of the exemplary SAR system, taking into accountboth the (solid) transmit events and the (dashed) nadir echo.

Fig. 12. Antenna architecture of beam-space MIMO SAR system, where theshaded area is for transmission.

three elevation subbeams is up to ±4.5◦, the element spacing inelevation is chosen to be 10 cm to avoid grating lobes arisingin the imaging region. Combining antenna height and elementspacing, it results in a number of 25 elements in elevation. Inorder to guarantee the required 115-km swath coverage, only astrip of the full area of the antenna is activated for transmission,while the whole antenna is used for reception. According to[28], the height of this transmission part is set to be 0.2 m.

In the end, we show the architecture of the planar phasedarray antenna in the system in Fig. 12 and further list the systemparameters that will be used in the following simulations inTable IV.

B. Simulation Results of DBF in Elevation

Since the separation between echoes from adjacent subpulsesis the prerequisite for the subsequent processing procedures,the validity of DBF in elevation in our scheme should bedemonstrated at first. In order to illustrate the necessity ofthe phase compensation in frequency domain, we compare theperformances achieved by two kinds of DBF configurations inelevation—one without this step and the other with this step.For the first case, the left column of Fig. 13 shows the antennaarray gain at different positions within the extent of subpulseat the swath center time τc (In order to make the comparisonbetween receive gains over the coverage of different subpulsesmore clear, we used the same coordinates, where the abscissaintrapulse time t denotes the position within the coverage ofthe subpulse. One can actually obtain the intact beam patternof each subbeam by juxtaposing three lines in each figureaccording to the transmission order of three subpulses). It can


TABLE IVSYSTEM PARAMETERS USED IN THE SIMULATION

be seen that, in the covered region of subpulse of interest,receive gain decays away from the subpulse center and severeloss of receive gain occurs at the edge positions, as comparedwith that at the center position. In addition, by comparing thereceive gain over the region of subpulse of interest and thaton the region of interfering subpulses as a whole, one canobserve that the latter is not significantly lower than the former,indicating that the acquired echoes of each subpulse will stillbe disturbed by echoes from other two subpulses to a certainextent.

By comparison, the right column of Fig. 13 shows theobtained results corresponding to the second case. It can be seenthat the receive gain has become constant in the whole regionof the subpulse of interest, while at the same time, interferingsignals from other two subpulses have been suppressed to amuch lower level compared with the corresponding results inthe left column (The pattern shown in the right column is notthe real beam pattern as it is the case in the left column, butcan be viewed as a kind of “effective beam pattern” that isformed by the combination of signal processing technique andthe real physical beam pattern). As a consequence, one cansafely conclude that better performance in separating echoesfrom different subpulses can be achieved by performing phasecompensation in DBF in elevation.

To quantitatively illustrate this improvement, we use theRatio of average receive gain on Interference to that on Signal(RIS) to describe the capabilities of each formed subbeamin suppressing echoes from interfering subpulses, which isdefined as

RISnm(τc) = 10 · log10(

gave_nm(τc)

gave_mm(τc)

)(30)

where gave_mm(τc) and gave_nm(τc) are, respectively, the aver-age receive gain on the useful signal and that on the interferingsignals for the mth output of elevation beamformer at timeinstant τc. They are given by

gave_nm(τc) =1

T

T/2∫−T/2

gnm(t|τc)dt (31a)

gave_mm(τc) =1

T

T/2∫−T/2

gmm(t|τc)dt (31b)

with gnm(t|τc) being the array gain of the mth subbeam overthe extent of the nth subpulse, as shown in Fig. 13.

Table V reports a comparison of RISnm measured in the twocases for the sample SAR system already considered in theprevious analysis (see Table IV). Note that the acquired RISs inthe case where phase compensation is performed (Case II) aremuch lower than those obtained in Case I, which makes clearthat much better performances have been achieved at the swathcenter time instant τc by integrating the phase compensationinto the DBF processing in elevation.

At last, we compare the performances of the two types ofDBF configurations in terms of RIS in the entire range of echotime τ (From the system parameters listed in Table IV, we cancalculate that the range of τ is [4.1 ms, 4.47 ms]). Fig. 14 showsthe plots of RIS versus echo time τ associated with three outputends of a digital beamformer in elevation. It can be seen thatRIS will be decreased by circa 15–20 dB by means of thisphase compensation. The achieved RISs are less than −30 dBfor almost the entire echo-receiving period, and it can thereforebe considered that echoes from different subpulses have beeneffectively separated.

This section presented the simulation results for DBF inelevation in our scheme, which systematically combines thetechniques of echo tracking, null steering, and phase com-pensation. Moreover, the validity of our method in separatingdifferent subpulse echoes is well supported by these results.

C. Simulation Results of DBF in Azimuth

After the effective separation between echoes from differentsubpulses associated with different transmit subbeams, azimuthprocessing will then be carried out according to the procedureshown in Fig. 10. In order to illustrate the advantage of STWESAR over conventional single-input single-output (SISO) SAR(only middle azimuth subaperture is activated for transmission


Fig. 13. Comparison of the formed beam patterns at the swath center time τc under two types of DBF configurations. The left column shows the results obtainedwithout performing phase compensation. The right column illustrates the improvement induced by phase compensation. The top row shows the contrast forsubbeam1, the middle row is the case for subbeam2, and the bottom row for subbeam3.

and receiving) and single-input multiple-output (SIMO) SAR1

(Here, pulse duration in both SISO SAR and SIMO SAR isset to be 150 μs, which is the same as that in STWE SAR.Furthermore, the PRF value is chosen to be 1050 Hz, andMAR technique [12]–[14] is employed in SIMO SAR to resolveazimuth ambiguities due to nonuniform sampling), we first give

1Concerning SIMO SAR mode, we refer to the case where the middleazimuth subaperture is used to emit a radar pulse with a broad azimuth beam,while all three azimuth subapertures are used to receive echoes and then MARis performed to resolve azimuth ambiguities.

the azimuth compression results for a point target under threeworking modes for comparison, using system parameters listedin Table IV.

Fig. 15 shows the corresponding compression results. Bycomparing Fig. 15(a) and (b), it can be seen that the azimuthambiguity has been suppressed to a lower level as a wholeby SIMO SAR. This is due to the extra spatial sampling inazimuth, which is equivalent to the increase of PRF. Then, byfurther introducing STWE and DBF in elevation on the basis ofSIMO SAR, one can observe that the first-order aliased ghostimages that appear in Fig. 15(b) no longer exist in Fig. 15(c),


TABLE VCALCULATED RISS AT THE SWATH CENTER TIME

Fig. 14. RIS versus echo-receiving time τ under two kinds of DBF configurations. (a) shows the corresponding results for subbeam1, (b) shows the results forsubbeam2, and (c) shows the results for subbeam3.

Fig. 15. Azimuth compression results for a point target under (a) SISO mode, (b) SIMO mode, and (c) STWE mode.

although higher order ghost images are still present. This resultis in agreement with the function of azimuth beamformers thatwe derived in Section IV, which is to remove the first-orderazimuth ambiguity components.

To further prove the validity of the technique, we utilize 2-Dimaging results of point target array to show the advantage ofSTWE SAR over SISO SAR and SIMO SAR. The simulation

scene consists of nine point targets, and the distribution of themis shown in Fig. 16. The coordinate origin is set to be the centerof the swath.

In Fig. 17, the imaging results of point targets under threemodes are provided and compared. We can see that the interfer-ence of azimuth aliasing in SISO SAR is quite obvious, as themagnitude of first-order ghost images is not much lower than


Fig. 16. Coordinates of distributed target with nine point scatterers.

that of real image. In the SIMO SAR case, although interferencehas been alleviated to a certain extent, these ghost images arestill visible. In the STWE SAR case, one can observe that thefirst-order ghost images that lie on the upper and lower sidesof the real image in Fig. 17(a) and (b) are totally removed byour processing scheme, as shown in Fig. 17(c). Therefore, thiscomparison confirms the applicability of the proposed 2-D DBFtechnique for the real target imaging.

VI. PERFORMANCE ANALYSIS FOR STWE SAR SYSTEM

From the introduction of STWE mode and the antennaconfiguration shown in Fig. 12, one can see that the effectivetransmit aperture area of STWE SAR is Na times that of SIMOSAR. Therefore, STWE SAR will experience a transmit gainNa with respect to SIMO SAR, and consequently, the transmit-ted power of the former mode can be reduced to 1/Na withoutaltering the SNR. With this in mind, we will next investigatethe system performance associated with STWE SAR in AASRand RASR. Furthermore, we will compare them with theircounterparts in SIMO SAR to illustrate the benefits reaped bySTWE SAR.

A. AASR

For the SIMO SAR mode, the goal of multiaperture azimuthprocessing is to extract all of the signal components fromthe aliased frequency spectrum and then combine them toreconstruct the complete Doppler bandwidth for a high azimuthresolution. The azimuth ambiguity components, however, willnot be effectively suppressed due to the limited DOFs. Usingthe SIMO SAR system model assumed in the preceding section(the middle azimuth subaperture radiates pulses and all threesubapertures receive the reflected echoes), we illustrate theazimuth processing scheme for SIMO SAR in Fig. 18. Ascan be seen, each set of azimuth beamformers will retain oneof three signal components without any distortion or scalingwhile removing the other two completely at the same time.Subsequently, these azimuth subbands will be combined in fre-quency domain. However, by comparing the originally aliasedspectrum in the upper side with the finally reconstructed spec-trum in the lower side, we can observe that azimuth ambiguitycomponents are still existing in the acquired broad spectrum.

For the STWE SAR mode, echoes from different subpulseswill be separated by DBF in elevation before multiapertureazimuth processing. Since these subpulses are transmitted along

azimuth subbeams, which correspond to narrow Doppler band-width, there will be no more need of removing aliased signalcomponents. As a consequence, the extra DOF offered by mul-tiple azimuth apertures can be utilized to suppress ambiguitycomponents, as shown in Fig. 9.

The aforementioned analysis gives an intuitive reason for amuch lower AASR achieved by STWE SAR as compared withthat of SIMO SAR. In the following, we will further prove thisconclusion quantitatively by deriving explicit expressions forAASR of both SIMO SAR [27] and STWE SAR.

1) AASR of SIMO SAR: As the full Doppler spectrum of thesignal is reconstructed undistortedly, the signal power in SIMOSAR can then be directly given by

ESIMO−sig =

B/2∫−B/2

|ASIMO(fa)|2 dfa (32)

where B is the Doppler bandwidth inverse proportional to thelength of azimuth subaperture

B = 0.8862v

da(33)

and ASIMO(fa) is the two-way azimuth antenna pattern inSIMO SAR expressed in terms of azimuth frequency variablefa as

ASIMO(fa) ≈ sinc2(da2v

fa

). (34)

The ambiguity power, however, is determined by two factorsjointly. The first one is the corresponding antenna pattern onthe ambiguity components, and the second one is the scalingimpact of multiaperture filter functions on these components.By extending the calculation results for the two-subaperturecase in [27], we can obtain the weighting coefficients for thecase of three azimuth subapertures, as given by (35a)–(35c),shown at the bottom of the next page, where W1, W2, and W3

are the weighting coefficients on subaperture1, subaperture2,and subaperture3, respectively.

By defining W = [W1 W2 W3]T, the azimuth ambiguity

power can be evaluated as

ESIMO−amb = 2

+∞∑k=1

B/2∫−B/2

∣∣WThazi(fa + k · PRF )

×ASIMO(fa + k · PRF )|2 dfa (36)

where hazi(fa) is the azimuth steering vector in terms offrequency variable fa and is expressed as

hazi(fa)=[exp(j2π/2v ·da ·fa) 1 exp(−j2π/2v ·da ·fa)]T(37)

(the middle subaperture is set to be the reference subaperture).Let SF (k, fa) = WT · hazi(fa + k · PRF) denote the scalingfactor; the explicit expression for SF (k, fa) can then be derived


Fig. 17. Imaging results of point targets for (a) SISO SAR, (b) SIMO SAR, and (c) STWE SAR.

from (36) and (37) as follows, see (38), shown at the bottom ofthe page.

From (36), one can see that the scaling impact representedby SF (k, fa), along with the antenna pattern ASIMO(fa +k · PRF ), should be taken into account when evaluating theazimuth ambiguity power for SIMO SAR. Additionally, notethat scaling coefficients are different from frequency interval tofrequency interval, as indicated by (38).

2) AASR of STWE SAR: For STWE SAR, the derivation ofAASR is slightly different from that in the SIMO SAR case,since the finally obtained Doppler spectrum in STWE SARis the combination of azimuth spectrum of different subpulseechoes. Consequently, the signal and ambiguity power asso-ciated with the echoes of each subpulse should be evaluatedfirst, and then, the total signal and ambiguity power can bedetermined by the sum of the signal power and the sum ofambiguity power, respectively.

As each azimuth beamformer given in (27) will not influencethe signal component in the corresponding subpulse echoes, theexpressions for the signal power associated with echoes from

three subpulses can be directly derived as

E1−sig =

fdc+Bd/2∫fdc−Bd/2

|A1(fa)|2 dfa

=

Bd/2∫−Bd/2

|A1(fa + fdc)|2 dfa (39a)

E2−sig =

Bd/2∫−Bd/2

|A2(fa)|2 dfa (39b)

E3−sig =

−fdc+Bd/2∫−fdc−Bd/2

|A3(fa)|2 dfa

=

Bd/2∫−Bd/2

|A3(fa − fdc)|2 dfa (39c)

W1 =

⎧⎪⎪⎨⎪⎪⎩

exp(−jπ/v·da·(fa+2PRF ))−exp(−jπ/v·da·(fa+PRF ))j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈

[− 3

2PRF,− 12PRF

]exp(−jπ/v·da·(fa−PRF ))−exp(−jπ/v·da·(fa+PRF ))

j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈[− 1

2PRF, 12PRF

]exp(−jπ/v·da·(fa−PRF ))−exp(−jπ/v·da·(fa−2PRF ))

j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈[12PRF, 3

2PRF] (35a)

W2 =

⎧⎪⎪⎨⎪⎪⎩

exp(jπ/v·da·PRF )−exp(−jπ/v·da·PRF )j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈

[− 3

2PRF,− 12PRF

]exp(−j2π/v·da·PRF )−exp(j2π/v·da·PRF )j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈

[− 1

2PRF, 12PRF

]exp(jπ/v·da·PRF )−exp(−jπ/v·da·PRF )j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈

[12PRF, 3

2PRF] (35b)

W3 =

⎧⎪⎪⎨⎪⎪⎩

exp(jπ/v·da·(fa+PRF ))−exp(jπ/v·da·(fa+2PRF ))j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈

[− 3

2PRF,− 12PRF

]exp(jπ/v·da·(fa+PRF ))−exp(jπ/v·da·(fa−PRF ))

j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈[− 1

2PRF, 12PRF

]exp(jπ/v·da·(fa−2PRF ))−exp(jπ/v·da·(fa−PRF ))

j·4 sin(π/v·da·PRF )·[1−cos(π/v·da·PRF )] , fa ∈[12PRF, 3

2PRF] (35c)

SF (k, fa) =

⎧⎪⎪⎨⎪⎪⎩

sin(π/v·da(k−2)PRF )−sin(π/v·da(k−1)PRF )+sin(π/v·daPRF )2 sin(π/v·daPRF )·[1−cos(π/v·daPRF )] , fa ∈

[− 3

2PRF,− 12PRF

]sin(π/v·da(k+1)PRF )−sin(π/v·da(k−1)PRF )−sin(2π/v·daPRF )

2 sin(π/v·daPRF )·[1−cos(π/v·daPRF )] , fa ∈[− 1

2PRF, 12PRF

]sin(π/v·da(k+1)PRF )−sin(π/v·da(k+2)PRF )+sin(π/v·daPRF )

2 sin(π/v·daPRF )·[1−cos(π/v·daPRF )] , fa ∈[12PRF, 3

2PRF] (38)


Fig. 18. Multiaperture azimuth processing for SIMO SAR.

TABLE VILOCATION OF SIGNAL AND AMBIGUITY COMPONENTS

where Bd is the Doppler bandwidth of each transmit sub-beam and A1(fa), A2(fa), and A3(fa) represent the two-way antenna patterns associated with subpulse1, subpulse2, andsubpulse3, respectively. They are given by

A1(fa) = sinc

[La

2v· (fa − fdc)

]· sinc

[da2v

· fa]

(40a)

A2(fa) = sinc

(La

2v· fa)· sinc

(da2v

· fa)

(40b)

A3(fa) = sinc

[La

2v· (fa + fdc)

]· sinc

[da2v

· fa]. (40c)

Then, the total signal power for STWE SAR can finally beobtained as

ESTWE−sig = E1−sig + E2−sig + E3−sig. (41)

Similarly to the ambiguity evaluation in SIMO SAR case,the residual ambiguity power for the STWE SAR case is alsogoverned by the two-way azimuth antenna pattern, as well asthe scaling coefficients. In Table VI, we show the correspondingazimuth frequency intervals where the useful signal and ambi-guity components are originally located for each subpulse.

With the knowledge about the location of ambiguity com-ponents, along with the azimuth beamformer weight vector


for STWE SAR, we obtain the respective ambiguity powerassociated with three subpulse echoes as

E1−amb=

+∞∑k=1

⎧⎨⎩

Bd/2∫−Bd/2

∣∣∣∣pT1 hazi(fa+fdc+k · PRF )

×A1(fa+fdc+k ·PRF )

∣∣∣∣2

dfa

+

Bd/2∫−Bd/2

∣∣∣∣pT1 hazi(fa+fdc−k · PRF )

×A1(fa+fdc−k ·PRF )

∣∣∣∣2

dfa

⎫⎬⎭

(42a)

E2−amb=2

+∞∑k=1

Bd/2∫−Bd/2

∣∣∣∣pT2 hazi(fa+k ·PRF )

×A2(fa+k ·PRF )

∣∣∣∣2

dfa (42b)

E3−amb=

+∞∑k=1

⎧⎨⎩

Bd/2∫−Bd/2

∣∣∣∣pT3 hazi(fa−fdc+k ·PRF )

×A3(fa−fdc+k ·PRF )

∣∣∣∣2

dfa

+

Bd/2∫−Bd/2

∣∣∣∣pT3 hazi(fa−fdc−k ·PRF )

×A3(fa−fdc−k ·PRF )

∣∣∣∣2

dfa

⎫⎬⎭.

(42c)

[Due to the unsymmetrical characteristics of the two-way az-imuth antenna pattern associated with subpulse1 and subpulse3,their kth-order azimuth ambiguity power is comprised of twounequal parts corresponding to the positive and negative signs,respectively, as indicated by (42a) and (42.c).]

Note that for the same k, the corresponding scaling factorsfor three subpulses are identical. For convenience, we useSFSTWE(k, fa) to represent it, which is given by

SFSTWE(k, fa)

=cos(π/v · da · k · PRF )− cos(π/v · da · PRF )

1− cos(π/v · da · PRF ). (43)

Subsequently, the total ambiguity power can be obtained as

ESTWE−amb = E1−amb + E2−amb + E3−amb. (44)

With the derived results given by (32), (36), (41), and (44),we present the AASR plots for both SIMO SAR and STWESAR with the same system parameters (Fig. 19). It can beobserved that AASR has been reduced by approximately 13 dBand approached −30 dB by STWE SAR with our processingscheme.

Fig. 19. AASR versus PRF for SIMO SAR and STWE SAR.

Fig. 20. RASR versus ground range for SIMO SAR.

B. RASR

In the elevation dimension of SIMO SAR, the SCan-On-REceive (SCORE) technique is employed to form a digitalbeam in range dimension to receive pulse echoes. This receivebeam is much narrower than the transmit beam, and hence,the normalized receive gain on range ambiguity will be muchreduced. On the other hand, the useful signal reflected fromthe scene will always be received by the maximum gain asthe receive beam is steered to track pulse echoes. The RASRperformance of SIMO SAR will then be greatly satisfactory ascompared with that of traditional spaceborne SAR.

In Fig. 20, we present the RASR performance for SIMO SARwith the PRF value of 1050 Hz, as selected in Section V-C. Itcan be observed that the maximum RASR has been already aslow as −53 dB and the average RASR over the whole scene canbe computed to be −59.41 dB.

In the STWE SAR case, the evaluation of range ambiguityis slightly complex compared with that of the SIMO SAR, asthe total range ambiguity consists of not only common rangeambiguity as in SIMO SAR but also extra range ambiguity,which is the mutual interference between echoes from adjacent


Fig. 21. Three point targets whose echoes will partially overlap.

subpulses transmitted in one PRI. In the following, we willutilize a generic point target model to examine extra rangeambiguity in STWE SAR.

As shown in Fig. 21, let the slant ranges of three point targetssatisfy

R1 = R2 + c · T/2 = R3 + 2 · c · T/2 (45)

and thereby, the echoes of P1, P2, and P3 generated by sub-pulse1, subpulse2 and subpulse3, respectively, will overlap.For the echo of any point scatterer, the echoes of the othertwo can be considered as range ambiguities. From the simu-lation results of DBF in elevation (Section V-B), we note thatthe receive gains over the interference echoes correspondingto each formed subbeam will be time varying rather than aconstant. Hence, the traditional expression for RASR shouldbe modified to evaluate the extra range ASR (ERASR) in theSTWE case.

Take the echo from subpulse1 as signal, and then, the ERASRexpression associated with subpulse1 can be given by (46a),shown at the bottom of the page, where ϕ1, ϕ2, and ϕ3

denotes the incidence angle at P1, P2, and P3, respectively, andGrm1(τ |ϕ1) represents the receive gain of subbeam1 over theecho of point target Pm. Note that all three point targets are inthe swath, and hence, their echo signals enjoy the same transmit

gain; we ignore the term corresponding to the transmit gain inthe expression.

Similarly, one can acquire the expressions for ERASR2 andERASR3 as (46b) and (46c), shown at the bottom of the page.

Since each former receive subbeam is a narrow beam, theconventional RASR is actually quite small as compared withERASR. Therefore, one can use ERASR to approximate RASRfor each subbeam, as shown by

RASRi ≈ ERASRi (47)

where i ∈ [1, 3].By applying the aforementioned RASR expressions to the

evaluation of RASR for the whole swath, we obtain the corre-sponding RASR plots as shown in Fig. 22.

The RASR performance corresponding to the STWE SARsystem can then be approximately evaluated by averaging theRASR values associated with three subpulse echoes as

RASRSTWE ≈ 1

3(RASR1 +RASR2 +RASR3) (48)

and the obtained result for RASRSTWE is shown in Fig. 23. Itcan be seen that the maximum RASR value for STWE SAR isabout −29.5 dB. In addition, the average RASR value over thewhole swath is calculated to be −30.91 dB.

By comparing Figs. 20 and 23, one can observe that a muchlower RASR can be achieved by SIMO SAR than that by STWESAR. However, for the quality of SAR image, it is sufficient tosuppress range ambiguity below −25 dB, implying that SARimage would not degrade much even if the RASR level has risenup to −30.91 dB from −59.41 dB. Therefore, it is worthwhileachieving performance improvement in AASR and transmittedpeak power reduction at the cost of deterioration in RASR.Furthermore, by taking into account both RASR and AASR,we can evaluate the total ASR for SIMO SAR and STWE SAR.With the selected PRF of 1050 Hz, the corresponding resultsare −16.98 and −27.42 dB for SIMO SAR and STWE SAR,respectively. It can be seen that ASR has been improved bymore than 10 dB through STWE SAR as compared with thatof SIMO SAR.

ERASR1(ϕ1) =

∫〈T 〉 Gr21(τ |ϕ1)dτ/R

32 sin(ϕ2) +

∫〈T 〉 Gr31(τ |ϕ1)dτ/R

33 sin(ϕ3)∫

〈T 〉 Gr11(τ |ϕ1)dτ/R31 sin(ϕ1)

(46a)

ERASR2(ϕ2) =

∫〈T 〉 G12(τ |ϕ2)dτ/R

31 sin(ϕ1) +

∫〈T 〉 G32(τ |ϕ2)dτ/R

33 sin(ϕ3)∫

〈T 〉 G22(τ |ϕ2)dτ/R32 sin(ϕ2)

(46b)

ERASR3(ϕ3) =

∫〈T 〉 G13(τ |ϕ3)dτ/R

31 sin(ϕ1) +

∫〈T 〉 G23(τ |ϕ3)dτ/R

32 sin(ϕ2)∫

〈T 〉 G33(τ |ϕ3)dτ/R33 sin(ϕ3)

(46c)


Fig. 22. RASR plots versus ground range for (a) subpulse1’s echo, (b) subpulse2’s echo, and (c) subpulse3’s echo.

Fig. 23. RASR versus ground range for STWE SAR.

VII. CONCLUSION

For the sake of obtaining the benefit of ambiguity reduction,this paper derives the complete processing scheme for STWESAR. The scheme is based on 2-D DBF, namely, DBF in eleva-tion and DBF in azimuth. Echoes from different subpulses will

be first separated by DBF in elevation utilizing the fact that theyarrive along different angles. Then, they will be processed bytheir corresponding azimuth beamformers to remove azimuthambiguities and finally combined to obtain the broad Dopplerspectrum for a high azimuth resolution. Although the RASRlevel will be raised as a sacrifice, our scheme enables STWESAR superiority over SIMO SAR in two main aspects: 1) Thesystem AASR can be reduced to a much lower level, withoutcompromise in swath width and/or azimuth geometric reso-lution, and 2) the peak power requirements can be alleviatedto achieve a predefined SNR, as all subapertures in azimuthdirection are used to transmit the encoded pulse, rather thana single subaperture in the case of SIMO SAR. Moreover, thetotal ASR can be decreased significantly by STWE SAR withour processing scheme.

As we assume an ideal spherical model for the Earth inour derivation of elevation beamformers, it is reasonable toexpect a critical sensitivity of our method to topography. In thefuture, our work will thus mainly concentrate on two aspects:1) in-depth analysis of beam-pointing error that may occur inrough mountain areas when performing DBF in elevation andits impacts on the finally acquired results of echo separation and2) hardware design for implementing the DBF in elevation onsatellite.


Fig. 24. Phase shift with (a) ideal switching frequency and (b) one-third ofthe ideal switching frequency.

APPENDIX

In the ideal scenario, the switching frequency of phaseshifters should equal the sampling frequency of the ADC,therefore assuring a precise phase shift value for each sampleddata. Now, we consider the case where the switching frequencyis only 1/N of the sampling frequency, which means that eachset of N successive sampled data will be phase shifted byone weight coefficient. Fig. 24 shows this case with a systemexample for which the ratio of sampling frequency to switchingfrequency is 3.

For the upper ideal case in the figure, each sampled data willhave its own specific phase-shifting value, whereas three con-secutive sampled data have to share one phase-shift value forthe lower case. In addition, the relation between the weighingcoefficient w in the ideal case and its counterpart w′ in the caseof lower switching frequency is

w′(n) = w(3n− 2). (A1)

Then, we utilize a point target with off-boresight angle α0

and round-trip delay τ0 to analyze the target response for thetwo cases shown earlier. For the ideal case, the demodulationsignal arriving at the kth subaperture on elevation, after sam-pling and phase shifting, can be modeled as

sk(τ) = exp(−j2πfcτ0) · rect[τ − τ0T

]

· exp{jπKr(τ − τ0)

2}

· exp{−j2π

(k − 1)deλ

[sin (α(τ))− sin(α0)]

}

·+∞∑

n=−∞δ

(τ − n

Fs

)(A2)

where fc is the carrier frequency, T is the pulse duration, Kr isthe chirp rate, λ is the wavelength, dei is the inter-subaperturespacing on elevation, α(τ) is the off-boresight angle associatedwith the receive beam center, which is a function of timevariable τ as it tracks the pulse echo on the ground, and∑+∞

n=−∞ δ(τ − n/Fs) is the sampling function with samplingfrequency Fs.

Using the approximate linear relationship [26] between α(τ)and time variable τ yields an alternative expression of sk(τ) as

sk(τ) ≈ exp(−j2πfcτ0) · rect[τ − τ0T

]


2}

· exp{−j2π

(k − 1)deλ

· ∂α(τ)∂τ

∣∣∣∣τ=τc

(τ − τ0)

}

·+∞∑

n=−∞δ

(τ − n

Fs

)

≈ exp(−j2πfcτ0) · rect[τ − τ0T

]


2}

· exp {−j2π(k − 1)f0(τ − τ0)}

·+∞∑

n=−∞δ

(τ − n

Fs

)(A3)

where (∂α(τ)/∂τ)|τ=τc is the value of the first-order derivativeof α with respect to the time variable τ at the swath center timeτc, and for the reason of compact expression form, we definef0 = (de/λ) · (∂α(τ)/∂τ)|τ=τc .

For the second case shown in the lower part of Fig. 24,the corresponding signal at the kth subaperture s′k(τ) can begiven by

s′k(τ) ≈ exp(−j2πfcτ0) · rect[τ − τ0T

]


2}

·{exp{− j2π(k − 1)f0(τ − τ0)

}

·+∞∑

n=−∞δ

(τ − 3n

Fs

)

+ exp

{− j2π(k − 1)f0

(τ − τ0 −

1

Fs

)}

·+∞∑

n=−∞δ

(τ − (3n+ 1)

Fs

)

+ exp

{− j2π(k − 1)f0

(τ − τ0 −

2

Fs

)}

·+∞∑

n=−∞δ

(τ − (3n+ 2)

Fs

)}. (A4)


Define s(τ) = rect[τ/T ] · exp{jπKrτ2}, and then, s′k(τ)

can be expressed on an alternative form as

s′k(τ) = exp(−j2πfcτ0)

·{s(τ − τ0) · exp

{− j2π(k − 1)f0(τ − τ0)

}

·+∞∑

n=−∞δ(τ − 3n

Fs

)

+

{s

(τ − τ0 +

1

Fs

)

· exp{− j2π(k − 1)f0(τ − τ0)

}·

+∞∑n=−∞

δ

(τ − 3n

Fs

)}⊗ δ

(τ − 1

Fs

)

+

{s

(τ − τ0 +

2

Fs

)

· exp{− j2π(k − 1)f0(τ − τ0)

}·

+∞∑n=−∞

δ(τ − 3n

Fs

)}

⊗ δ(τ − 2

Fs

)}(A5)

where the symbol ⊗ stands for convolution operator. Utilizingthe explicit expression for the spectrum of LFM pulse [29] andthe property of Fourier transform, we obtain the spectrum ofs′k(τ) as

S ′k(f)≈ exp {−j2π(fc + f)τ0} ·

+∞∑n=−∞

rect

[f − n · Fs/3

KrT

]

· exp{jπ

[f + (k − 1)f0 − n · Fs/3]2

Kr

}

· exp{j2π

nFs

3τ0

}

·{1 + exp

{j2π (f + (k − 1)f0 − n · Fs/3) ·

1

Fs

}

· exp(−j2πf

1

Fs

)

+ exp

{j2π (f + (k − 1)f0 − n · Fs/3) ·

2

Fs

}

· exp(−j2πf

2

Fs

)}(A6)

(In fact, the precise envelope function is rect[(f + (k − 1)f0 −n · Fs/3)/KrT ] rather than rect[(f − n · Fs/3)/KrT ]. How-ever, since the term (k − 1)f0 is small enough compared withboth the signal bandwidth KrT and the repetition period Fs/3,its impact on the spectrum displacement can be neglected,and we ignore this term in the signal envelope function forconvenience.)

With proper manipulations, the term exp{jπ(([f + (k −1)f0 − n · Fs/3]

2)/Kr)} in (A6) can be factored as

exp

{jπ

[f + (k − 1)f0 − n · Fs/3]2

Kr

}

= exp

{jπ

f2

Kr

}· exp

{j2π

(k − 1)f0Kr

f

}

· exp{−j2π

n · Fs

3Krf

}

· exp{jπ

[(k − 1)f0 − n · Fs/3]2

Kr

}. (A7)

After the term exp{j2π(((k − 1)f0)/Kr)f} in (A7) hasbeen compensated by the filter function given by (22), thespectrum becomes

S ′k(f)≈ exp {−j2π(fc + f)τ0} ·

+∞∑n=−∞

rect

[f − n · Fs/3

KrT

]

· exp{jπ

f2

Kr

}· exp

{−j2π

n · Fs

3Krf

}

· exp{jπ

[(k − 1)f0 − n · Fs/3]2

Kr

}

·1− exp

{j2π[3(k−1)f0

Fs− n]}

1− exp{j2π (k−1)f0−n·Fs/3

Fs

}· exp

{j2π

nFs

3τ0

}. (A8)

From (A8), one can see that, in the case of lower switchingfrequency of phase shifter, the spectrum of the signal will berepetitive with a period of Fs/3. Thus, if Fs/3 is not above theNyquist rate, the aliasing occurs and the effect of paired echoeswill be observable in range dimension. The offset between thereal image and artifacts owing to paired echoes can further beevaluated to be n · Fs/3Kr in terms of range time, as indicatedby the linear phase term exp{−j2π(n · Fs/3Kr)f} in (A8).

With the system parameters in Table IV, the simulationresults of range compression for a point target at the scenecenter with the phase-shift switching frequency of 150 MHz(the same as sampling frequency Fs) and 50 MHz in Fig. 25.As can be observed clearly, a pair of artifacts has appearedwhen the phase-shift frequency is reduced to 50 MHz from150 MHz. The distance between real target image andghost image is 3125 range bins, which is equivalent to(Fs/3Kr)/(1/Fs). Therefore, the result well supports the con-clusion we obtained earlier.

To summarize, if the switching frequency of the phase shifterhas been lowered and does not satisfy the Nyquist samplingcriterion any longer, the linear frequency phase error will beinduced, giving rise to pairs of artifacts in the final SAR image.However, since the magnitude of these artifacts are often veryweak (In Fig. 25, the magnitude of the artifact is almost −80 dBbelow the target peak) and the quality of SAR image maynot be degraded severely, a tradeoff can be made between therequirements of the hardware onboard and the quality of SARimage.


Fig. 25. Range compression results of point target echo with the phase-shifter switching frequency of (a) 150 MHz and (b) 50 MHz.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable comments and suggestions to improve the qualityand readability of this paper.

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Fan Feng was born in Shaanxi, China, in 1984. Hereceived the B.S. degree in information engineeringfrom the University of Science and Technology ofChina, Hefei, China, in 2006. He is currently work-ing toward the Ph.D. degree in the field of multiple-input multiple-output and digital beamformingsynthetic aperture radar technology in the GraduateUniversity of Chinese Academy of Science, Beijing,China.

Shiqiang Li was born in Hebei, China, in 1967. Hereceived the B.S. degree from Beijing Institute ofTechnology, Beijing, China, in 1989 and the M.Sc.and Ph.D. degrees in electrical engineering from No.54 Institute of China Electronics Technology Groupand Institute of Electronics, Chinese Academyof Science (IECAS), Beijing, in 1992 and 2004,respectively.

He has been with IECAS since 2004 and be-came an Associate Professor of Communications andInformation System in 2007. His current research

interests are high-resolution and wide-swath spaceborne synthetic apertureradar (SAR) system design and simulation and multistatic spaceborne SARsystems. He has published six papers and two patents.

Weidong Yu (M’00) was born in Henan, China, in1969. He received the M.Sc. and Ph.D. degrees inelectrical engineering from Nanjing University ofAeronautics and Aerospace, Nanjing, China, in 1994and 1997, respectively.

He has been with the Institute of Electronics,Chinese Academy of Science (IECAS), since 1997and became a Professor of Communication and In-formation System in 2000. His research interestsare airborne and spaceborne synthetic aperture radar(SAR) system design and their signal processing. He

has published more than 50 papers and five patents. He has been the ChiefDesigner for several SAR systems and is currently the Deputy Director of theDepartment of Space Microwave Remote Sensing System, IECAS.

Shuo Wang was born in Liaoning, China, in 1981.He received the B.S. degree in information engineer-ing from Tianjin University, Tianjin, China, in 2004.He is currently working toward the Ph.D. degree inthe field of polarimetric spaceborne synthetic aper-ture radar technology in the Graduate University ofChinese Academy of Science, Beijing, China.

study on the processing scheme for space–time waveform encoding sar system based on...

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