Published in IET Radar, Sonar and Navigation Received on 26th September 2012 Revised on 11th December 2012 Accepted on 3rd January 2013 doi: 10.1049/iet-rsn.2012.0286 ISSN 1751-8784 Using omega-K algorithm for bistatic synthetic aperture radar image formation based on modified extended Loffeld’s bistatic formula Mohammad Zamani, Mahmood Modarres-Hashemi Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran E-mail: [email protected]Abstract: Bistatic synthetic aperture radar (BSAR) as a way of Earth remote sensing has been developed considerably in recent years, both theoretically and practically due to its unsubstitutable services. BSAR frequency-domain processing algorithms are efficient ways of image formation in comparison to ideal two-dimensional matched filtering and to relatively accurate and time-consuming time-domain algorithms. Among these frequency-domain algorithms, omega-K is the most precise. The starting and key step of frequency-domain algorithms is the derivation of bistatic spectrum. Recently, a new bistatic spectrum is reported, which is probably the latest and the most modified version of Loffeld’s bistaic formula (LBF), maintaining its accuracy even in azimuth-variant configurations with high squint angles. So far, this spectrum has only been used within range-Doppler algorithm to process BSAR data. The authors investigate the possibility and results of applying this modified version of LBF as a basis for omega-K algorithm. Two approaches, based on Stolt interpolation and inverse scaled Fourier transform, are examined and their effectiveness in general azimuth-variant geometry is validated through several simulations. The proposed implementations show higher performance in terms of image quality measurements as compared to extended LBF-based implementations. 1 Introduction Exploiting bistatic synthetic aperture radar (BSAR) systems through separate platforms of transmitter and receiver brings us the possibility of defining advanced imaging configurations, obtaining considerable scattering information and ensuring cost savings due to the possibility of sharing a transmitter between several receivers, capability of passive monitoring and subsequent advantages like covert operation and small size [1, 2]. In general, BSAR geometry is azimuth-variant, that is, with non-parallel trajectories or unequal velocities of platforms. Hence, its processing considerations are much more than monostatic counterparts [1]. Processing of BSAR raw data and image formation can be done in terms of time or frequency. Efficient and more favoured frequency-domain algorithms require the derivation of bistatic point target reference spectrum (BPTRS), which is responsible for their accuracy [3]. On the contrary to monostatic SAR, it is difficult to express azimuth time as a function of azimuth frequency with the use of conventional principle of stationary phase (PSP) in BSAR. It is due to the double squared roots property of the range equation [4]. According to the literature, techniques of solving this problem can be categorised into preprocessing, numerical and analytical methods [4]. Since analytical methods are relatively accurate, less computationally intensive and not limited to a specific geometry [4], they have attracted a lot of attention recently. The most important method in the development of analytical solutions may be the Loffeld’s bistatic formula (LBF) [5]. It is based on the determination of common point of stationary phase from the individual ones of transmitter and receiver with the assumption of their equal contributions to Doppler frequency history [5]. LBF accuracy is degraded in extreme BSAR configurations like space-surface BSAR [6]. In [6], an extension of LBF (ELBF) is proposed, which considers different weight factors for transmitter and receiver Doppler contributions, proportional to their time-bandwidth product (TBP). In high squint cases, disregarding the Doppler centroid influence reduces the precision of the spectrum [7]. In [8], a modification of ELBF for the general BSAR is introduced, whose focusing performance is significant in almost all bistatic configurations. We call it modified ELBF (MELBF) in this paper for the simplicity of future referring. Method of series reversion (MSR) [3] that controls the spectrum accuracy by the number of terms used in the power series is another interesting achievement in this area. By selecting one spectrum derivation among these options, we will have a clue of how to deal with possible choices of image formation algorithms that somehow imply the performance prospect. Based on LBF, some image formation algorithms like two-dimensional inverse scaled Fourier transform (ISFT) [9] and chirp scaling algorithm (CSA) [10] are reported. Based on ELBF, omega-K algorithm [11] and its modification using ISFT [12] are www.ietdl.org IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392 383 doi: 10.1049/iet-rsn.2012.0286 & The Institution of Engineering and Technology 2013
Transcript
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IE
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Published in IET Radar, Sonar and NavigationReceived on 26th September 2012Revised on 11th December 2012Accepted on 3rd January 2013doi: 10.1049/iet-rsn.2012.0286
T Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392oi: 10.1049/iet-rsn.2012.0286
ISSN 1751-8784
Using omega-K algorithm for bistatic syntheticaperture radar image formation based on modifiedextended Loffeld’s bistatic formulaMohammad Zamani, Mahmood Modarres-Hashemi
Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran
Abstract: Bistatic synthetic aperture radar (BSAR) as a way of Earth remote sensing has been developed considerably in recentyears, both theoretically and practically due to its unsubstitutable services. BSAR frequency-domain processing algorithms areefficient ways of image formation in comparison to ideal two-dimensional matched filtering and to relatively accurate andtime-consuming time-domain algorithms. Among these frequency-domain algorithms, omega-K is the most precise. Thestarting and key step of frequency-domain algorithms is the derivation of bistatic spectrum. Recently, a new bistatic spectrumis reported, which is probably the latest and the most modified version of Loffeld’s bistaic formula (LBF), maintaining itsaccuracy even in azimuth-variant configurations with high squint angles. So far, this spectrum has only been used withinrange-Doppler algorithm to process BSAR data. The authors investigate the possibility and results of applying this modifiedversion of LBF as a basis for omega-K algorithm. Two approaches, based on Stolt interpolation and inverse scaled Fouriertransform, are examined and their effectiveness in general azimuth-variant geometry is validated through several simulations.The proposed implementations show higher performance in terms of image quality measurements as compared to extendedLBF-based implementations.
1 Introduction
Exploiting bistatic synthetic aperture radar (BSAR) systemsthrough separate platforms of transmitter and receiver bringsus the possibility of defining advanced imagingconfigurations, obtaining considerable scattering informationand ensuring cost savings due to the possibility of sharing atransmitter between several receivers, capability of passivemonitoring and subsequent advantages like covert operationand small size [1, 2]. In general, BSAR geometry isazimuth-variant, that is, with non-parallel trajectories orunequal velocities of platforms. Hence, its processingconsiderations are much more than monostatic counterparts [1].Processing of BSAR raw data and image formation can be
done in terms of time or frequency. Efficient and morefavoured frequency-domain algorithms require thederivation of bistatic point target reference spectrum(BPTRS), which is responsible for their accuracy [3]. Onthe contrary to monostatic SAR, it is difficult to expressazimuth time as a function of azimuth frequency with theuse of conventional principle of stationary phase (PSP) inBSAR. It is due to the double squared roots property of therange equation [4]. According to the literature, techniquesof solving this problem can be categorised intopreprocessing, numerical and analytical methods [4]. Sinceanalytical methods are relatively accurate, lesscomputationally intensive and not limited to a specificgeometry [4], they have attracted a lot of attention recently.
The most important method in the development ofanalytical solutions may be the Loffeld’s bistatic formula(LBF) [5]. It is based on the determination of commonpoint of stationary phase from the individual ones oftransmitter and receiver with the assumption of their equalcontributions to Doppler frequency history [5]. LBFaccuracy is degraded in extreme BSAR configurations likespace-surface BSAR [6]. In [6], an extension of LBF(ELBF) is proposed, which considers different weightfactors for transmitter and receiver Doppler contributions,proportional to their time-bandwidth product (TBP). In highsquint cases, disregarding the Doppler centroid influencereduces the precision of the spectrum [7]. In [8], amodification of ELBF for the general BSAR is introduced,whose focusing performance is significant in almost allbistatic configurations. We call it modified ELBF (MELBF)in this paper for the simplicity of future referring. Methodof series reversion (MSR) [3] that controls the spectrumaccuracy by the number of terms used in the power series isanother interesting achievement in this area.By selecting one spectrum derivation among these options,
we will have a clue of how to deal with possible choices ofimage formation algorithms that somehow imply theperformance prospect. Based on LBF, some imageformation algorithms like two-dimensional inverse scaledFourier transform (ISFT) [9] and chirp scaling algorithm(CSA) [10] are reported. Based on ELBF, omega-Kalgorithm [11] and its modification using ISFT [12] are
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presented. MSR is used in non-linear chirp scaling [13] inazimuth-variant configuration, and omega-K [14] and rangeDoppler algorithm (RDA) [15] in azimuth-invariantconfigurations to process BSAR data. RDA for forwardlooking azimuth-invariant BSAR is developed based onMELBF [16]. In [17], BSAR processing is done by meansof CSA and RDA based on a novel spectrum which isderived from a two-dimensional PSP approach. Thisspectrum is similar to MELBF in terms of mathematicalexpression and hence, some mathematical simplificationsused in Section 2 are inspired by [16, 17].Omega-K is attractive to SAR community, since it can
handle wide apertures and highly squinted scenarios [14,18]. As demonstrated before, MELBF is an accurateBPTRS that can be applied within any image formationalgorithm. It does not have the focusing limitations ofprevious spectrum derivations that occurred whether inextreme BSAR configurations or in highly squinted cases.The use of MELBF as a basis for omega-K has not beenevaluated before and is the main subject of this work.In this paper, we investigate possible implementations ofomega-K algorithm based on MELBF spectrum. Wedevelop two different methodologies based on (1) accurateStolt interpolation and (2) efficient approximateinterpolation-free ISFT. The ISFT theory and its applicationto SAR processing have been described in [19] and the useof it as an integral part of approximated omega-K has beenpresented in [20].This paper is organised as follows: In Section 2, the signal
model and MELBF spectrum are described. Section 3discusses the accurate omega-K implementation with theso-called Stolt interpolation using MELBF spectrum.Modified interpolation-free omega-K using MELBF ispresented in Section 4. The simulation results are reportedin Section 5. Finally, some conclusions are drawn inSection 6.
2 Signal model and bistatic spectrum
The general BSAR geometry (similar to [8]) is shown inFig. 1. The point target P1 located in the x–y plane isilluminated by the composite footprint of transmitter andreceiver antennas. Platforms of receiver and transmitter,moving with velocities of vr and vt, respectively, have the
Fig. 1 Bistatic geometry
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ranges of the closest approach to P1, symbolised by R0r andR0t, respectively. In this paper, η and fη are symbols forazimuth time and azimuth frequency variables, while t and frepresent range time and range frequency. Zero-Dopplertimes of receiver and transmitter are denoted by η0r and η0t.The received signal from P1 after baseband demodulation
is expressed as [8]
s(t, h) = p t − Rt(h)+ Rr(h)
c
[ ]wa(h− hc)
× exp −j2p f0Rt(h)+ Rr(h)
c
{ } (1)
where p(t) is the transmitted signal, wa(η) is the compositeazimuth antenna pattern and ηc is the beam centre crossingtime. The notations of carrier wavelength and carrierfrequency of transmitted signal are λ and f0, respectively,and c is the speed of light. Instantaneous ranges fromreceiver and transmitter to the point target represented byRr(η) and Rt(η) can be obtained as follows [8]
Rr(h) =���������������������R20r + v2r h− h0r
( )2√
Rt(h) =���������������������R20t + v2t h− h0t
( )2√ (2)
The frequency-domain formulation of (1) can be yielded byusing one of the existing BPTRSs. MELBF spectrum, ourapproach in this paper, can be explained by the followingequations [8]
S( f , fh) = P(f )Wa( fh) exp(−jC( f , fh))
C( f , fh) = 2p( fhrh0r + fhth0t) + 2pR0r
cFr + 2p
R0t
cFt
(3)
where the transmitted signal spectrum and Doppler frequencyenvelope are defined by P( f ) andWa( fη), respectively. fηr andfηt as Doppler contributions of receiver and transmitter,respectively, are drawn by [8]
fhr = kr( fh − fdcr − fdct) + fdcr
fht = kt( fh − fdcr − fdct) + fdct(4)
They satisfy the equation: fηr + fηt = fη. According to (4), theDoppler contributions of transmitter and receiver aregenerally different, depending on the Doppler centroids atthe composite beam centre crossing time as well asweighting factors. The receiver Doppler contributionweighting factor, kr, which is the ratio of the TBP of thereceiver to the total TBP, is [8]
kr =TBPr
TBPt + TBPr
= v2r / lR0r
( )( )cos3(usqr)T 2
img
v2t / lR0t
( )( )cos3(usqt)T 2
img + v2r / lR0r
( )( )cos3(usqr)T 2
img
(5)
where θsqr and θsqt are the squint angles of receiver andtransmitter at ηc, respectively, and Timg is the observationtime of the target by the composite footprint of transmitter
IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392doi: 10.1049/iet-rsn.2012.0286
Fig. 2 Block diagram of omega-K
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and receiver. Similarly, for the transmitter, we have [8]
kt =TBPt
TBPt + TBPr
= v2t / lR0t
( )( )cos3(usqt)T 2
img
v2t / lR0t
( )( )cos3(usqt)T2
img + v2r / lR0r
( )( )cos3 (usqr)T
2img
(6)
The Doppler centroids of receiver, fdcr, and transmitter, fdct,can be written as [8]
fdcr =f + f0c
vr sin(usqr)
fdct =f + f0c
vt sin(usqt)(7)
Obviously, the total Doppler centroid is fdc = fdct + fdcr.Meanwhile, Fr and Ft are defined as [8]
Fr =���������������������f + f0
( )2− cfhrvr
( )2√
Ft =���������������������f + f0
( )2− cfhtvt
( )2√ (8)
Doppler bandwidth, Δfη, can be approximately formulated by[17]
Dfh �v2t cos
3 usqt
( )lR0t
+v2r cos
3 usqr
( )lR0r
⎛⎝
⎞⎠Timg (9)
The parameters which are effective in obtaining MELBFspectrum were clarified above. The spectrum derivation is afunction of zero-Doppler times of transmitter and receiver,which do not have a linear relation in general BSARgeometry [9]. As offered in [9] or [11], by the linearexpression of η0t over η0r and the zero-offsetreceiver-to-target range variable, r, the space variations ofspectrum can be distinguished and the foundation ofresultant image formation algorithm is facilitated.
h0t = p10 + p11r + p12h0r (10)
where
p10 = h0t
∣∣r=0,h0r=0
p11 =∂h0t
∂r
∣∣∣∣r=0,h0r=0
p12 =∂h0t
∂h0r
∣∣∣∣r=0,h0r=0
r = R0r − Rm
(11)
and Rm is the closest range from the scene centre to thereceiver.Substituting (10) in the bistatic phase,Ψ( f, fη) and
accompanying it by some standard analysis and
IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392doi: 10.1049/iet-rsn.2012.0286
simplifications, we will have [17]
C = 2p h0r(kr + ktp12)fh + (p10 + p11r)ktfh
[
+ R0r
c
ar1ar2
Drf + R0t
c
at1at2
Dtf + R0r
cFr +
R0t
cFt
](12)
where
ac =ktvr sin(usqr) − krvt sin(usqt)
l
ar1 =l
vr(krfh + ac)
ar2 =l
vrac
at1 =l
vt(ktfh − ac)
at2 = − l
vtac
Dr =��������1− a2
r1
√
Dt =��������1− a2
t1
√
(13)
So far, the signal model and the interested spectrum have beenbriefly discussed. Deliberately, we tried to use the symbols ofparameters identical to their corresponding references forbetter understanding. As mentioned earlier, MELBFspectrum was employed in RDA [16], but not in otheralgorithms. However, RDA and CSA have been developedbased on a similar spectrum derived by usingtwo-dimensional PSP too [17]. In the next section, we willuse MELBF in the omega-K algorithm and propose a newimage formation method.
3 Focusing procedure for the parallelconfiguration
The block diagram of the proposed omega-K algorithm,which is to be implemented based on MELBF spectrum, isshown in Fig. 2. It is suggested to be utilised in bistaticconstellations with nearly parallel trajectories of transmitterand receiver. It consists of the following major steps:A.1 By performing range and azimuth FFTs, the received
signal is transformed into the two-dimensional frequencydomain. Since MELBF is our choice of BPTRS, (3) and(12) are valid after this step. We can use the followinglinearisation operations in the case of parallel configuration[9, 11, 17], which relates the transmitter and receiver’sclosest ranges to the point target. This is because of thesame reason as that in (10) and ease of understanding and
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developing the focusing procedure.
R0t = p20 + p21r
R0r = r + Rm
(14)
where
p20 = R0t
∣∣r=0,h0r=0
p21 =∂R0t
∂r
∣∣∣∣r=0,h0r=0
(15)
Equation (12) can be rewritten using (14)
C= 2p h0r(kr + ktp12)fh+ (p10+ p11r)ktfh+r+Rm
c
ar1ar2
Drf
[
+ p20+ p21r
c
at1at2
Dtf + r+Rm
cFr +
p20+ p21r
cFt
](16)
Ψ is decomposed into three components: space-invariantterm, Ψiv, range-variant term, Ψrv, and azimuth scalingterm, Ψsc. They are as follows (see (17))Each term has its distinctive characteristics and hence, they
should be compensated in the corresponding space.A.2 Reference function multiplication (RFM) is
accomplished to fulfil some purposes includingtransformation of data into the baseband by using aphase factor exp(−j2π((Rm + p20)/c)f ) [11, 21], rangecompression and the space-invariant term removal, which isresponsible for bulk range cell migration (RCM),range-invariant range-azimuth coupling and range-invariantazimuth modulation. RFM filter is given by
HRFM(f , fh)= P∗(f ) exp(jciv) exp −j2pRm+ p20
cf
( )(18)
It is noteworthy to add that since ψiv is not range-dependentand azimuth-dependent, it could be simply removed intwo-dimensional frequency domain as above. Theremainder of bistatic signal after RFM is
Although the reference target is focused completely, othertargets are not, since they have non-zero Ψrv.
C = Civ +Crv +Csc
Civ = 2pp10ktfh + 2pRm
c
ar1ar2
Drf + 2p
Rm
c
Crv = 2p1+ p21( )
r
c
cp11ktfh + ar1ar2/D([
Csc = 2p kr + ktp12( )
h0rfh
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A.3 Stolt interpolation, which is a non-linear mapping oforiginal range frequency variable f to a new one, f′, correctsthe differential RCM, differential range-azimuth couplingand differential azimuth modulation, all being dependent onrange. It can be expressed as
cp11ktfh + ar1ar2/ Dr
( )( )f + Fr + p21 at1at2/ Dt
( )( )f + p21Ft
(1+ p21)
� f0 + f ′
(20)
Thus, Ψrv can be modified to the following linear function ofthe new range frequency variable
C′rv = 2p
1+ p21( )
r
cf0 + f ′
( )(21)
This step concludes the focusing of non-reference targets.A.4 To transform the data to the range-Doppler domain, a
range inverse fast Fourier transform (IFFT) is applied. Thesignal after this step takes the following form
S3(t, fh) = Rx t − r 1+ p21( )
c
( )Wa( fh) exp −jCsc
( )(22)
where Rx is the compressed signal envelope in range.A.5 The final step of image formation is transforming the
compressed data back to the two-dimensional time domainwith an azimuth IFFT. To remove the azimuth scaling termthat appears in general azimuth-variant configurations andregister the target accurately, we could use inverse scaledFFT (ISFT) with the scaling factor (kr + ktp12). So, we obtain
s4(t, h) = Rx t − r 1+ p21( )
c
( )Ry h− h0r
( )(23)
with Ry as the compressed signal envelope in azimuth.
4 Focusing procedure for the generalconfiguration
In this section, the modified version of previousimplementation is derived. It is based on the approximationof non-linear Stolt interpolation with a shift and a scalingwhich could be processed individually, evading expensiveinterpolation operation. The scaling and shift are done byan ISFT in the range direction with azimuthfrequency-dependent scaling factor and a phase multiply,respectively. Similar approaches have been traced for otherBPTRS derivations in [9, 17, 21]. The proposed
Fr + 2pp20c
at1at2
Dtf + 2p
p20c
Ft
r
)f + Fr + p21 at1at2/Dt
( )f + p21Ft
(1+ p21)
] (17)
IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392doi: 10.1049/iet-rsn.2012.0286
Fig. 3 Block diagram of approximated interpolation-free omega-K
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implementation for general azimuth-variant geometry, whichis shown in Fig. 3, has the following main steps.B.1 Transformation of the raw data into the
two-dimensional frequency domain is done. By consideringgeneral geometry, we can modify (14) to [9, 11]
R0t = p20 + p21r + p22h0r
R0r = r + Rm
(24)
where p20 and p21 are defined in (15) and
p22 =∂R0t
∂h0r
∣∣∣∣r=0,h0r=0
(25)
Replacing (24) in (12) makes bistatic phase to become(see (26))The phase includes three components: the space-invariantterm, Ψiv, which is independent of range and azimuthlocations of the target, the range-variant term, Ψrv and theterm Ψav which is azimuth-variant. (see (27))B.2 Similar to (18) and (19), RFM is carried out in
two-dimensional frequency domain to remove Ψiv. Thenon-reference targets will have the remaining RCM,range-azimuth coupling and azimuth modulation that willbe encountered later. Next, Fr and Ft in Ψrv are expanded
C = 2p h0r kr + ktp12( )
fh + p10 + p11([
+ p20 + p21r + p22h0r
c
at1at2
Dtf + r
C = Civ +Crv +Cav
Civ = 2pp10ktfh + 2pRm
c
ar1ar2
Drf + 2p
Rm
cF
Crv = 2p(1+ p21)r
c
cp11ktfh + ar1ar2/(D([
Cav = 2p kr + ktp12( )
h0rfh + 2pp22h0r
c
at1a
Dt
IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392doi: 10.1049/iet-rsn.2012.0286
by the first-order Taylor series around f = 0. We arrive at
Fr � Drf0 +1− ar1ar2
Drf
Ft � Dtf0 +1− at1at2
Dtf
(28)
Regarding these approximations, we have, in fact, ignored thesecond and higher order terms of the series, that is,range-variant range-azimuth coupling. This is the keyapproximation of this implementation in comparison to theprior one. Using (28) in Ψrv formula expressed in (27) gives
Crv = 2p(1+ p21)r
c
× cp11ktfh + Drf0 + p21Dtf0(1+ p21)
+ (1/Dr)f + p21(1/Dt)f
(1+ p21)
[ ](29)
Ψav is expanded with the first-order two-dimensional Taylorseries around f = 0 and fη = fdc|f =0. Since the desired outputof the algorithm is a magnitude image, the constant termsare neglected and we have the following equation where fdctis computed at f = 0
Cav = 2p(kr + ktp12)h0rfh
+ 2pp22h0r
c
− c2/v2t( )
ktfdct������������������f 20 − c2/v2t
( )f 2dct
√ fh
+ 2pp22h0r
c
1����������������1− l2/v2t
( )f 2dct
√ f
(30)
B.3 The second term in (29), being a linear function of f,represents the range-variant RCM and can be considered asan azimuth frequency-dependent scaling in the rangefrequency domain. ISFT along the range frequency isperformed to correct the range-variant RCM. It willtransform the data to the range-Doppler domain. Thescaling factor is
ara(fh) =1/Dr
( )+ p21/Dt
( )1+ p21( )
r)ktfh +
r + Rm
c
ar1ar2
Drf
+ Rm
cFr +
p20 + p21r + p22h0r
cFt
] (26)
r + 2pp20c
at1at2
Dtf + 2p
p20c
Ft
r))f + Fr + p21 at1at2/ Dt
( )( )f + p21Ft
(1+ p21)
]
t2 f + 2pp22h0r
cFt
(27)
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which depends on azimuth frequency. After this step, weobtain (see (31))It should be reminded that as azimuth-dependent
RCM, which is the third term in (30), is small, theeffect of azimuth frequency-variant scaling factor isnegligible and its value for the Doppler centroid has beensubstituted.B.4 The first term in (29) is residual azimuth modulation.
This term is range-variant as well as azimuthfrequency-variant and can be removed using a simple phasemultiply in the range-Doppler domain. This can be written as(see (32))B.5 Range FFT is used to transform the data back to
two-dimensional frequency domain, giving the followingresult (see (33))
S3(t, fh) = Rx t − r 1+ p21( )
c− p22
ch0r √
⎛⎜⎝
× exp −j2pr
c(cp11ktfh + Drf0 +
[
× exp −j 2p kr + ktp12( )
h0rfh
⎡⎢⎣
⎛⎜⎝
S4(t, fh) = S3(t, fh) exp j2pr
c(cp11ktfh + D
[
= Rx t − r 1+ p21( )
c− p22
ch0r √
⎛⎜⎝
× exp −j 2p(kr + ktp12)h0rfh
⎡⎢⎣
⎛⎜⎝
S4(f , fh) = Wa(fh)|P(f )|2 exp −j2pfr 1+ p(
c
⎛⎜⎝
⎛⎜⎝
× exp −j 2p kr + ktp12( )
h0rfh +
⎡⎢⎣
⎛⎜⎝
S5(f , h) = |P(f )|2Ry h− h0r
( )exp −j2pf
r 1+(⎛⎜⎝
⎛⎜⎝
S6(f , h) = S5(f , h) exp j2pfp22c
h0r1����������������
1− l2/v2t( )
f 2dct
√ara fdc
(⎛⎜⎝
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B.6 The scaling factor of azimuth frequency
aaz = kr + ktp12 +p22c
− c2/v2t( )
ktfdct�����������������f 20 − c2/v2t
( )f 2dct
√is a constant and can be corrected using another ISFT. Theresult can be formulated as (see (34))However, if correct registration is not mandatory, an azimuthIFFT can be used, instead. It should be mentioned that largemagnitudes of scaling make using ISFT difficult.B.7 Azimuth-dependent RCM, which is the third term in
(30), appears in the exponential term in (34). It can beremoved by a simple phase multiply in azimuth time-rangefrequency domain. (see (35))
1����������������1− l2/v2t
( )f 2dctara fdc
( )⎞⎟⎠Wa( fh)
p21Dtf0)]
+ 2pp22h0r
c
− c2/v2t( )
ktfdct������������������f 20 − c2/v2t
( )f 2dct
√ fh
⎤⎥⎦⎞⎟⎠
(31)
rf0 + p21Dtf0)]
1����������������1− l2/v2t
( )f 2dctara fdc
( )⎞⎟⎠Wa( fh)
+ 2pp22h0r
c
− c2/v2t( )
ktfdct������������������f 20 − c2/v2t
( )f 2dct
√ fh
⎤⎥⎦⎞⎟⎠
(32)
21
)+ p22
ch0r
1����������������1− l2/v2t
( )f 2dct
√ara(fdc)
⎞⎟⎠⎞⎟⎠
2pp22h0r
c
− c2/v2t( )
ktfdct������������������f 20 − c2/v2t
( )f 2dct
√ fh
⎤⎥⎦⎞⎟⎠
(33)
p21)
c+ p22
ch0r
1����������������1− l2/v2t
( )f 2dct
√ara fdc
( )⎞⎟⎠⎞⎟⎠ (34)
)⎞⎟⎠ = |P(f )|2Ry h− h0r
( )exp −j2pf
r 1+ p21( )
c
( )( )(35)
IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392doi: 10.1049/iet-rsn.2012.0286
Table 1 Simulation parameters for the scenario I
Parameter Value Parameter Value
PRF, Hz 2000 reference target R0r, m 3231simulation time,s
PRF, Hz 5800 reference target R0r, m 6403simulationtime, s
2 reference target R0t, km 700
carrier freq.,GHz
5 reference target sin(θsqr) 0
antenna length,m
0.5 reference target sin(θsqt) 0.033
pulse duration,μs
10 angle betweentransmitter and receiver
path, rad
(π/3)
bandwidth,MHz
25 xn, m [4700 47004200 4200]
receivervelocity, m/s
150 yn, m [100 − 10050 − 50]
transmittervelocity, m/s
4000 range oversample 4
receiver height,m
5000 reference target Doppleroversample
6.61
transmitterheight, km
700 reference target Dopplerbandwidth, Hz
878
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B.8 Range IFFT is performed to complete the focusingprocedure. The final signal will be
s6(t, h) = Rx t − r 1+ p21( )
c
( )Ry h− h0r
( )(36)
Fig. 5 Focused targets in the scenario II
5 Simulation results
The correctness and performance evaluation of our algorithmhave been accomplished by simulations. To do this, weconsider three scenarios.First, a parallel airborne BSAR scenario is considered with
simulation parameters of Table 1 to compare the capability ofthe proposed procedures (A and B) in the processing of a pointtarget. The target coordinates symbolised by (xn, yn),hypothetical scene reference target coordinates, BSAR flightparameters and other necessary information are listed inTable 1.The responses of two procedures to the point target are
shown in Fig. 4 and their quality measurements are alsospecified in this figure. The utilised quality criteria areimpulse response width (IRW), peak sidelobe ratio (PSLR)and integrated sidelobe ratio (ISLR). It can be seen that
Fig. 4 Comparison of the two procedures
a Stolt interpolation-basedb ISFT-based
IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392 389doi: 10.1049/iet-rsn.2012.0286 & The Institution of Engineering and Technology 2013
Table 3 Point targets image quality measurements for thescenario II
Table 4 Simulation parameters for the scenario III
Parameter Value Parameter Value
PRF, Hz 2500 reference target R0r, m 3231simulation time,s
5 reference target R0t, m 5045
carrier freq.,GHz
10 reference target θsqr 17.2°
antenna length,m
0.5 teference target θsqt 11.2°
pulse duration,μs
10 xn, m 3400
bandwidth, MHz 50 yn, m 1200receivervelocity, m/s
100 range oversample 4
transmittervelocity, m/s
100 reference targetDoppler oversample
7.73
receiver height,m
1200 reference targetDoppler bandwidth, Hz
324
transmitterheight, m
1200 reference targetcoordinates
(x = 3000 m,y = 1000 m)
Table 5 Point target quality measurements for the scenario III
Parameter MELBF-based (proposedmethod)
ELBF-based
azimuth IRW(samples)
7.69 7.72
azimuth PSLR, dB − 12.57 − 9.33azimuth ISLR, dB − 9.46 − 6.80range IRW(samples)
4.01 3.98
range PSLR, dB − 12.58 − 12.73range ISLR, dB − 9.99 − 9.96
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both procedures reach relatively similar performance. Thisresult is generally true except for very highly squinted casesin which the performance of procedure (B) is degraded.Under these conditions, accurate Stolt interpolation(procedure (A)) is preferred. However, exactimplementation of Stolt interpolation increases theprocessing time considerably. Since both focusingprocedures have the same basis, they have similarperformance in most cases, and also procedure (B) can beused for more general BSAR geometries. This procedure isselected for reporting the next simulation outcomes; in fact,it covers the other one too.In the next simulation, we consider space-surface BSAR
with non-parallel trajectories of transmitter and receiver.The scene in x–y plane (refer to Fig. 1) consists of fourpoint targets whose coordinates along with other simulationparameters are listed in Table 2. The transmitted signal ischirp and the hypothetical reference target is assumed to beat (x = 4000 m, y = 0 m).The result without the azimuth scaling correction, that is,
using azimuth IFFT in B6 step, is shown in Fig. 5. FromFig. 5, it is clear that the azimuth relative locations oftargets to the reference target are compressed. In this case,the azimuth scaling is noticeably severe (aaz = 0.16).However, in most cases, especially for airborne orspaceborne BSAR, the azimuth scaling is trivial.
Fig. 6 Point spread functions of the targets in the scenario II
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Point target quality measurements for all targets are givenin Table 3. The PSLR and ISLR are within 2 and 1 dB ofthe theoretical values of − 13.3 and − 10 dB forrectangular weighting, respectively, showing a goodfocusing quality. The azimuth and range IRWs meet the
IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392doi: 10.1049/iet-rsn.2012.0286
Fig. 7 Comparison of different implementations of omega-K
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theoretical limits, which are approximately equal to thecorresponding oversampling rates. The point targetresponses are examined in more detail in Fig. 6. It shouldbe mentioned that for proper measurements and display, aninterpolation by a factor of 32 in the vicinity of the peak isemployed.In [8], it is demonstrated that MELBF spectrum is
generally more accurate than ELBF spectrum, particularlyin high squint configurations. Hence, we expect that anyimage formation algorithm based on MELBF has betterimaging results. Additionally, unlike LBF or ELBF, whichneed blocking operation to compensate the bistaticdeformation phase term, MELBF spectrum does not sufferfrom this inefficiency. Avoiding block preprocessingoperation or selecting broader range-azimuth subsections inELBF-based image formation algorithms, which is for thepurpose of reducing computational complexity, makesthem erroneous even in lower squint angles. To illustratethis, in the third scenario, we also simulate the omega-Kalgorithm with ELBF spectrum, similar to the work donein [11, 12]. Here, we consider an azimuth-invariantairborne BSAR transmitting a chirp signal (Table 4). Thesimulated point target is away from the reference target by400 m in x and 200 m in y-direction and no blocking isused. The squint angles of receiver and transmitter arenearly moderate.The quality measurements for both methods (the proposed
MELBF-based omega-K and ELBF-based omega-K ) areshown in Table 5. We discover that while both algorithmshave acceptable performance in range, just the proposedMELBF-based omega-K algorithm works appropriately inazimuth. To clarify the event, the azimuth impulseresponses are compared in Fig. 7, where 4000 samplesaround the peak are highlighted.It is important to note that in our proposed algorithm, kr and
kt are assumed to be constant for the whole scene. But they areweakly dependent on squint angles as well as range. Thisapproximation makes a phase error which should be withinthe acceptable region [− (π/4), (π/4)] [18]. This conditionenforces a maximum allowable scene size in range andazimuth regions. If our desired scene is broader than theseregions, it should be divided into several blocks andprocessing should be done independently.
IET Radar Sonar Navig., 2013, Vol. 7, Iss. 4, pp. 383–392doi: 10.1049/iet-rsn.2012.0286
It can be easily shown that the computational complexity ofthe proposed algorithm is of the order O(N2log2(N )) whereN2 is the number of SAR raw image data array. Thiscomputation requirement is of the same order with otherfrequency-domain approaches and less than time-domainback-projection algorithm of the order O(N3) [8].
6 Conclusion
In this paper, we have proposed the omega-K algorithm that isbased on a recent accurate bistatic point target referencespectrum (MELBF). Two focusing procedures, based onStolt interpolation and ISFT, are analytically derived for thegeneral and nearly parallel configurations, respectively.Although Stolt interpolation-based method is more accurate,ISFT method seems less time-expensive. The proposedalgorithm was verified by simulation. We also establishedthe superiority of the proposed algorithm in comparison toELBF-based one in high squint scenarios or when blockingoperation is neglected.
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