DTIC ADA469563: On the Design of SAR Apertures using the Cramer-Rao Bound

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19-06-2007 Technical Paper 1 July 2005 – 1 July 2007
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER

On the design of SAR apertures using the Cramer-Rao Bound 5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER
61102F
6. AUTHOR(S) 5d. PROJECT NUMBER
2311
Robert, Linnehan, David Bready, John K. Schindler, Leonid Perlovsky, Muralidhar 5e. TASK NUMBER
Rangaswamy HE
5f. WORK UNIT NUMBER
01
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
AFRL/SNHE REPORT
80 Scott Drive
Hanscom AFB MA 01731-2909
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Electromagnetics Technology Division Source Code 437890 ACRONYM(S)
AFRL-SN-HS
Sensors Directorate
Air Force Research Laboratory
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80 Scott Drive
NUMBER(S)
Hanscom AFB MA 01731-2909
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Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES
ESC Clearance Number: 05-0017. Published: IEEE Transactions on Aerospace and Electronic Systems, Vol. 43, No. 1, January
2007
14. ABSTRACT
The Cram´er-Rao inequality is applied to the likelihood function of the synthetic aperture radar (SAR) scatterer parameter vector to
relate the choice of flight path to estimation performance. Estimation error bounds for the scatterer parameter vector (including
height) are developed for multi-dimensional synthetic apertures, and quantify the performance enhancement over a limited sector of
the image plane relative to standard-aperture single-pass SAR missions. An efficient means for the design and analysis of SAR
waveforms and flight paths is proposed using simulated scattering models that are limited in size. Comparison of the error bounds to
those for standard-aperture SAR show that estimates of scatterer range and cross-range positions are accurate for multi-dimensional
aperture SAR, even with the additional estimator for height. Furthermore, multi-dimensional SAR is shown to address the layover
problem.
15. SUBJECT TERMS
Cramer-Rao Bounds, synthetic aperture radar, SAR, multi-dimensional apertures
16. SECURITY CLASSIFICATION OF: 17.LIMITATION 18.NUMBER 19a. NAME OF RESPONSIBLE PERSON
OF ABSTRACT OF PAGES Leonid Perlovsky
a. REPORT b. ABSTRACT c.THIS PAGE 19b. TELEPHONE NUMBER (include area code)
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Prescribed by ANSI Std. Z39.18

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I. INTRODUCTION
Designs for enhanced resolution of synthetic
On the Design of SAR aperture radar (SAR) systems have been addressed
in many ways. In [1], the design of a microstrip
Apertures using the antenna architecture was considered for spaced-based
SAR applications. In [2], an innovative Cassegrain
Cram´er-Rao Bound antenna was shown to improve system performance
for space-based SAR systems. In [3], the authors
proposed SAR enhancement by introducing a
three-dimensional modular filtering architecture
in the preprocessing step. In [4], SAR system
ROBERTLINNEHAN,StudentMember,IEEE performance was improved by the joint specification
AirForceResearchLaboratory
and design of the SAR system and platform. Many
DAVIDBRADY,Member,IEEE other approaches to improving SAR performance
NortheasternUniversity
have been considered, including digital filtering
JOHNSCHINDLER,Fellow,IEEE [5], searching by parallel supercomputers [6],
AnteonCorporation
and classification by support vector machines
LEONIDPERLOVSKY,SeniorMember,IEEE
[7].
MURALIDHARRANGASWAMY,Fellow,IEEE To date, little work has been published which
AirForceResearchLaboratory
approaches SAR enhancement via aperture
adjustment. Standard SAR employs a 1-D aperture
(a synthetic linear array) and assumes a nominal
flat ground when processing 2-D range/cross-range
TheCrame´r-Raoinequalityisappliedtothelikelihood
images. This removes the range/height ambiguity
functionofthesyntheticapertureradar(SAR)scatterer
simply by not estimating height, resulting in the
parametervectortorelatethechoiceofflightpathto
so-called layover problem [8, 9]. In [9] and [10]
estimationperformance.Estimationerrorboundsforthe the layover problem is thoroughly addressed using
scattererparametervector(includingheight)aredeveloped cross-track, multibaseline SAR interferometry. In
formulti-dimensionalsyntheticapertures,andquantifythe a sense, this is an aperture adjustment, requiring
performanceenhancementoveralimitedsectoroftheimage the use of multiple antennas, multiple platforms, or
planerelativetostandard-aperturesingle-passSARmissions.An multiple passes. Reference [11] describes a technique
efficientmeansforthedesignandanalysisofSARwaveformsand where a curved flight path with sufficient angular
flightpathsisproposedusingsimulatedscatteringmodelsthat diversity creates a stereo pair in spotlight mode
arelimitedinsize.Comparisonoftheerrorboundstothosefor that is then used to measure scatterer height and
resolve layover. This too is an aperture adjustment
standard-apertureSARshowthatestimatesofscattererrangeand
that provides vertical excursion of the radar platform
cross-rangepositionsareaccurateformulti-dimensionalaperture
for scatterer height estimation. By contrast, we
SAR,evenwiththeadditionalestimatorforheight.Furthermore,
examine the performance of nonlinear flight
multi-dimensionalSARisshowntoaddressthelayoverproblem.
paths forming 2-D apertures (sparse synthetic
planar arrays) that decouple the range/height
ambiguity, resolving the layover problem while
providing the potential for 3-D imagery. Unlike
interferometry, this can be done using single-pass,
ManuscriptreceivedSeptember21,2005;revisedApril4,2006; monostatic SAR without making multiple data
releasedforpublicationMay25,2006. collections. Crame´r-Rao bounds (CRBs) do not
IEEELogNo.T-AES/43/1/895039. depend on the methods or techniques to process
the observation, hence the design is not restricted
RefereeingofthiscontributionwashandledbyV.C.Chen.
to splitting data into stereoscopic components
ThisworkwassupportedinpartbytheAFOSR. or spotlighting. In this work we also examine a
Authors’addresses:R.Linnehan,L.Perlovsky,andM. 3-D synthetic aperture where excursions in the
Rangaswamy,AirForceResearchLaboratory,SensorsDirectorate, range and height dimensions can further decouple
80ScottDr.,HanscomAFB,MA01731-2909,E-mail:
the range/height ambiguity and improve overall
([email protected]);D.Brady,Dept.ofElectrical
performance.
andComputerEngineering,NortheasternUniversity,Boston,MA;
J.Schindler,AnteonCorporation,HanscomAFB,MA. Previous literature describes the use of CRBs for
radar parametric estimation techniques. In [9] and
0018-9251/07/$25.00°c 2007IEEE [12], the authors derive CRBs for the interferometric
344 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 1 JANUARY 2007
phases to evaluate phase estimation techniques.
In [13], the author computes bounds for various
target parameters, including amplitude, phase
and spatial angle. In [14], CRBs are used to
evaluate spectral estimation methods applied to
curvilinear SAR. In this work we consider the joint
estimation problem, and compute the CRBs for all
parameters given any aperture that is synthesized. We
illustrate that the CRB provides a computationally
efficient means to quantify the influence of SAR
waveforms, flight paths, and scattering models on
the parameter estimation problem. In Section II
we introduce the SAR signal and scattering
models used in simulations. Section III depicts a Fig. 1. Illustration of coordinate system used in this work. Radar
mathematical method for computing the CRBs. In platform is at position (xac(t),yac(t),hac(t)) relative to centroid of
2-D scatterer grid on (x,y)-plane. For the case of standard SAR
Section IV we describe experimental simulations
1-D aperture, y and h are constants.
that use bounds to compare standard SAR with ac ac
multi-dimensional aperture SAR. Finally, conclusions
and future directions of the research are offered in positions over the synthetic array and k indexes the
Section V. frequency components that synthesize the emitted
pulse in fast-time [17]. Although in practice a
n
is frequency and aspect angle dependent, we assume
II. SCATTERING AND SIGNAL MODELS
constant, isotropic scattering throughout this
work.
In this work we consider a finite number of
The observation model is
scatterers positioned on a 2-D grid. The centroid of
this grid defines the origin of an (x,y,h) Cartesian
r(t,k)=s(t,k)+n(t,k) (2)
coordinate system, with the grid lying in the (x,y)
plane. Scatterer n is located at cross-range xn, range where the set fn(t,k)g is comprised of independent
yn, and height hn, and at time t the radar platform is and identically distributed complex circular Gaussian
located at (xac(t),yac(t),hac(t)). For the special case of random variables.
a 1-D aperture, both y and h would be constants,
ac ac We adopt the following vector notation
but for general 3-D apertures all three parameters
are functions of t. Fig. 1 illustrates this coordinate r=[r(1,1):::r(1,N):::r(N,1):::r(N,N)]T
k t t k
system.
g =[g (1,1):::g (1,N):::g (N,1):::g (N,N)]T
For each time-frequency sample index (t,! ), with n n n k n t n t k
k (3)
t2f1,2,:::,Ng and k2f1,2,:::,Ng the noiseless
t k G=[g :::g ]
received signal s(t,k) is given by 1 N
a=[ja j:::ja j]T:
N 1 N
ej!kt¢s(t,k)=ej!kt¢ janjgn(t,k) The signal and noise vectors, s and n, are similarly
Xn=1 defined. With this notation, the observation vector
may be represented as
1 (1)
g (t,k)= ej(Án¡¯k2Rn(t))
n Rn2(t) r=s+n
=Ga+n: (4)
R (t)= ¢x2(t)+¢y2(t)+¢h2(t)
n n n n
q
where ¢x (t)=x ¡x (t), ¢y (t)=y ¡y (t), and
n n ac n n ac
III. CRB DERIVATION
¢h (t)=h ¡h (t). Here the summation is over the N
n n ac
scatterers, R (t) is the range from the radar to scatterer
n
The CRB proceeds from the development of
n at time t, and ¯ is the wavenumber for frequency
k
! . The complex reflectivity of scatterer n is a , with maximum likelihood estimation (MLE) [18]. The
k n
magnitude ja j and phase Á . We disregard the leading effectiveness of the maximum likelihood procedure
n n
factors in the first line of (1) in the sequel, as they is revealed by computing the variance of the estimate,
are known to the receiver, and the distribution of a task frequently difficult to perform [19]. However,
the additive noise is invariant to these rotations. a lower bound on the variance is often easier to
In SAR nomenclature, t indexes the slow-time compute using CRB. The result not only applies
LINNEHANETAL.:ONTHEDESIGNOFSARAPERTURESUSINGTHECRAME´R-RAOBOUND 345
to MLE but is in fact valid for any unbiased or yields the following FIM components:
asymptotically unbiased estimator.
@2` 2
For a vector of unknown parameters , CRB E ¡ = RefgHg g
@ja j@ja j ¾2 m n
establishes a lower bound on the error covariance ½ n m ¾ 0
matrix of unbiased estimates of [18]. The error @2`
E ¡ =0
covariance matrix C for the estimator ˆ is bounded ½ @janj@vn¾
by the inverse of the Fisher information matrix @2` 4ja j
(FIM) J, E ¡@ja j@v = ¾2m ImfgHn¨vmgmg for m6=n
½ n m¾ 0
C=Ef[ ˆ¡ ][ ˆ¡ ]Tg¸J¡1 (5) @2` 8ja j2
E ¡ = n gH¨ ¨ g
@v2 ¾2 n vn vn n
where Ef:::g is the expectation operator, and the ½ n¾ 0
inequality is to be interpreted in the positive-definite @2` 8ja j2
E ¡ = n gH¨ ¨ g
sense. The (i,j)th element of J is computed by @v @w ¾2 n vn wn n
½ n n¾ 0 (10)
averaging a second-order mixed partial derivative of
@2` 8ja jja j
the log-likelihood function `, E ¡ = n m RefgH¨ ¨ g g
@v @w ¾2 n vn wm m
½ n m¾ 0
@2 @2` 2ja jja j
J =E ¡ ` : (6) E ¡ = n m RefgHg g
ij @ @ @Á @Á ¾2 n m
( i j ) ½ n m¾ 0
@2`
The ith diagonal element of J is known as the E ¡ =0
@Á @ja j
Fisher information for the parameter . The ½ n n ¾
i
off-diagonal elements of J, when i6=j, is called the @2` 2ja j
E ¡ =¡ n ImfgHg g for m6=n
cross-information between the parameters and . @Á @ja j ¾2 n m
i j ½ n m ¾ 0
Given statistical independence of the white noise
@2` 4ja jja j
components, their joint probability density function E ¡ =¡ n m RefgH¨ g g:
@Á @v ¾2 n vm m
has the circular-complex Gaussian form [20] yielding ½ n m¾ 0
the likelihood function of the received signal Fisher information for the uniformly distributed
random phases Á of each scatterer are computed
n
1
f(rjs)= e¡(1=¾02)(r¡s)H(r¡s) (7) but treated as nuisance parameters. It was shown by
(¼¾2)NtNk Gini [22] that assuming knowledge of a parameter
0
and excluding it in the FIM lowers, or at least has
where H in the exponent indicates the Hermitian
no affect on, the variance bounds for the remaining
(conjugate transpose) of the column vector, and ¾2=2
0 parameters. In this sense, the CRBs for the parameters
is the variance of each in-phase and quadrature white
of interest are larger when the random phases are
noise term.
considered unknown and included in J.
Components of the FIM follow directly from A compact description of the FIM for SAR is
the Gaussian form of the likelihood function and achievable by defining
the well-known Slepian-Bangs formula [21].
v˜ =2ja j¨ g
We may concisely present these results if we n n vn n (11)
define V=[v˜ v˜ :::v˜ ]
1 2 N
¨ =diag ¢vn(1)¯ ¢vn(1)¯ ¢¢¢¢vn(1)¯ where v˜n can be x˜n, y˜n or h˜n, and V can be X, Y or H.
vn R (1) 1 R (1) 2 R (1) Nk Also, we define
· n n n
¢v (2) ¢v (N) ©=¡[ja jg ja jg :::ja jg ] (12)
n ¯ ¢¢¢ n t ¯ (8) 1 1 2 2 N N
R (2) 1 R (N) Nk
n n t ¸ and construct the block matrix
where v assumes the parameters x, y, or h. We make M=[X Y H ©]: (13)
the following approximation
Fisher information and cross-information among the
@s(t,k) 2¯ ja j scattering positions and phases are given by the matrix
¼¡j k n ¢v (t)g (t,k) (9)
@vn Rn(t) n n J = 2 RefMHMg: (14)
which follows since the derivatives @R¡2(t)=@v are xyhÁ ¾02
n n
relatively small in magnitude for all t. The matrix containing information of scattering
Below denote v and w as two distinct positional magnitudes is
parameters x, y, or h, and ` is the natural log of the 2
J = RefGHGg: (15)
likelihood function in 7. The Slepian-Bangs formula jaj ¾2
0
346 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 1 JANUARY 2007
Finally, the matrix P yields cross-information among
the scattering magnitudes and positions and is of
dimensions 4N£N
2
P= ImfMHGg: (16)
¾2
0
The complete FIM for SAR has order 5N and the
following form Fig. 2. With the target center located, we increase grid size and
examine effects additional scatterers have on its CRB. CRB
J P
xyhÁ asymptotes and minimal number of scatterers N used to replicate
J= : (17)
PT J clutter around target is determined.
· jaj¸
The CRBs for each scatterer are on the diagonal of
J¡1 in the form Typical SAR processing estimates scattering
amplitudes for predetermined locations on a large
diag(J¡1)=[CRB CRB :::CRB CRB :::
x1 x2 xN y1 patch of ground. The number of estimated positions
CRB :::CRB :::CRB :::CRB ] of the ground patch depends on the pixel size desired
yN hN ÁN jaNj by the operator. Generally speaking, pixel dimensions
(18)
are based on the resolution in range and cross-range
where CRB denotes the CRB for parameter q for the provided by the radar. The image of a large swath
qn
nth scatterer. could include millions of scattering measurements.
Thus in many cases the large computational
requirements necessitates off-line formation of the
IV. MULTI-DIMENSIONAL APERTURES
high-resolution images.
The following subsections describe a Jointly computing the CRBs for parameters
computationally efficient method of simulating a SAR x , y , h , ja j and the nuisance parameter Á of
n n n n n
process in order to compute CRBs on the scattering a single scatterer involves the inversion of a 5£5
parameters of interest. First we describe a way to matrix J. Although a telling exercise, it may be more
greatly reduce the number of scatterers modeled informative to observe lower error bounds for many
around one scatterer of interest so the effects of scattering elements within the desired ground patch.
clutter may be examined. We then describe a method Thorough analysis of a radar system or the design
to undersample the aperture and the transmitted of a new system would entail computing CRBs for
waveform, while averting the effects of grating millions of scatterers, not just one, i.e., observing
lobes. The resulting simulation is used to evaluate how surrounding clutter affects the CRBs. However,
the legitimacy of multi-dimensional apertures and constructing and inverting such a large FIM is not
compare performance with standard SAR processing. feasible using a desktop computer for analysis.
One method to overcome this complexity is to
simply surround a scatterer of interest, hereafter
A. Minimized Scattering Model
known as the target, with just enough scatterers to
In this paper we emphasize Ku-band SAR which approach the effects of an infinitely large area of
might be used on an agile unmanned air vehicle clutter and yet small enough to limit the size of J.
(UAV) to produce high resolution images. The radar The number of scatterers needed to meet these criteria
is assigned the following parameters: depends, among other things, on the spacing among
the scatterers and their strength relative to the targets,
i.e., the signal-to-clutter ratio (SCR). This number
center frequency f =16:7 GHz is determined by computing the CRB for the target
c
bandwidth B=1:67 GHz alone and examining the increase in this bound as the
synthetic aperture L=1176 m number of surrounding scatterers increases. At some
aircraft velocity v =56 m/s
ac point the additional clutter no longer influences the
aircraft height h =3000 m
ac CRBs of the target and an asymptote is reached. We
nominal range to target R =6708 m
0 start by computing the CRB on estimating ja j for the
n
target only. Then we increase the number of cells to a
3£3 grid and compute the CRB on estimating ja j for
Based on these values, approximate resolution in n
all N=9 scatterers. Then we increase the cells to a
range and azimuth are, respectively, [17]
5£5 grid, and then to a 7£7 and so on (see Fig. 2),
c ¸R computing CRBs for all scatterers.
± ¼ =0:090 m, ± ¼ 0 =0:051 m (19)
r 2B a 2L Fig. 3 shows the CRB for only the target
where c is the speed of light. magnitude as the grid size increase. Spacing among
LINNEHANETAL.:ONTHEDESIGNOFSARAPERTURESUSINGTHECRAME´R-RAOBOUND 347
Fig. 3. Bound for reflectivity, CRB , is seen to asymptote as
jaj
number of scatterers increases (depending on scatterer density).
Fig. 4. Magnitude response in cross-range and range dimensions
with first quasi-grating lobes indicated by arrows. Reducing the
PRF to 2 pulses/s and sampling the pulse every 100 MHz
the cells is tried at 0.20 m and at 0.25 m. We can
positions the grating lobes beyond the region of interest in this
see the bound approaches an asymptote at about the
experiment.
3£3 grid size for the larger spacing, and at about the
7£7 when the scatterers are more closely spaced. The
collected and the range to target at broadside,
difference indicates that target parameters are more
difficult to resolve when the surrounding scatterers NN
¾2= t k : (20)
are closer together. For simulations where the effects 0 SNR¢R4
0
of clutter are examined we will construct a 3£3 or
7£7 depending on the average spacing among the
C. Vertical Aperture
scatterers. This will create a clutter patch large enough
to replicate an infinitely large area around the target.
Standard SAR range resolution is achieved by
The target will likely have the largest CRBs since it
narrowing an uncompressed received pulse that has
is located among the most dense clutter. Influence
sufficient bandwidth. Resolution in cross-range is
from surrounding scatterers on the target parameter
achieved with a narrow beamwidth produced over
estimates is inversely proportional to their distance
the synthetic aperture created by linear cross-range
from the target [16].
motion. However, as would be expected, introducing
height estimation results in an ill-conditioned matrix
J if some aperture in the vertical dimension is not
B. Undersampled Aperture and Chirp
included. This is intuitively known and mathematically
satisfied by examining the matrix J for a single
Typical pulse repetition frequencies (PRF) of xyh
scatterer,
Ku-band SAR are on the order of 1 kHz requiring
tens of thousands of pulses to achieve the desired x˜Hx˜ x˜Hy˜ x˜Hh˜
1 1 1 1 1 1
resolution. However, in our simulations we show that 2
the PRF can be substantially reduced. Although the Jxyh= ¾22y˜H1x˜1 y˜H1y˜1 y˜H1h˜13: (21)
undersampled, or sparse, aperture causes quasi-grating 0 6h˜Hx˜ h˜Hy˜ h˜Hh˜ 7
lobes in the azimuth dimension, these will not impair 6 1 1 1 1 1 17
4 5
the analysis if they are physically beyond the region Because ¢y1 and ¢h1 are constants and not a function
of interest. The sampling rate of the linear FM pulse of t using standard, side-looking SAR, the determinant
can also be reduced to further ease computational of (21) is zero. Another way of stating this is the
complexity. The pulse sampling rate is minimized correlation coefficient between y˜ and h˜ is unity,
1 1
while assuring that quasi-grating lobes in range
y˜Hh˜
dimension are beyond the geometric limits of the ½ = 1 1 =1:0: (22)
yh
scattering model (§»0:75 m). Fig. 4 shows the y˜Hy˜ h˜Hh˜
magnitude response of the white noise matched filter 1 1 1 1
q
in the azimuth (a) and range (b) dimensions (given In order to reduce ½ , some variation in ¢h is
yh 1
the operational parameters mentioned above) with a required. This might lead one to believe that simply
PRF of 2 pulses/s and a frequency sampling period directing the radar platform at some non-zero
of 100 MHz. The first quasi-grating lobes are well angle relative to the ground plane will satisfy the
beyond the clutter swath used in the simulations. requirement of both azimuthal and vertical apertures.
The signal-to-noise ratio (SNR) accumulated over Fig. 5 shows a linear flight path adjusted to an
the aperture is selected to be 10 dB. The noise power angle ° and the radar’s position relative to a single
is appropriately scaled based on the amount of data scatterer located at ground range ¢y. Note here
348 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 1 JANUARY 2007
Fig. 6. 2-D aperture in form of sinusoidal flight path
parameterized by spatial amplitude A and frequency f .
ac ac
Fig. 5. Linear aperture at angle ° relative to horizon creates
singular FIM J . Scatterer height cannot be estimated along with
xyh
range in this case.
that both ¢x and ¢h are functions of t. However,
further investigation reveals that this FIM is also
ill-conditioned. We again examine the block matrix of
J in (21), and since the factors defined for x˜, y˜ and h˜
are identical in the single scatterer case except for the
geometric differentials, the matrix can be simplified to Fig. 7. Examples of flight paths that employ 1-D, 2-D, and 3-D
apertures.
¢xT¢x ¢xT¢y ¢xT¢h
1 1 1 1 1 1
Jxyh»2¢yT1¢x1 ¢yT1¢y1 ¢yT1¢h13: (23) demonstrated in Fig. 6. Ultimately the 2-D aperture
¢hT¢x ¢hT¢y ¢hT¢h reduces ½ and this lower correlation begins to
6 1 1 1 1 1 17 yh
4 5 mitigate the range/height ambiguity, allowing these
Looking again at Fig. 5 we see that the height above
parameters to be estimated.
the scatterer at any position t along the array is equal
For our simulations we try to maintain nearly
to the current change in height from the original
identical distances between array positions when
position on the aperture minus the radar height at
creating 2-D apertures. This is achieved by
broadside h (b). Thus we can define
n determining the separation distance d between array
¢h (t)¡¢h (b) positions based on the PRF and the speed of the
tan(°)= n n (24)
¢x (t) aircraft, and using the general formula for computing
n
the arc length of a function f(x) between consecutive
and
positions [23] (see appendix)
¢h (b)
tan(Á)= n : (25)
¢y b
n d= 1+[f0(x)]2dx: (27)
Then the height at any t is Za
p
A third dimension included in the aperture further
¢h (t)=tan(Á)¢y ¡tan(°)¢x (t): (26)
n n n enhances performance relative to the 2-D case. We
Since ° and Á are constants for a linear array, the row accomplish this by applying a cosine function to the
and column associated with ¢h in (23) are linear flight path in the range dimension (see Fig. 7). This
1
combinations of the other two rows and columns, is analogous to a 3-D curve traced out by a circularly
yielding a singular matrix of rank 2. Therefore, not polarized wave. As a result the correlation coefficients
only is vertical aperture required for height estimation among the geometric parameters, ½ , ½ , and ½ are
yh xy xh
but some nonlinearity in the aperture formation, further decreased, subsequently reducing the CRBs for
offering spatial diversity in a 2-D plane is also needed. all parameters of interest.
One method of vertical excursion that achieves a A question arises as to whether an acceptable
2-D aperture is a sinusoidal flight path, parameterized ambiguity function is maintained when the synthetic
by spatial amplitude A , spatial frequency f , aperture includes additional dimensions. Fig. 8(a)—(c)
ac ac
and phase relative to its broadside position as show the magnitude response for standard 1-D SAR
LINNEHANETAL.:ONTHEDESIGNOFSARAPERTURESUSINGTHECRAME´R-RAOBOUND 349
Fig. 8. Ambiguity functions for 1-D, 2-D, 3-D apertures. Mainlobes of multi-dimensional cases retain shape similar to that of standard
SAR. (a) Magnitude response for 1-D aperture. (b) Magnitude response for 2-D aperture. (c) Magnitude response for 3-D aperture.
and for the alternate 2-D and 3-D flight paths. It is
evident from these plots that the mainlobe of the point
spread function sufficiently retains its shape, and the
grating lobes are still physically beyond the region of
interest.
D. Comparison of CRBs for Multi-Dimensional
Apertures
Computations were performed on a 7£7 cell
grid (49 scatterers) with the target scatterer having
complex reflectivity a located at the center. The
t
cells are arranged so that their centers are nominally Fig. 9. Scattering model realization with target in center cell (}),
0.25 m apart. To avoid any irregularity in the results and clutter randomly positioned within surrounding cells.
because of the periodic placement of the scatterers,
their positions within each cell is random, uniformly realizations. The model for standard SAR assumes
distributed in range and cross-range. The limits on a linear, side-looking flight path and no height
the distribution are 1/4 of the cell spacing so in estimation, so from (13) we have
this case the closest two scatterers can be to one M =[X Y ©]: (28)
another is 0.125 m. This minimum separation is 1-D
still greater than the range and azimuth resolutions Excluding H in the FIM implies that scatterer heights
according to (19). The heights of all the scatterers are known for the 1-D aperture case. As stated
are Gaussian distributed with mean height 1 m and previously, the processing assumes a nominal flat
standard deviation ¾ =0:125 m. Finally, the real ground level and thus the CRBs for the remaining
h
and imaginary components of scatterer amplitude parameters should inherently be smaller relative to
are randomly generated using independent, standard those of the multi-dimensional aperture cases that do
Gaussian distributions. Because the results are so include height estimation.
heavily dependent on the realization, especially the The number of pulses (total transmitted power)
geometric parameters, the average of a series of and aircraft velocity is consistent in all three aperture
random realizations is computed in the following models. The 1-D aperture is slightly longer in
experiment. An illustration of one scattering azimuth, producing a more narrow beam and thus is
realization is seen in Fig. 9. The experiment compares expected to have an even lower CRB for cross-range.
the CRBs of standard 1-D aperture SAR with those of The accumulated SNR is also equivalent for the three
2-D and 3-D synthetic apertures. The displayed data different flight paths, i.e., the average slant range
are averaged for 200 randomly generated scattering to the target R (t) over the array positions at t is
tg
350 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 1 JANUARY 2007
Fig. 10. 1-D (}) and 2-D aperture (¢) labeled with pulse
numbers.
Fig. 11. Target (}) and a “laid over” forward scatterer that
the same for the three apertures. The 1-D and 2-D inhibits accurate measurements of both magnitudes.
apertures are seen in detail in Fig. 10 on geometric
axes with array positions indexed by pulse numbers only. The results are correlated with the range error
(positions along a 3-D aperture are similarly indexed). variances in Fig. 12(b), scaled by a factor of about 3.
Fig. 12(a)—(d) show CRBs for each parameter as a Again it appears this bound is reduced by including
function of pulse number over the synthetic aperture range excursion in the aperture.
formation, correlating to the pulse numbers seen in Examining the results in Fig. 12(d) the average
the aperture plots of Fig. 10. Log values of the CRBs magnitude CRB is much worse for the standard SAR
allow the three curves to be displayed together for case. This is not related to SNR; as stated above the
comparison. The beginning portions of the apertures accumulated SNR for the multi-dimensional paths
do not contain enough data to condition the matrix J, equals that of the linear path. The poor average
and thus displayed CRBs start at pulse number 15. error variance in amplitude estimates occurs with
In Fig. 12(a) the CRB for cross-range target standard SAR because a small percentage of the
estimation is larger for the 2-D (dashed) and 3-D random realizations result in layover of the target
(dotted) flight paths compared with standard as with an adjacent scatterer. This is worthy of further
expected. However, the values for the bounds at investigation into a particular realization where
the completion of the apertures (43rd pulse) are the 1-D aperture cannot resolve the target from
all on the order of 10¡4 which is sufficiently small a nearby scatterer. Fig. 11 shows only the three
given the radar cross-range resolution. An ancillary relevant scatterers for this discussion, the target at
observation is the smoothness of the curve for the position (x,y,h), and the adjacent scatterers directly
t t t
standard aperture relative to the others. This occurs
forward and aft at positions (x ,y ,h ) and
fwd fwd fwd
because the linear aperture exhibits a more consistent
(x ,y ,h ), respectively. The placement of the
aft aft aft
data collection process that steadily improves SNR
forward scatterer at a height h happens to have a
and azimuth resolution. For instance, the 2-D aperture fwd
broadside range, R =6707:68 m within 0.02 m
bounds in Fig. 12(a) decrease sharply after pulse 15 0,fwd
of the target broadside range R =6707:70 m, for
through 22, and then remains flat until pulse 35. This 0,t
all three apertures. Even at the radar positions for
corresponds to the geometry in Fig. 10 where the
the first and last pulses (the beginning and end of
aircraft is not contributing as much to the azimuthal
the synthetic array), the range to the target and the
aperture in the center portion of the plot.
forward scatterer are within 0.01 m of each other,
Bounds for range estimates are shown in
whereas the aft scatterer maintains about a 0.1 m
Fig. 12(b). Examining the adequacy of range CRBs
difference in range. The results from this layover
we see the standard SAR results in a range CRB of
realization of the target CRBs for the three apertures
1:62£10¡4 m2, which is clearly acceptable given the
are shown below in Table I.
resolution of 0.09 m in range. The 2-D aperture offers
We see that the the range and cross-range bounds
a range CRB of 3:54£10¡3 m2, and thus a 95%
for the 1-D aperture are an order of magnitude worse
confidence interval of 0.119 m. Given that the average
than those for the 2-D and 3-D case. Estimation
spacing among the scatterers in this model is 0.25 m
of magnitude significantly degenerates for the 1-D
(more than twice the range resolution) this is sufficient
aperture since CRB =37:16. The magnitude
to resolve the target in range for any realization. The jatj
CRB for the forward scatterer is CRB =
3-D aperture yields a notable improvement in the jaj,fwd
range CRB compared with the 2-D case. 37:11, suggesting that the magnitude of these
Fig. 12(c) shows the minimum error variance of scatterers cannot be resolved, whereas for the aft
height estimation for the multi-dimensional apertures scatterer CRBjaj,aft=0:052, suggesting this adjacent
LINNEHANETAL.:ONTHEDESIGNOFSARAPERTURESUSINGTHECRAME´R-RAOBOUND 351