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Iterative Adaptive Approaches toMIMO Radar Imaging
William Roberts, Student Member, IEEE, Petre Stoica, Fellow, IEEE,Jian Li*, Fellow, IEEE, Tarik Yardibi, Student Member, IEEE, and Firooz A. Sadjadi
Abstract—Multiple-input multiple-output (MIMO) radar canachieve superior performance through waveform diversity overconventional phased-array radar systems. When a MIMO radartransmits orthogonal waveforms, the reflected signals from scat-terers are linearly independent of each other. Therefore, adaptivereceive filters, such as Capon and amplitude and phase estima-tion (APES) filters, can be directly employed in MIMO radarapplications. High levels of noise and strong clutter, however,significantly worsen detection performance of the data-dependentbeamformers due to a shortage of snapshots. The iterativeadaptive approach (IAA), a non-parametric and user parameter-free weighted least-squares algorithm, was recently shown to offerimproved resolution and interference rejection performance inseveral passive and active sensing applications. In this paper, weshow how IAA can be extended to MIMO radar imaging, in boththe negligible and non-negligible intra-pulse Doppler cases, andwe also establish some theoretical convergence properties of IAA.In addition, we propose a regularized IAA algorithm, referredto as IAA-R, which can perform better than IAA by accountingfor unrepresented additive noise terms in the signal model.Numerical examples are presented to demonstrate the superiorperformance of MIMO radar over single-input multiple-output(SIMO) radar, and further highlight the improved performanceachieved with the proposed IAA-R method for target imaging.
Index Terms—MIMO radar, phased-array radar, iterativeadaptive approach (IAA), regularized IAA, radar imaging, intra-pulse Doppler.
I. INTRODUCTION
A multiple-input multiple-output (MIMO) antenna arraysystem can be used in radar applications to provide higherresolution and better sensitivity (for detecting slowly-moving
Copyright (c) 2008 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].
This material is based upon work supported in part by the SMARTfellowship program, the Army Research Office under Grant No. W911NF-07-1-0450, the National Science Foundation under Grant No. CCF-0634786,the National Aeronautics and Space Administration (NASA) under GrantNo. NNX07AO15A, the Swedish Research Council (VR), and the EuropeanResearch Council (ERC). Opinions, interpretations, conclusions, and recom-mendations are those of the authors and are not necessarily endorsed by theUnited States Government.
William Roberts and Tarik Yardibi are with the Department of Electricaland Computer Engineering, University of Florida, Gainesville, FL 32611-6130, USA. Email: [email protected] and [email protected].
*Jian Li is with the Department of Electrical and Computer Engineering,University of Florida, Gainesville, FL 32611-6130, USA. Phone: (352)392-2642; Fax: (352) 392-0044; Email: [email protected]. Please address allcorrespondence to Jian Li.
Petre Stoica is with the Department of Information Technology, UppsalaUniversity, Uppsala, Sweden. Phone: 46-18-471-7619; Fax: 46-18-511925;Email: [email protected].
Firooz A. Sadjadi is with Lockheed Martin Corp, 3333 Pilot Knob Road,Eagan, MN 55121, USA. Email: [email protected].
targets) than a phased-array radar system (see, e.g., [1]-[12]and the references therein). The waveform diversity affordedby the MIMO radar can serve to increase the flexibility atthe transmitter [6]-[12]. For example, probing signals can bedesigned to approximate a desired transmit beampattern and tominimize undesirable cross-correlation terms (see, e.g., [13]-[17] and the references therein). When orthogonal waveformsare transmitted, the parameter identifiability of the radar,meaning the maximum number of targets that can be uniquelyidentified, is vastly improved. With careful construction of theradar’s antenna structure, in fact, the parameter identifiabilitycan be increased by a factor of M , where M is the numberof transmitting antennas, over the corresponding phased-arraysystem [18]. Furthermore, when the radar transmits M or-thogonal waveforms, the virtual array of the radar system is afilled array with an aperture length up to M times that of thereceive array [10], [11]. This advantage of MIMO radar canbe exploited to achieve an M -fold improvement in the spatialimaging resolution over the conventional phased-array radar[10], [11].
In this paper, we consider MIMO radar imaging. The goalof radar imaging is to provide an estimate of the radar crosssections (RCS) of targets at precise angular locations anddistances relative to the radar. When motion (either of theradar or of targets within a scene of interest) is present,the relative speeds of objects can also be estimated throughDoppler considerations.
Irrespective of the array system, data-independent ap-proaches, such as delay-and-sum (DAS) (or matched filtering),can be used for radar imaging. However, DAS suffers from lowresolution and high sidelobe level problems. With a narrow-band MIMO radar, orthogonal waveforms from the transmitantennas hit targets in the scene of interest at different timeinstants, thus undergoing different phase shifts. Therefore, thereflected signals at the receiver are linearly independent ofeach other when the number of targets per range and Dopplerbin is less than or equal to the number of orthogonal probingwaveforms. Due to this independence, adaptive beamform-ing techniques, which fail to work with coherent or highlycorrelated signals, can be effectively employed to estimatetarget parameters with MIMO radar systems [19]. The existingadaptive approaches, such as Capon [20] and amplitude andphase estimation (APES) [21], can be used to mitigate cluttereffects and improve resolution over conventional DAS. Whennoise (and clutter) levels in the received signal are high,however, the performance of data-dependent approaches mightdegrade significantly unless the number of snapshots is rather
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large.Recently, a non-parametric and user parameter-free algo-
rithm, referred to as the iterative adaptive approach (IAA),1
was presented for applications including passive array pro-cessing, multiple-input single-output (MISO) communicationchannel estimation and single antenna radar systems [22] .IAA was shown to perform well for cases of few or even asingle snapshot and closely-spaced sources, and it was shownto outperform existing sparse signal reconstruction algorithms.In [25], IAA was extended to the case of spectral analysisfor non-uniformly sampled real-valued data, and was shownto attain significantly better performance than the traditionalperiodogram approach to spectral estimation. IAA was furtheradapted to single antenna range-Doppler imaging in [26],specifically for the case when the transmitted signal is atrain of probing pulses. Improved resolution was achievedusing IAA compared to that obtained using a data-independentinstrumental variables (IV) filter.
Previous descriptions of IAA for radar imaging have beenrestricted to the single antenna case. In this paper, we broadenthe application of IAA to include MIMO antenna arrays anddemonstrate the superior imaging performance of MIMO radarover its phased-array counterpart. In Section II, we extendIAA for MIMO radar applications in the negligible intra-pulse Doppler case, and in Section III, we incorporate intra-pulse Doppler effects into the algorithm. In Section IV, wepropose the regularized IAA algorithm, namely IAA-R, whichimproves the robustness of IAA by explicitly accounting forthe additive noise terms. By assuming a statistical modelof the received signal, we derive the maximum likelihoodbased version of IAA-R (IAA-R-ML) in Section V. We showthat IAA-R can be viewed as an approximation to IAA-R-ML, which is guaranteed to be locally convergent. We alsoprovide a theoretical local convergence analysis of IAA (inthe appendix). Numerical simulations, which we present inSection VI, demonstrate the superior performance of a MIMOsystem over that of a single-input multiple-output (SIMO)antenna array system, as well as the superior performanceof IAA-R over DAS and several data-adaptive beamformingtechniques. Also, the advantages of IAA-R over IAA, forMIMO radar applications, are highlighted. Conclusions areprovided in Section VII.
Notation : We denote vectors and matrices by boldfacelowercase and uppercase letters, respectively. ‖·‖2 denotes theEuclidean norm, ‖·‖2F denotes the matrix Frobenius norm, (·)T
denotes the transpose operation, (·)∗ denotes the conjugatetranspose operation, vec(·) refers to the vectorization operation(i.e, stacking the columns of a matrix on top of each other),tr(·) denotes the trace operation, ¯ denotes the Hadamard(elementwise) matrix product, | · | refers to the determinantoperation, and IN represents the N × N identity matrix.R ∈ R L×M and R ∈ C L×M denote a real and complex-valued L×M matrix R, respectively. α denotes the estimateof the parameter α.
1Note that the algorithm was named as IAA-APES in [22]. Herein, we referto the algorithm as IAA for conciseness.
II. NEGLIGIBLE DOPPLER CASE
This section formulates the MIMO spotlight synthetic aper-ture radar (SAR) [27] imaging problem and describes howIAA can be extended to estimate the target parameters, specif-ically the amplitude, range, and angle of each target present.The targets are assumed to be stationary and therefore theintra-pulse Doppler effects are neglected. The non-negligibleDoppler case will be analyzed in the next section.
A. Problem Formulation
Consider a MIMO radar system with M transmit antennasand N receive antennas. Let
xm =[
xm(1) . . . xm(L)]T
, m = 1, . . . , M, (1)
represent the length L transmitted signal from the mth transmitantenna. Let
X =[
x1 x2 . . . xM
](2)
consist of the transmitted signals from all the transmit antennas(X ∈ C L×M ). Further, let
X =[
X0(P−1)×M
], (3)
where X ∈ C (L+P−1)×M denotes the zero-appended trans-mitted waveform matrix, 0(P−1)×M is a (P − 1)×M matrixof zeros, and P −1 refers to the maximum delay (in samplingintervals) between the reflected signals from various range binsand the first received signal (from the closest range bin tothe radar). For a radar collecting data from N positions, thereceived signal from collection position n, that is synchronizedwith the arrival of the first reflected signal, can be expressedas:
D∗(n) =P∑
p=1
K∑
k=1
αp,kak(n)bTk (n)X∗Jp + E∗(n) ,
n = 1, . . . , N , (4)
where the complex scattering coefficients of the targets,which are directly proportional to their corresponding radarcross section (RCS), are represented by {αp,k}, p denotes therange index, k denotes the angle index, K is the number ofpotential scatterers in each range bin and {E(n)} representsthe additive noise. The terms {ak(n)} and {bk(n)} refer to thereceive and transmit steering vectors, respectively. For uniformlinear arrays, they can be described by:
ak(n) =[e−
j2π((n−1)dn)sin(θk)λ0 e−
j2π((n−1)dn+dr)sin(θk)λ0
. . . e−j2π((n−1)dn+(N−1)dr)sin(θk)
λ0
]T
, (5)
and
bk(n) =[e−
j2π((n−1)dn)sin(θk)λ0 e−
j2π((n−1)dn+dt)sin(θk)λ0
. . . e−j2π((n−1)dn+(M−1)dt)sin(θk)
λ0
]T
, (6)
3
where dt and dr refer to the distances between adjacenttransmitting and receiving antennas, respectively, λ0 representsthe carrier wavelength of the radar system, dn refers to theseparation between collection positions, and θk denotes theimpinging angle (relative to the array normal) of targets in thekth angle bin. Finally, Jp ∈ R (L+P−1)×(L+P−1) is a shiftmatrix used to describe the received signals from differentrange bins, and it can be written as:
Jp =
p︷ ︸︸ ︷1 0
. . .1
0
. (7)
Note that
X∗Jp =
p︷ ︸︸ ︷0 · · · 0 x∗1(1) · · · x∗1(L) 0 · · · 0
......
...0 · · · 0 x∗M (1) · · · x∗M (L) 0 · · · 0
,
p = 1, . . . , P. (8)
To further simplify notation, we accumulate the receivedsignals from each of the look positions as:
d =
vec(D∗(1))...
vec(D∗(N))
. (9)
In a similar way, we can define a matrix Y that contains theknown quantities (the steering vectors, the transmit waveformsand the shift matrix) in the received signal as follows:
Y =[
y1,1 . . . y1,K y2,1 . . . yP,K
], (10)
where
yp,k =
yp,k(1)...
yp,k(N)
, p = 1, . . . , P, k = 1, . . . , K,
(11)and
yp,k(n) = vec[ak(n)bT
k (n)X∗Jp
], p = 1, . . . , P,
k = 1, . . . ,K, n = 1, . . . , N . (12)
Consequently, by using (10)-(11), (9) can be expressed as:
d = Yα + e , (13)
where
α =[
α1,1 . . . α1,K α2,1 . . . αP,K
]T, (14)
and
e =
vec(E∗(1))...
vec(E∗(N))
. (15)
The problem of interest, then, is to accurately estimate thetarget reflection coefficients α from the measurement vectord and known matrix Y.
The DAS estimates of the target parameters are given by:
αp,k =y∗p,kd
y∗p,kyp,k, p = 1, . . . , P, k = 1, . . . , K. (16)
The basic assumption behind DAS is that y∗p,kyp,k is rela-tively large compared to y∗p,kyp′,k′ , where (p′, k′) 6= (p, k),p′ = 1, . . . , P and k′ = 1, . . . ,K, so that the signal-of-interest is passed undistorted while the contribution from theinterfering signals is reduced. In other words, in order forDAS to work properly, the columns of Y should be close tobeing orthogonal. However, this condition is rarely satisfiedin practice and hence DAS usually suffers from high sidelobelevel and low resolution problems.
B. IAA
IAA iteratively refines the DAS target estimates to achievehigher resolution and better interference suppression. Here,we show how IAA can be extended to estimate the targetparameters in the MIMO radar problem defined in (13) (westill refer to the extended algorithm as IAA for simplicity).
Let the target parameter of interest be αp,k. Treating thepossible targets corresponding to {αp′,k′}, where (p′, k′) 6=(p, k), p′ = 1, . . . , P and k′ = 1, . . . , K, as interferences, wedefine the interference covariance matrix as:
Qp,k =P∑
p′=1
K∑
k′=1
|αp′,k′ |2yp′,k′y∗p′,k′ ,
(p′,k′)6=(p,k)
p = 1, . . . , P, k = 1, . . . ,K. (17)
The data covariance matrix is defined as:
R =P∑
p=1
K∑
k=1
|αp,k|2yp,ky∗p,k , (18)
where R ∈ C (NN(L+P−1))×(NN(L+P−1)).IAA minimizes the following weighted least-squares cost
function (see, e.g., [28], [29]) with respect to the reflectioncoefficient, αp,k, of the target of interest:
(d− αp,kyp,k)∗Q−1p,k(d− αp,kyp,k) . (19)
The minimization of (19) yields the following estimate for thetarget parameters:
αp,k =y∗p,kQ
−1p,kd
y∗p,kQ−1p,kyp,k
, p = 1, . . . , P, k = 1, . . . , K. (20)
Noting that
Qp,k = R− |αp,k|2yp,ky∗p,k , (21)
and applying the matrix inversion lemma to (21) (see, e.g.,[29]) yield,
y∗p,kQ−1p,k =
y∗p,kR−1
1− |αp,k|2y∗p,kR−1yp,k. (22)
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TABLE IIAA FOR MIMO SAR IMAGING.
initialize
αp,k =y∗p,kd
y∗p,k
yp,k, p = 1, . . . , P , k = 1, . . . , K
repeat
R =
P∑p=1
K∑k=1
|αp,k|2yp,ky∗p,k
αp,k =y∗p,kR−1d
y∗p,k
R−1yp,k, p = 1, . . . , P , k = 1, . . . , K
until (a certain number of iterations is reached)
Consequently, Q−1p,k in (20) can be replaced with R−1, which
needs to be computed only once. The IAA estimate of αp,k
then becomes:
αp,k =y∗p,kR
−1d
y∗p,kR−1yp,k, p = 1, . . . , P, k = 1, . . . , K. (23)
Since R depends on {αp,k}, which are unknown, IAA isimplemented as an iterative algorithm and the DAS estimatesare used for initializing IAA. For most practical applications,convergence occurs after typically no more than 10 iterations[22]. The algorithm is summarized in Table 1. A local conver-gence analysis of IAA is provided in the appendix. Note that Rin IAA (see (18)) is computed using the previous estimates ofthe target parameters {αp,k} and not from the measurements das done in adaptive array processing algorithms. Note also thatsetting R = INN(L+P−1) in (23) results in the DAS estimate.
C. Incorporating BIC
To produce sparsity in the IAA result, the Bayesian in-formation criterion (BIC), a model order selection tool [30],can be used. Given the IAA estimate for a scene, the BICapproach selects a target with range-angle indices (p, k), whichminimizes the following criterion:
BIC(p,k)(η) = 2NN(L + P − 1)
× ln
∥∥∥∥∥∥∥d−
∑
(p,k)∈{J (η)⋃
(p,k)}yp,kαp,k
∥∥∥∥∥∥∥
2
2
+ 4ηln(2NN(L + P − 1)) , (24)
where η denotes the number of targets currently selected (η =1 for the first iteration). The value of 4 in the penalty termof (24) is chosen to reflect the number of unknowns for eachtarget: amplitude (complex valued), range, and angle. Also,J (η) refers to the set of target indices already selected at thecurrent iteration (J (1) = {∅}, where {∅} denotes the emptyset), and (p, k) represents a remaining target point in the scene(i.e., (p, k) /∈ J (η)). At each iteration of the algorithm, a newtarget point (p, k) is selected such that, along with the setof indices J (η) currently selected, BIC(p,k)(η) is minimized.This procedure is repeated until the function in (24) does notdecrease anymore. Reflection coefficients not present in J (η)at the end of this procedure are set to zero and are assumed not
to represent true targets in the scene. Note that the second termon the right side of (24) does not matter to peak selection; itmatters only when (24) is used to select the number of peaksto retain.
III. NON-NEGLIGIBLE DOPPLER CASE
This section considers the case of a stationary verticalMIMO array and mobile targets, and further extends IAA todeal with Doppler considerations in a multiple antenna system.In this case, the signal model incorporates Doppler shifts onthe received signals and the target parameters of interest arenow amplitude, range, angle and Doppler.
A. Problem Formulation
In order to describe IAA for the Doppler-present case, thereceived signal model has to be reformulated. Denote thereceived data as Z∗ ∈ C N×(L+P−1), which is synchronizedwith the arrival of the first reflected signal. Z can be expressedas:
Z∗ =P∑
p=1
K∑
k=1
H∑
h=1
αp,k,hakbTk X∗
D(ωh)Jp + E∗D , (25)
where “D” is used to denote Doppler, p is the range index,k is the angle index, h is the Doppler index, H denotes thenumber of bins in the Doppler interval of interest and ωh
denotes the angular Doppler frequency corresponding to thehth Doppler bin, h = 1, . . . , H . The steering vectors, {ak}and {bk}, and the noise term, E∗D, are defined similarly totheir corresponding terms in (4), except that the dependencyon n is removed since the radar is assumed to be stationary (letn = 1 in (5) and (6)). The complex scattering coefficients ofthe targets, {αp,k,h}, now include a third dimension (indexedby h) to model the targets’ speeds. We let
xm(ωh) = xm ¯ d(ωh) , m = 1, . . . , M, h = 1, . . . , H,(26)
with xm defined in (1), and
d(ωh) =[
1 ejωh . . . ejωh(L−1)]T
, h = 1, . . . , H.(27)
We then define XD(ωh) similarly to (3) by letting
XD(ωh) =[
XD(ωh)0(P−1)×M
], (28)
where X is replaced by the Doppler shifted signal XD(ωh) ∈C L×M :
XD(ωh) =[
x1(ωh) x2(ωh) . . . xM (ωh)]
. (29)
To write the signal model given in (25) more compactly, letz = vec(Z∗). z can then be represented in the following form:
z = YDαD + eD , (30)
where
YD =[
y1,1,1 y1,1,2 . . . yP,K,H
], (31)
αD =[
α1,1,1 α1,1,2 . . . αP,K,H
]T, (32)
5
with
yp,k,h = vec[akbT
k X∗D(ωh)Jp
], p = 1, . . . , P,
k = 1, . . . ,K, h = 1, . . . , H. (33)
The complex noise in (30) is simply defined as eD = vec(E∗D).Similarly to (16), the DAS estimate of the target parameters
is given by:
αp,k,h =y∗p,k,hz
y∗p,k,hyp,k,h, p = 1, . . . , P,
k = 1, . . . ,K, h = 1, . . . , H. (34)
In order for DAS to show good performance, y∗p,k,hyp,k,h
should be large relative to y∗p,k,hyp′,k′,h′ , where (p′, k′, h′) 6=(p, k, h), p′ = 1, . . . , P , k′ = 1, . . . , K, and h′ = 1, . . . ,H ,which is hardly possible in practice.
B. IAA
To describe IAA for a MIMO angle-range-Doppler imagingradar, a similar approach to the one taken in Section II-B willbe used. As in (18), we can model the covariance matrix ofthe received signal by:
RD =P∑
p=1
K∑
k=1
H∑
h=1
|αp,k,h|2yp,k,hy∗p,k,h , (35)
where RD ∈ C (N(L+P−1))×(N(L+P−1)). We represent thecovariance matrix of the interference to the particular target,αp,k,h, as:
Qp,k,h = RD − |αp,k,h|2yp,k,hy∗p,k,h . (36)
We again consider the weighted least-squares cost function:
(z− αp,k,hyp,k,h)∗Q−1p,k,h(z− αp,k,hyp,k,h) . (37)
Minimization of (37) with respect to the RCS related ampli-tude of the target of interest αp,k,h yields the update formula:
αp,k,h =y∗p,k,hQ
−1p,k,hz
y∗p,k,hQ−1p,k,hyp,k,h
, p = 1, . . . , P,
k = 1, . . . ,K, h = 1, . . . , H. (38)
The matrix inversion lemma can again be used to replaceQ−1
p,k,h in (38) with R−1D to decrease computation time. The
resulting algorithm is summarized in Table 2.
C. Incorporating BIC
Sparsity can be incorporated by following the methodologydescribed in the negligible Doppler case in Section II-C. Thecriterion in (24) can be rewritten for the revised signal modelas:
BIC(p,k,h)(η) = 2N(L + P − 1)
× ln
∥∥∥∥∥∥∥z−
∑
(p,k,h)∈{J(η)⋃
(p,k,h)}yp,k,hαp,k,h
∥∥∥∥∥∥∥
2
2
+ 5ηln(2N(L + P − 1)). (39)
TABLE IIIAA FOR ANGLE-RANGE-DOPPLER IMAGING WITH A MIMO ARRAY.
initialize
αp,k,h =y∗
p,k,hz
y∗p,k,h
yp,k,h, p = 1, . . . , P
k = 1, . . . , K, h = 1, . . . , Hrepeat
RD =
P∑p=1
K∑k=1
H∑h=1
|αp,k,h|2yp,k,hy∗p,k,h
αp,k,h =y∗
p,k,hR−1
D z
y∗p,k,h
R−1D yp,k,h
, p = 1, . . . , P,
k = 1, . . . , K, h = 1, . . . , Huntil (a certain number of iterations is reached)
The peak selection procedure is identical to the one describedin Section II-C with the only difference being the inclusion ofthe Doppler dimension in the analysis. We use 5 in the penaltyterm of (39) to account for the unknown target parameters:amplitude (complex valued), range, angle, and Doppler.
IV. REGULARIZED IAA (IAA-R)
In practice, a radar system may illuminate only a restrictedangular region (for example, −30◦ to 30◦) relative to theradar by adjusting the transmit beampattern of each transmitantenna. Then the scanning grid does not need to cover theentire region (−90◦ to 90◦), and it could be limited to theangular region of interest (−30◦ through 30◦, for example)only. Even though the reflected power at locations not cor-responding to illuminated targets is small, these estimatesstill contribute to the rank of the covariance matrix usedin IAA. As the size of the scanning grid, K, decreases,the condition number of R in (18) and RD in (35) couldincrease to unfavorable levels. In the negligible Doppler case,R is invertible only if PK ≥ NN(L + P − 1). For thenon-negligible Doppler case, invertibility of RD requires thatPKH ≥ N(L + P − 1). Moreover, R in (18) and RD in(35) do not explicitly consider the contribution of the noiseterms e and eD, respectively. To resolve these issues, weregularize the R and RD with diagonal matrices Σ and ΣD,respectively, whose diagonal elements represent the unknownnoise powers and are computed automatically. The regularizedIAA approach, referred to hereafter as IAA-R, fits naturallywithin the user parameter-free framework of the existing IAAalgorithm.
When intra-pulse Doppler effects are neglected, we canwrite the regularized version of (18) as:
R =P∑
p=1
K∑
k=1
|αp,k|2yp,ky∗p,k + Σ , (40)
where the noise power estimates along the diagonal of Σ aredenoted by {σ2
l }NN(L+P−1)l=1 . IAA-R is implemented as in
Table 1, except that the number of unknowns is now increasedto KP + NN(L + P − 1) from KP . The “steering vector”corresponding to σl is the lth column of INN(L+P−1), whichwe denote as vl. Consequently, the update estimate for σl is
6
given by:
σ2l =
∣∣∣∣v∗l R
−1dv∗l R−1vl
∣∣∣∣2
, l = 1, . . . , NN(L + P − 1). (41)
The noise power estimates can be initialized as all zeros or as asmall (relative to the strength of the targets) constant number.
In the non-negligible Doppler case, very similarly, we canwrite the regularized version of (35) as:
RD =P∑
p=1
K∑
k=1
H∑
h=1
|αp,k,h|2yp,k,hy∗p,k,h + ΣD , (42)
where the noise power estimates, {σ2l }N(L+P−1)
l=1 , are againcontained along the diagonal of ΣD. The IAA-R estimate forσl is then:
ˆσ2l =
∣∣∣∣v∗l R
−1D z
v∗l R−1D vl
∣∣∣∣2
, l = 1, . . . , N(L + P − 1), (43)
where vl denotes the lth column of IN(L+P−1).IAA-R performs well with irregularly sampled scanning
grids and with arbitrary sensor spacings, whereas the inversionof R in (18) and RD in (35) without the regularizing termmight be problematic in such cases. Certainly, the techniquecould be similarly applied to other active and passive sensingapplications considered in [22] if desired.
V. MAXIMUM LIKELIHOOD BASED IAA-R (IAA-R-ML)
In the previous sections, we developed an extension to theIAA algorithm which used a deterministic model for the signalof interest. We now adopt a stochastic interpretation of the sig-nal model and use maximum likelihood (ML) techniques to es-timate the target parameters. Again performing regularizationon the covariance matrix, we identify this approach as IAA-R-ML. Like IAA-R, IAA-R-ML is non-parametric and userparameter-free. For conciseness, we formulate the algorithmfor the non-negligible Doppler case, as the negligible Dopplerversion of the algorithm can be similarly synthesized.
We assume the received signal, z, to have a complex mul-tivariate Gaussian distribution with zero mean and covariancematrix, RD, given in (35), so that the likelihood of z has theform:
p(z|RD) =1
πN(L+P−1)|RD|e−z∗R−1
D z . (44)
Maximization of the logarithm of the likelihood with respectto the unknown terms in RD is equivalent to minimization ofthe following cost function:
z∗R−1D z + ln|RD| . (45)
Using the matrix inversion lemma, the relationship in (36),and the properties of the determinant operation (specifically,that |I + AB| = |I + BA|), we obtain:
|RD| = |Qp,k,h|(1 + |αp,k,h|2y∗p,k,hQ−1p,k,hyp,k,h) , (46)
and
R−1D = Q−1
p,k,h −|αp,k,h|2Q−1
p,k,hyp,k,hy∗p,k,hQ−1p,k,h
1 + |αp,k,h|2y∗p,k,hQ−1p,k,hyp,k,h
, (47)
where (p, k, h) represents any target of interest. For notationalconvenience, let βp,k,h = |αp,k,h|2. By using (46) and (47), theminimization of (45) with respect to βp,k,h (for fixed Qp,k,h)becomes equivalent to minimizing:
f(βp,k,h) = ln(1 + βp,k,hy∗p,k,hQ−1p,k,hyp,k,h)
− βp,k,hz∗Q−1p,k,hyp,k,hy∗p,k,hQ
−1p,k,hz
1 + βp,k,hy∗p,k,hQ−1p,k,hyp,k,h
. (48)
To minimize (48), we set the first derivative of (48) withrespect to βp,k,h to zero and solve for βp,k,h:
βp,k,h =y∗p,k,hQ
−1p,k,h(zz∗ −Qp,k,h)Q−1
p,k,hyp,k,h
(y∗p,k,hQ−1p,k,hyp,k,h)2
. (49)
Taking the second derivative of (48) and inserting the aboveestimate of βp,k,h, we find that
f ′′(βp,k,h) =(y∗p,k,hQ
−1p,k,hyp,k,h)2
(1 + βp,k,hy∗p,k,hQ−1p,k,hyp,k,h)2
, (50)
which is strictly positive. Hence, the estimate of βp,k,h in (49)is the global minimizer of (48). Though unlikely, it is possiblethat the estimate for βp,k,h given in (49) could be negative. Toenforce nonnegativity of the power estimates, the IAA-R-MLestimate is then given by:
βp,k,h = max(
0, βp,k,h
). (51)
Since βp,k,h is the unique minimizer of f(βp,k,h) and sincethe first derivative of f(βp,k,h) is greater than zero forβp,k,h > βp,k,h, we can conclude that βp,k,h minimizesf(βp,k,h) subject to βp,k,h ≥ 0.
As we did in the previous sections, we replace Q−1p,k,h in
(49) and (51) with R−1D via the matrix inversion lemma (to
reduce computation):
βp,k,h = max
(0,
y∗p,k,hR−1D (zz∗ −RD)R−1
D yp,k,h
(y∗p,k,hR−1D yp,k,h)2
+ βp,k,h
). (52)
Since this estimate depends on RD and βp,k,h = |αp,k,h|2,we must again adopt an iterative approach; {αp,k,h} can beinitialized using DAS. Furthermore, we recalculate RD aftereach αp,k,h is updated.
We can rewrite the estimate in (52) as:
βp,k,h = max
(0,
y∗p,k,hR−1D (zz∗)R−1
D yp,k,h
(y∗p,k,hR−1D yp,k,h)2
+βp,k,h − 1y∗p,k,hR
−1D yp,k,h
). (53)
The term on the far right of (53) is just the estimate forβp,k,h attained with a standard Capon beamformer (SCB) [29](assuming RD is invertible). If βp,k,h is close to the SCBestimate (which should be the case at least locally around thetrue values), then the estimate given in (53) is approximately
7
equal to the IAA-R estimate. In this way, we may view IAA-Ras an approximation to IAA-R-ML.
IAA-R-ML is a cyclic algorithm that maximizes the log-likelihood function, and is therefore locally convergent. SinceIAA-R is an approximation to IAA-R-ML, we expect that itshares similar convergence properties. A local convergenceanalysis for IAA-R, like the one for IAA presented in theappendix, is not possible since IAA-R takes into account thenoise covariance matrix, whereas the analysis in the appendixis concerned with the convergence to the true parameters inthe noise-free case.
VI. NUMERICAL RESULTS
This section uses numerical examples to demonstrate thesuperior performance of IAA-R over that of conventional DASand of several adaptive beamforming algorithms as well asof the extended IAA algorithm without regularization. Theadvantage of a MIMO radar over a SIMO radar will alsobe illustrated. The MIMO radar under consideration containsM = 5 transmit antennas spaced at dt = 2.5λ0 and N = 5receive antennas spaced at dr = 0.5λ0. In this way, i.e., witha sparse transmit array and filled receive array, we effectivelycreate a filled virtual array with NM = 25 antennas [10],[11], [18]. The SIMO system under consideration, on the otherhand, contains N = 5 receive antennas spaced at dr = 0.5λ0
and M = 1 transmit antenna. The total transmitted power ofthe MIMO radar is the same as that of the SIMO system.
For each of the figures shown in these examples, we fixthe amplitude scale used to represent the targets; we performthresholding to the targets that fall outside of these boundaries.In this way, we are able to accurately compare and illustratethe sidelobe levels that result from the different estimationapproaches.
A. MIMO SAR Imaging with Negligible Doppler
A MIMO array for spotlight SAR imaging in the negligibleintra-pulse Doppler case is considered. In other words, weassume that the scene of interest is stationary and that theradar platform is moving; we neglect the intra-pulse Dopplershifts that result from the radar’s motion (as done most oftenin practice [31], [32]). If the motion of the radar cannot beneglected, then we could adjust the signal model to account forthe radar’s velocity. The true response consists of 29 targetsspread randomly across P = 24 range bins with at least 2◦
separation between target scatterers. The amplitudes of thetargets are selected independently from a uniform distributionon the unit interval [0, 1]. We provide the true target imagein Figure 1(a). The angular interval of interest ranges from−30
◦to 30◦, with 1◦ angular grid size, i.e., K = 61. We
will assume circularly symmetric independent and identicallydistributed (i.i.d.) additive complex Gaussian noise with zero-mean and variance σ2. We let dn, the separation betweencollection positions, be 12.5λ0.
1) Example 1: For the first example, we restrict the numberof data collection positions, N , to one and the length of thetransmit waveform is fixed at L = 64. The signal-to-noise
ratio (SNR), which is defined as 10 log10
(tr(X∗X)
Lσ2
), is set at
20 dB.
Figures 2(a)-(b) demonstrate the performance of DAS andIAA-R with a SIMO radar. In Figure 2(a), a widely usedpseudo-noise (PN) sequence [33], which has reasonably goodauto-correlation properties, is transmitted and the target pa-rameters are estimated at the receiver using DAS. The samePN sequence is transmitted for the case in Figure 2(b), but nowIAA-R is used for estimating the target parameters instead ofDAS. The results obtained using a MIMO radar are shownin Figures 2(c)-(e). For each of these examples, orthogonalHadamard waveforms, scrambled with a PN sequence toimprove the waveforms’ merit factors, are transmitted [33].In Figures 2(c)-(d), DAS and IAA-R, respectively, are used toestimate the target parameters. The IAA-R estimate when theBIC algorithm is applied is shown in Figure 2(e).
From Figure 2, we observe that the MIMO system offersmuch improved resolution over the SIMO system, even whenusing DAS as the receive filter. On the other hand, it isobserved that IAA-R successfully refines the DAS estimatesin both the SIMO and MIMO cases to effectively reduce thesidelobe levels. IAA-R shows quite good estimation accuracyin the MIMO case. To evaluate the performance of IAA-Rwith BIC for the SIMO and MIMO systems, we compare theprobability of detection (PD) and the probability of false alarm(PFA) for each case; we let PD = (number of targets detected)/ (total number of true targets) and PFA = (total number offalse positives) / (total number of grid points not occupiedby targets). We classify a target as successfully identified ifBIC chooses a point within 1◦ of the true target location (andwithin the same range bin). For the SIMO result in Figure2(c), PD = 65.5% and PFA = 1.0%. For the MIMO resultin Figure 2(f), PD = 96.6% and PFA = 0%. As BIC doesnot provide satisfactory results for the IAA-R result from theSIMO system, we ommit these images in the next examples.We furthermore neglect to show the results of applying BIC tothe DAS results, as the poor resolution of these images, withBIC applied, would lead to an unacceptably low PD [25].
In Figure 3, we show the results of using IAA withoutregularization for the MIMO case. For Figure 3(a), the radarscans an angular interval of interest ranging from −30
◦to
30◦, with 1◦ angular grid size, as in the previous images.Since, in this case, the condition number of R in (18) reachesunfavorable levels (due to K being small relative to the entireangular region), IAA suffers from poor performance. In Figure3(b), on the other hand, the scanning region is increased to−90
◦through 90◦ (the entire scanning region), again with 1◦
angular grid size. In this image, however, only the angularregion of interest (−30
◦to 30◦) is shown. The condition
number of R in (18) is significantly reduced in this case (sinceK is much larger), and the performance of IAA improvesdrastically. Thus, we conclude that IAA works well when theentire angular region is considered. If the targets are knownto exist within a smaller region relative to the whole scanninggrid, then the actual scanning grid can be decreased to reducecomputational costs by using IAA-R instead to avoid problemsthat might arise with the inversion of R.
8
2) Example 2: In the second example, we consider the per-formance of IAA-R when the noise level is raised. We restrictour attention to the MIMO array due to its better performancein the previous example compared to a SIMO system. Forthis example, the SNR was set at 15 dB, L = 32, and N = 3.We again transmit orthogonal Hadamard waveforms scrambledwith a PN sequence.
Figures 4(a)-(c) show the DAS, Capon, and APES estimatesof the scene, respectively. To form a MIMO radar image usingthese techniques, we first form a synthetic aperture usingthe returns from the N positions, and then perform rangecompression to form the matrix Dp:
Dp =[
D∗p(1)YMF, D∗
p(2)YMF, . . . , D∗p(N)YMF
],
p = 1, . . . , P , (54)
where YMF represents the matched filter (YMF = X(X∗X)−1)and D∗
p(n) ∈ C N×L denotes the portion of the receivedsignal (from position n) synchronized with the return fromthe pth range bin. We can then apply Capon or APES toDp to obtain an estimate for targets at the pth range bin[19], [34]. Data-adaptive methods, such as Capon and APES,perform poorly when the number of data snapshots, whichis represented by the number of columns in (54) (for thisexample, MN = 15), is not significantly greater than thenumber of array sensors (N = 5) [21]. In this case, theestimated signal covariance matrix, which depends directly onthe number of data snapshots, can differ significantly from thetrue covariance matrix. When only a single snapshot is attained(M = N = 1 in (54)), adaptive beamforming methods fail,since the sample covariance matrix becomes singular. IAA-R(and IAA), on the other hand, can perform well even when thenumber of snapshots is one. For this example, we see muchbetter results using IAA-R and IAA-R with BIC (comparedto Capon and APES), as shown in Figures 4(d) and 4(e),respectively.
In Figure 4(f), a plot of the mean-squared error (MSE)of IAA-R versus the iteration number is shown, with MSEdefined as:
‖B0 −B(i)IAA-R‖2F
‖B0‖2F, (55)
where i denotes the iteration number, B0 denotes the groundtruth and B(i)
IAA-R denotes the IAA-R range-angle image es-timate at iteration i. Iteration 0 denotes the MSE of DAS(as IAA-R is initialized with DAS). Note that the MSE valuedecreases monotonically with the iteration number and appearsto converge after only a few iterations of IAA-R.
B. MIMO Range-Angle-Doppler Imaging with Non-negligibleDoppler
This subsection considers the non-negligible intra-pulseDoppler case; the antenna array (with the same structure asbefore) is assumed vertical and stationary, and the targetsare assumed mobile. The ground truth consists of 4 targetsplaced within P = 24 range bins. We represent the Dopplershift, in degrees, of a target in a particular Doppler bin asΦh = ωhL(180◦/π), h = 1, . . . , H . The Doppler interval of
interest, which ranges from −90◦ to 90◦, is divided into 181bins resulting in 1◦ Doppler grid size. All of the scatterersare located at a Doppler angle of 1◦, which corresponds toa target moving at approximately 40 m/s for X-band radar(a faster object, which would have a larger Doppler angle,would be easier to detect). We again let K = 61 and theangular interval range from −30◦ to 30◦ with 1◦ grid size.L is set at 32. The targets are positioned at an angle of 1◦
relative to the array normal. We provide the true target imagein Figure 1(b). As before, i.i.d. circularly symmetric complexGaussian noise is assumed with a 20 dB SNR. The stationaryground clutter return is 20 dB stronger than the targets andis placed at a Doppler angle of 0◦ and at −9◦ relative to thearray normal. In the following examples, the dynamic rangeis chosen to extend from -25 dB to 25 dB and the amplitudeestimates were coerced into this interval through hard limiting.
1) Example 1: The DAS and IAA-R estimates using aSIMO array and a PN transmit signal are shown in Figures5(a)-(b). In these figures, we examine a range-Doppler sliceof the target estimates taken at 1◦ relative to a broadsidescan. The results obtained using a MIMO array are shownin Figures 5(c)-(e), using DAS, IAA-R, and IAA-R with BIC,respectively. For these images, we again transmit Hadamardwaveforms scrambled with PN sequences. From the range-Doppler results, we can observe that the DAS images forboth the SIMO and MIMO arrays have significantly lowerresolution than the corresponding IAA-R images, and none ofthe targets can be identified using DAS. Furthermore, the IAA-R image obtained using a MIMO array demonstrates superiorperformance over that obtained using a SIMO array; the IAA-R with BIC algorithm produces a sparse result and identifiesall of the targets in the scene.
2) Example 2: In the second example, the angle-range slice(chosen at a certain Doppler shift) of the estimated targetparameters is considered. Particularly, the slice correspondingto the 1◦ Doppler bin, which corresponds to the Doppler shiftof our target responses, is selected. For each image shown,the transmit waveforms used correspond to their respectiveimages shown in Figure 5. The DAS and IAA-R results witha SIMO array are shown in Figures 6(a)-(b), respectively. Forthe MIMO radar case, the DAS and IAA-R results are shownin Figures 6(c)-(d), respectively. Due to the strong presence ofthe ground return in this angle-range slice, the BIC result wasnot satisfactory and thus the result is not shown. As evidenced,the poor resolution and accuracy of DAS is further emphasizedfrom the angle-range perspective. In addition, the improvedangular resolution that results from using the MIMO array, ascompared to the SIMO antenna array, is demonstrated in theIAA-R result. The IAA-R result again shows notably betterperformance than DAS.
In Figures 6(a) and 6(c), a null (approximately zero ampli-tude) occurs in the amplitude estimates at 14◦, across all rangebins. When we apply a DAS receive filter matched to 14◦,a zero occurs in the filter response at precisely −9◦. Thus,the contribution from the clutter ground return is removedfrom target estimates at 14◦. In Figure 6(e), we show thebeampattern response of the DAS filter, which is steeredto 14◦. As evidenced, the ground clutter return offers no
9
contribution to these target estimates.
C. Complexity Analysis
The per iteration computational complexity of IAA-R in thenegligible and non-negligible Doppler cases is O((NN(L +P−1))2(PK)) and O((N(L+P−1))2(PKH)), respectively,whereas the total complexity of DAS is O(NN(L + P −1)(PK)) and O(N(L + P − 1)(PKH)) for the negligibleand non-negligible Doppler cases, respectively. The superiorperformance of IAA-R over all the other considered methodsjustifies its increased computational cost, especially in view ofthe fact that IAA-R can be used to update all grid points ofinterest and noise variances in parallel in each iteration.
VII. CONCLUSIONS
In this paper, we have presented a user parameter-free andnon-parametric iterative adaptive approach, namely IAA, forMIMO radar imaging applications with both negligible intra-pulse Doppler (e.g., spotlight SAR stationary target imaging)and with non-negligible intra-pulse Doppler (mobile targetimaging). We further provided a local convergence analysisof IAA. A regularized version of IAA, IAA-R, was proposedin order to improve the performance of IAA and to make itwork with incomplete scanning regions, which is usually thecase in radar applications. Compared to a SIMO phased-arrayradar, application of a MIMO radar system resulted in higherangular resolution as well as higher Doppler resolution. Fur-thermore, IAA-R demonstrated superior performance for botharray systems, as compared to DAS, adaptive beamformingtechniques, and the IAA algorithm with incomplete scanningregions and without regularization. By incorporating the BICalgorithm, we were able to produce further sparsity in theIAA-R result. Finally, we also derived the maximum likelihoodbased IAA-R-ML, which assumes a statistical model on thereceived signal and is guaranteed to be locally convergent.We showed how IAA-R can be viewed as an approximationto IAA-R-ML.
APPENDIXLOCAL CONVERGENCE ANALYSIS OF IAA
Consider a generic data model, similar to the one in (13):
d = Aα + e , (56)
where d ∈ CM×1, A ∈ CM×N (N > M ), α ∈ CN×1,e ∈ CM×1, and
A =[
a1 . . . aN
]. (57)
The analysis below assumes that e is negligible. In the absenceof noise, d is assumed to have the form:
d = Aα , (58)
where A ∈ CM×K , (K < M ), rank(A) = K, and
α =[
α1 . . . αK
]T, (59)
with αk 6= 0 ∀k. Let
A =[
a1 . . . aK
]. (60)
We assume that there exist n1, . . . , nK such that ak = ank,
k = 1, . . . , K. Also, we assume that the columns of A thatare different from those of A do not belong to the range spaceof A, i.e., an 6∈ Range(A) for n 6= nk, k = 1, . . . , K. Theseassumptions are typical of the sparse estimation approaches[35].
Consider the following regularized IAA algorithm (to avoidnumerical ill-conditioning, see below for details). Let (simi-larly to (18))
R = APA∗ + ρI , (61)
where ρ > 0 is a small scalar and
P =
|α1|2 0
. . .0 |αN |2
. (62)
Let i denote the iteration index. Then (see (20) and (21)), forn = 1, . . . , N ,
αi+1n =
a∗n(Qin)−1d
a∗n(Qin)−1an
, (63)
andQi
n =∑N
k=1,k 6=n |αik|2aka∗k + ρI
= Ri − |αin|2ana∗n .
(64)
Remarks: (a) The regularization parameter ρ in (61) can bechosen only slightly larger than the value needed to avoid anill-conditioning message during the numerical implementation.In practical applications, one may choose ρ = 0 first untilthe ill-conditioning warning occurs, at which point one mayreplace the ρ = 0 with a small ρ > 0. (b) Using the matrixinversion lemma, we can show that (Qi
n)−1 in IAA can bereplaced by (Ri)−1 (see (22) and (23)). However, as suggestedby the calculations below, the computational advantage thatfollows from this replacement is offset by the fact that, closeto convergence, the use of (Qi
n)−1 in IAA turns out to lead toa numerically more stable algorithm. In practical applications,one may use (Ri)−1 at the beginning of the iterations untilthe first ill-conditioning warning occurs, at which point onemay replace (Ri)−1 with ρ(Qi
n)−1 (see (71) below).Consider first the stationary points of IAA. These points
satisfy the equation:
αn =a∗nQ−1
n da∗nQ−1
n an
, n = 1, . . . , N . (65)
Note first that αn = 0, n = 1, . . . , N , is not one of thestationary points. To see this, note that setting αn = 0 inthe right hand side of (65) yields
αn =a∗nd‖an‖2 = 0 , n = 1, . . . , N , (66)
which is generally a contradiction. This result starkly contrastswhat happens for most sparse algorithms (such as FOCUSS,etc.), for which αn = 0 is a stationary point [35]. Moreimportantly, we note that the true α gives a stationary point(as ρ → 0), i.e.,
αn ={
αk , n = nk , k = 1, . . . , K0 , n 6= nk , k = 1, . . . , K .
(67)
10
To see this, first observe that the matrices Qn correspondingto (67) are given by:
Qn ={
AkPkA∗k + ρI , n = nk , k = 1, . . . , K
APA∗ + ρI , n 6= nk , k = 1, . . . , K ,(68)
where Ak is the matrix A without the kth column,
P =
|α1|2 0
. . .0 |αK |2
, (69)
and Pk is the above matrix without the kth row and column.Next, let us consider a general matrix having the form of
(68):Q = BB∗ + ρI , (70)
where B∗B > 0. For ρ → 0, we have that
ρQ−1 = I− 1ρB(I + 1
ρB∗B)−1B∗
= I−B(ρI + B∗B)−1B∗ρ→0−−−→ I−B(B∗B)−1B∗ = P⊥B ,
(71)
where P⊥B is the orthogonal projector onto the null space ofB∗.
It follows from (68) and (71), as well as the assumptionsmade, that (for ρ tending to zero):
αn =
a∗nkP⊥
AkAα
a∗nkP⊥
Akank
= αk , n = nk , k = 1, . . . , K
a∗nP⊥AAα
a∗nP⊥Aan
= 0 , n 6= nk , k = 1, . . . , K .
(72)Note that a∗nk
P⊥Ak
ank6= 0 because ank
6∈ Range(Ak), andsimilarly for a∗nP⊥
Aan. With these observations, the proof that
(67) is a stationary point of IAA (for ρ → 0) is concluded.Remark: If we replaced Qn with R in the IAA algorithm,
then the denominator in the first equation of (72) would havebeen a∗nk
P⊥Aank
= 0, which points to the ill-conditioning thatmight be caused by such a replacement as briefly mentionedin Remark (b) following (64).
Next, we prove the local convergence of IAA to the truevalues. Let us denote the true values in (67) by {α0
n}:
α0n =
{αk , n = nk , k = 1, . . . , K0 , n 6= nk , k = 1, . . . , K .
(73)
Assume that IAA is initialized as follows:
αn ={
ck , n = nk , k = 1, . . . , KO(ε) , n 6= nk , k = 1, . . . , K ,
(74)
where O(ε) denotes a term that tends to zero as ε, when ε → 0,and {ck} are arbitrary non-zero constants. Also, assume thatthe regularization parameter is given by ρ = ε2. We will show,under these assumptions, that one step of IAA yields, for εapproaching zero:
αn = α0n + O(ε2) , (75)
which implies that IAA is locally convergent with at least aquadratic rate (note that the attribute “local” here refers tothe initialization of the zero components of {α0
n}; the non-zero components of this vector can be initialized arbitrarily!).
To prove (75), first observe from (74) that:
|αn|2 = O(ε2), n 6= nk , k = 1, . . . , K . (76)
Let
Qn =N∑
k=1,k 6=n
|αk|2aka∗k + ρI , (77)
let S denote an orthonormal basis of Range(An), and let Gcomprise an orthonormal basis of the null space of A∗
n. (Weomit the dependence of S and G on n to simplify the notation.)Using this notation, we can write (under the assumptionsmade):
Qn = AnCA∗n + BnDB∗
n + ε2I , (78)
where An = A for n 6= nk (k = 1, · · · , K) by convention,Bn is the matrix whose columns are equal to the vectors{ak}k 6=n in (77) that do not appear in A, C is a diagonalmatrix with diagonal elements {ck}K
k=1,k 6=n corresponding tothe {ak} in An, and D is a diagonal matrix made fromthe terms {|αk|2} corresponding to the {ak} in Bn; henceD = O(ε2) (see (76)).
Next, we note that there exist constant matrices E, F andH such that:
An = SH , (79)
and
Bn =[
S G] [
EF
]. (80)
Inserting (79) and (80) into (78), we can rewrite the expressionfor Qn as:
Qn = SHCH∗S∗ +[
S G] [
EDE∗ EDF∗
FDE∗ FDF∗
]
×[
S∗
G∗
]+ ε2
[S G
] [S∗
G∗
]
=[
S G] [
W1 W3
W∗3 W2
] [S∗
G∗
],
(81)where
W1 = HCH∗ + EDE∗ + ε2I = HCH∗ + O(ε2) (82)
tends to a nonsingular matrix, namely HCH∗, as ε → 0,
W2 = FDF∗ + ε2I = O(ε2) (83)
is a nonsingular matrix for any ε > 0, and
W3 = EDF∗ = O(ε2) . (84)
[W1 W3
W∗3 W2
]−1
=[
(W1 −W3W−12 W∗
3)−1 −(W1 −W3W−1
2 W∗3)−1W3W−1
2
−W−12 W∗
3(W1 −W3W−12 W∗
3)−1 (W2 −W∗
3W−11 W3)−1
]. (85)
11
For any ε2 > 0, the inverse of the partitioned matrix in (81)exists and is given by (85) at the bottom of the previous page.Observe that the (1,1), (1,2), and (2,1) blocks of the abovematrix tend to constant matrices as ε → 0, whereas the (2,2)block tends to infinity as 1/ε2. It follows from this observationalong with (81) and (85) that:
ε2Q−1n =
[S G
] [O(ε2) O(ε2)O(ε2) ∆
] [S∗
G∗
]
= G∆G∗ + O(ε2) , (86)
where
∆ = ε2(W2 −W∗3W
−11 W3)−1
=[I + F
(1ε2
D)
F∗]−1
+ O(ε2) (87)
tends to a constant positive-definite matrix as ε2 → 0.Making use of (86) in the main equation of the IAA
algorithm yields:
αn =a∗nε2Q−1
n da∗nε2Q−1
n an
=a∗nG∆G∗da∗nG∆G∗an
+ O(ε2) , (88)
from which we can obtain (75) by a calculation similar to (72).The proof of (75) is thus concluded.
In Figures 7(a)-(b), we provide an example to illustrate theconvergence behavior of IAA and its dependence on ρ, asdiscussed above. We simulate a 1-D passive array with N = 10receiving antennas separated at dr = 0.5λ0 and with only asingle data snapshot. This antenna array scans from −90◦ to90◦, relative to the array normal, with 1◦ separation betweenadjacent scanning points. Two target signals, each with unitpower, are located at 7◦ and 14◦, respectively. We neglect noisein this example. In Figure 7(a), we show the DAS estimate forthis case. In Figure 7(b), we show the result of IAA after 10iterations. When applying IAA, we use ρ = 0 until the firstill-conditioning message for R occurs in MATLAB. At thatpoint, we switch to using the minimum ρ needed to avoid ill-conditioning. For this case, we do not need ρ until the verylast iteration, at which point we chose ρ = 10−14, which wasapproximately the minimum value needed to avoid an errormessage. As shown, IAA provides a much improved resultover the DAS image, which serves to illustrate the convergenceanalysis in this appendix.
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[18] J. Li, P. Stoica, L. Xu, and W. Roberts, “On parameter identifiabilityof MIMO radar,” IEEE Signal Processing Letters, vol. 14, no. 12, pp.968–971, December 2007.
[19] L. Xu, J. Li, and P. Stoica, “Target detection and parameter estimation forMIMO radar systems,” IEEE Transactions on Aerospace and ElectronicSystems, vol. 44, no. 3, pp. 927–939, July 2008.
[20] J. Capon, “High resolution frequency-wavenumber spectrum analysis,”Proceedings of the IEEE, vol. 57, pp. 1408–1418, August 1969.
[21] J. Li and P. Stoica, Eds., Robust Adaptive Beamforming. New York,NY: John Wiley & Sons, 2005.
[22] T. Yardibi, J. Li, P. Stoica, M. Xue, and A. B. Baggeroer,“Source localization and sensing: A nonparametric iterative adap-tive approach based on weighted least squares,” to appear in IEEETransactions on Aerospace and Electronic Systems, available athttp://www.sal.ufl.edu/eel6935/2008/IAA.pdf.
[23] W. Roberts, T. Yardibi, J. Li, X. Tan, and P. Stoica, “Sparse signalrepresentation for MIMO radar imaging,” 42nd Asilomar Conference onSignals, Systems and Computers (invited), Pacific Grove, CA, October2008.
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[28] J. Li and P. Stoica, “An adaptive filtering approach to spectral estimationand SAR imaging,” IEEE Transactions on Signal Processing, vol. 44,no. 6, pp. 1469–1484, June 1996.
[29] P. Stoica and R. L. Moses, Spectral Analysis of Signals. Upper SaddleRiver, NJ: Prentice-Hall, 2005.
12
Angle(deg)
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(a) (b)Fig. 1. True target images for: (a) negligible Doppler examples and (b) non-negligible Doppler examples.
[30] P. Stoica and Y. Selen, “Model-order selection: a review of informationcriterion rules,” IEEE Signal Processing Magazine, vol. 21, no. 4, pp.36–47, July 2004.
[31] C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia, andP. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A SignalProcessing Approach. Norwell, MA: Kluwer Academic Publishers,1996.
[32] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight SyntheticAperture Radar Signal Processing Algorithms. Norwood, MA: ArtechHouse Inc., 1995.
[33] J. G. Proakis, Digital Communications. McGraw-Hill Inc., third edition,1995.
[34] J. Li, P. Stoica, and X. Zheng, “Signal synthesis and receiver designfor MIMO radar imaging,” IEEE Transactions on Signal Processing,vol. 56, no. 8, pp. 3959–3968, August 2008.
[35] I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction fromlimited data using FOCUSS: A re-weighted minimum norm algorithm,”IEEE Transactions on Signal Processing, vol. 45, no. 3, pp. 600–616,1997.
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Fig. 2. Spotlight SAR images for N=1, L = 64 and SNR = 20 dB: (a) DAS with a SIMO array, (b) IAA-R with a SIMO array, (c) IAA-R with a SIMOarray and BIC applied, (d) DAS with a MIMO array, (e) IAA-R with a MIMO array and (f) IAA-R with a MIMO array and BIC applied. All levels shownare in dB. ‘O’ denotes a true target and ‘X’ represents a target estimate.
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Fig. 3. MIMO SAR images for N=1, L = 64 and SNR = 20 dB: (a) IAA without regularization and with a scanning grid from −30◦ to 30◦ and (b) IAAwithout regularization and with a scanning grid from −90◦ to 90◦. All levels shown are in dB. ‘O’ denotes a true target and ‘X’ represents a target estimate.
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Fig. 5. Range-Doppler images at 1◦ relative to a broadside scan for L = 32 and SNR = 20 dB: (a) DAS with a SIMO array, (b) IAA-R with a SIMO array,(c) DAS with a MIMO array, (d) IAA-R with a MIMO array and (e) IAA-R with a MIMO array and BIC applied. All levels shown are in dB. ‘O’ denotesa true target and ‘X’ represents a target estimate.
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16
Jian Li (S’87-M’91-SM’97-F’05) received theM.Sc. and Ph.D. degrees in electrical engineeringfrom The Ohio State University, Columbus, in 1987and 1991, respectively.
From April 1991 to June 1991, she was anAdjunct Assistant Professor with the Departmentof Electrical Engineering, The Ohio State Univer-sity, Columbus. From July 1991 to June 1993, shewas an Assistant Professor with the Departmentof Electrical Engineering, University of Kentucky,Lexington. Since August 1993, she has been with the
Department of Electrical and Computer Engineering, University of Florida,Gainesville, where she is currently a Professor. In Fall 2007, she was onsabbatical leave at MIT, Cambridge, Massachusetts. Her current researchinterests include spectral estimation, statistical and array signal processing,and their applications.
Dr. Li is a Fellow of IEEE and a Fellow of IET. She is a memberof Sigma Xi and Phi Kappa Phi. She received the 1994 National ScienceFoundation Young Investigator Award and the 1996 Office of Naval ResearchYoung Investigator Award. She was an Executive Committee Member of the2002 International Conference on Acoustics, Speech, and Signal Processing,Orlando, Florida, May 2002. She was an Associate Editor of the IEEETransactions on Signal Processing from 1999 to 2005, an Associate Editorof the IEEE Signal Processing Magazine from 2003 to 2005, and a memberof the Editorial Board of Signal Processing, a publication of the EuropeanAssociation for Signal Processing (EURASIP), from 2005 to 2007. She hasbeen a member of the Editorial Board of Digital Signal Processing – A ReviewJournal, a publication of Elsevier, since 2006. She is presently a member of theSensor Array and Multichannel (SAM) Technical Committee of IEEE SignalProcessing Society. She is a co-author of the papers that have received the Firstand Second Place Best Student Paper Awards, respectively, at the 2005 and2007 Annual Asilomar Conferences on Signals, Systems, and Computers inPacific Grove, California. She is a co-author of the paper that has received theM. Barry Carlton Award for the best paper published in IEEE Transactionson Aerospace and Electronic Systems in 2005. She is also a co-author ofthe paper that has received the Best Student Paper Award at the 2009 SPIEDefense, Security, and Sensing Conference in Orlando, Florida.
William Roberts (S’09) received the B.S. and M.Sc.degrees in electrical engineering from the Universityof Florida in 2006 and 2007, respectively. He isworking towards the Ph.D. degree in the Departmentof Electrical and Computer Engineering, Universityof Florida, Gainesville. His research interests includestatistical signal processing and radar signal process-ing. He is a Student Member of the IEEE.
Tarik Yardibi (S’08) received the B.S. degree fromHacettepe University, Ankara, Turkey in 2004 andthe M.S. degree from Bilkent University, Ankara,Turkey in 2006, both in electrical engineering. Heis currently pursuing the Ph.D. degree at the Spec-tral Analysis Laboratory at University of Florida,Gainesville. His research interests include statisticalsignal processing, sensor array processing, aeroa-coustic measurements and sparse signal representa-tions.
Petre Stoica (F’94) received the D.Sc. degreein automatic control from Polytechnic Institute ofBucharest (BPI), Bucharest, Romania, in 1979 andan honorary doctorate degree in science from Upp-sala University (UU), Uppsala, Sweden, in 1993.
He is a professor of systems modeling with theDivision of Systems and Control, the Departmentof Information Technology at UU. Previously, hewas a professor of system identification and signalprocessing with the Faculty of Automatic Controland Computers at BPI. He held longer visiting posi-
tions with Eindhoven University of Technology, Eindhoven, The Netherlands;Chalmers University of Technology, Gothenburg, Sweden (where he held aJubilee Visiting Professorship); UU; the University of Florida, Gainesville, FL;and Stanford University, Stanford, CA. His main scientific interests are in theareas of system identification, time series analysis and prediction, statisticalsignal and array processing, spectral analysis, wireless communications, andradar signal processing.
Dr. Stoica has published nine books, ten book chapters, and some 500papers in archival journals and conference records. The most recent bookhe coauthored, with R. Moses, is Spectral Analysis of Signals (Prentice-Hall, 2005). He is on the editorial boards of six journals: the Journal ofForecasting; Signal Processing; Circuits, Signals, and Signal Processing;Digital Signal Processing: A Review Journal; Signal Processing Magazine;and Multidimensional Systems and Signal Processing. He was a co-guesteditor for several special issues on system identification, signal processing,spectral analysis, and radar for some of the aforementioned journals, as wellas for IEE Proceedings. He was corecipient of the IEEE ASSP Senior Awardfor a paper on statistical aspects of array signal processing. He was alsorecipient of the Technical Achievement Award of the IEEE Signal ProcessingSociety. In 1998, he was the recipient of a Senior Individual Grant Awardof the Swedish Foundation for Strategic Research. He was also co-recipientof the 1998 EURASIP Best Paper Award for Signal Processing for a workon parameter estimation of exponential signals with time-varying amplitude,a 1999 IEEE Signal Processing Society Best Paper Award for a paper onparameter and rank estimation of reduced-rank regression, a 2000 IEEE ThirdMillennium Medal, and the 2000 W. R. G. Baker Prize Paper Award fora paper on maximum likelihood methods for radar. He was a member ofthe international program committees of many topical conferences. From1981 to 1986, he was a Director of the International Time-Series Analysisand Forecasting Society, and he was also a member of the IFAC TechnicalCommittee on Modeling, Identification, and Signal Processing. He is also amember of the Royal Swedish Academy of Engineering Sciences, an honorarymember of the Romanian Academy, and a fellow of the Royal StatisticalSociety.
17
Firooz A. Sadjadi received the BSEE from PurdueUniversity in 1972, MSEE in 1974, and the DEEin 1976 from the University of Southern California(USC). He worked at the USC Image ProcessingInstitute and the Image and Pattern Analysis Lab-oratory of the University of Tennessee, and was aconsultant to the Oak Ridge National Laboratory.During 1983-1993, he was with the Honeywell sys-tems and Research Center as a principal researchscientist. In 1993 he joined the Lockheed MartinCorporation where he is a senior research staff
scientist. His interests are in theoretical and experimental research relatedto signal and image processing, pattern recognition, target tracking andinformation fusion.
He was the Guest Editor for the Optical Engineering Journal SpecialIssues on Performance Evaluations of Signal and Image Processing Systems,February 1991, Automatic Target Recognition, December 1992, and for theIEEE Computer Special Section on Applications of Computer Vision, July1997. He currently serves on the Editorial Board of the Information FusionJournal. He is the author of more than 150 journal and conference publications,is an author of 7 book chapters, and holds 10 US and International Patents.He is the editor of several books, Automatic Target Recognition Systems(2000), Sensor and Data Fusion (1996), Performance Evaluations of Signaland Image Processing Systems (1993), and The Physics of Automatic TargetRecognition published by Springer in 2007. He has served on a number ofgovernmental and academic research panels and currently serves as the TwinCities Chapter of the IEEE Aerospace and Electronic Systems Society. Hehas received a number of technical awards including the Honeywell TechnicalAchievement Award, and the Lockheed Martin Evening of the Stars Award.He is a Fellow of the International Society for Optical Engineering (SPIE), isa Senior Member of the IEEE, and is a member of Sigma Xi, Optical Societyof America and Mathematical Association of America.