Amulti-input multi-output (MIMO) radar system, unlike standard phased-array radar,can transmit, via its antennas, multiple probing signals that may be correlated oruncorrelated with each other. While the companion article by Blum et al., to appear inthe November issue, exploited the diversity offered by widely separated transmit/receiveantenna elements, we focus on the merits of the waveform diversity allowed by trans-
mit and receive antenna arrays containing elements that are colocated. For the latter type of MIMOradar systems, we provide an overview of recent results showing that the waveform diversity enablesthe MIMO radar superiority in several fundamental aspects, including: 1) significantly improvedparameter identifiability, 2) direct applicability of adaptive arrays for target detection and parameterestimation, and 3) much enhanced flexibility for transmit beampattern design. Specifically, we showthat 1) the maximum number of targets that can be uniquely identified by the MIMO radar is up to Mt
times that of its phased-array counterpart, where Mt is the number of transmit antennas, 2) theechoes due to targets at different locations can be linearly independent of each other, which allows the
[Jian Li and Petre Stoica]
MIMO Radar withColocated Antennas
Digital Object Identifier 10.1109/MSP.2007.904812
direct application of many adaptive techniques to achieve highresolution and excellent interference rejection capability, and 3)the probing signals transmitted via its antennas can be opti-mized to obtain several transmit beampattern designs withsuperior performance. For example, the covariance matrix of theprobing signal vector transmitted by the MIMO radar can beoptimized to maximize the power around the locations of thetargets of interest and also to minimize the cross-correlation ofthe signals reflected back to the radar by these targets, therebysignificantly improving the performance of adaptive MIMO radartechniques. Additionally, we demonstrate the advantages of sev-eral MIMO transmit beampattern designs, including a beampat-tern matching design and a minimum sidelobebeampattern design, over their phased-arraycounterparts. In conclusion, the two articles inthis issue show that MIMO radar is a fertileresearch ground that merits further investiga-tion, including reaping the full benefits of bothtypes of diversity covered in the two articles.
INTRODUCTIONMIMO radar is an emerging technology that isattracting the attention of researchers and prac-titioners alike. Unlike a standard phased-arrayradar, which transmits scaled versions of a singlewaveform, a MIMO radar system can transmit viaits antennas multiple probing signals that maybe chosen quite freely (see Figure 1). This wave-form diversity enables superior capabilities com-pared with a standard phased-array radar. In[1]–[3], for example, the diversity offered bywidely separated transmit/receive antenna ele-ments is exploited. Many other papers, including,for instance, [4]–[21], have considered the meritsof a MIMO radar system with colocated antennas.For colocated transmit and receive antennas, theMIMO radar paradigm has been shown to offerhigher resolution (see, e.g., [4], [5]), higher sen-sitivity to detecting slowly moving targets [6],better parameter identifiability [11], [19], anddirect applicability of adaptive array techniques[11], [13]. Waveform optimization has also beenshown to be a unique capability of a MIMO radarsystem. For example, it has been used to achieveflexible transmit beampattern designs (see, e.g.,[7], [15]–[17]) as well as for MIMO radar imagingand parameter estimation [10], [20], [21].
We provide herein an overview of our recentresults showing that this waveform diversityenables the MIMO radar superiority in severalfundamental aspects. Without loss of generality,we consider targets associated with a particularrange and Doppler bin. Targets in adjacent rangebins contribute as interferences to the range binof interest.
First we address a basic aspect on MIMO radar—its param-eter identifiability, which is the maximum number of targetsthat can be uniquely identified by the radar. We show that thewaveform diversity afforded by MIMO radar enables a muchimproved parameter identifiability over its phased-array coun-terpart, i.e., the maximum number of targets that can beuniquely identified by the MIMO radar is up to Mt times thatof its phased-array counterpart, where Mt is the number oftransmit antennas. The parameter identifiability is furtherdemonstrated in a numerical study using both the Cramér-Rao bound (CRB) and a least-squares method for targetparameter estimation.
[FIG1] (a) MIMO radar versus (b) phased-array radar.
MIMO Transmit Array
MIMO Receive Array
Targets
(a)
(b)
Transmit Phased-Array
Receive Phased-Array
Targets
Combinations of {sm(t)}
sMt(t )s1(t )
β1s(t ) β2s(t )
WMts(t )W1s(t )
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We then consider an adaptive MIMO radar scheme that canbe used to deal with multiple targets. Linearly independentwaveforms can be transmitted simultaneously via the multipletransmit antennas of a MIMO radar. Due to the different phaseshifts associated with the different propagation paths from thetransmitting antennas to targets, these independent waveformsare linearly combined at the targets with different phase factors.As a result, the signal waveforms reflected from different targetsare linearly independent of each other, which allows for thedirect application of Capon and of other adaptive array algo-rithms. We consider herein applying the Capon algorithm toestimate the target locations and an approximate maximumlikelihood (AML) method recently introduced in [22] to deter-mine the reflected signal amplitudes.
Finally, we show that the probing signal vector transmittedby a MIMO radar system can be designed to approximate adesired transmit beampattern and also to minimize the cross-correlation of the signals bounced from various targets ofinterest—an operation that hardly would be possible for aphased-array radar. An efficient semidefinite quadratic pro-gramming (SQP) algorithm can be used to solve the signaldesign problem in polynomial time. Using this design, we cansignificantly improve the parameter estimation accuracy of theadaptive MIMO radar techniques. In addition, we consider aminimum sidelobe beampattern design. We demonstrate theadvantages of these MIMO transmit beampattern designs overtheir phased-array counterparts. Due to the significantly largernumber of degrees of freedom (DoF) of a MIMO system, we canachieve much better transmit beampatterns with a MIMOradar, under the practical uniform elemental transmit powerconstraint, than with its phased-array counterpart.
PROBLEM FORMULATIONConsider a MIMO radar system with Mt transmit antennas andMr receive antennas. Let xm(n) denote the discrete-time base-band signal transmitted by the mth transmit antenna. Also, let θdenote the location parameter(s) of a generic target, for exam-ple, its azimuth angle and its range. Then, under the assump-tion that the transmitted probing signals are narrowband andthat the propagation is nondispersive, the baseband signal at thetarget location can be described by the expression (see, e.g., [7],[15] and [23], Chapter 6)
Mt∑m=1
e− j2π f0τm(θ)xm(n) � a∗(θ)x(n), n = 1, · · · , N, (1)
where f0 is the carrier frequency of the radar, τm(θ) is the timeneeded by the signal emitted via the m th transmit antenna toarrive at the target, (·)∗ denotes the conjugate transpose, Ndenotes the number of samples of each transmitted signal pulse,
x(n) = [x1(n) x2(n) · · · xMt(n)]T, (2)
and
a(θ) = [e j2π f0τ1(θ) e j2π f0τ2(θ) · · · e j2π f0τMt(θ)]T, (3)
with (·)T denoting the transpose. By assuming that the transmitarray of the radar is calibrated, a(θ) is a known function of θ .
Let ym(n) denote the signal received by the mth receiveantenna; let
y(n) = [y1(n) y2(n) · · · yMr(n)]T, n = 1, · · · , N, (4)
and
b(θ) = [e j2π f0 τ1(θ) e j2π f0 τ2(θ) · · · e j2π f0 τMr(θ)]T, (5)
where τm(θ) is the time needed by the signal reflected by thetarget located at θ to arrive at the mth receive antenna. Then,under the simplifying assumption of point targets, the receiveddata vector can be described by the equation (see, e.g., [3] and[14])
y(n) =K∑
k=1
βkbc(θk)a∗(θk)x(n) + ε(n), n = 1, · · · , N, (6)
where K is the number of targets that reflect the signals backto the radar receiver, {βk} are the complex amplitudes propor-tional to the radar cross sections (RCSs) of those targets, {θk}are their location parameters, ε(n) denotes the interference-plus-noise term, and (·)c denotes the complex conjugate. Theunknown parameters, to be estimated from {y(n)}N
n=1 , are{βk}K
k=1 and {θk}Kk=1.
PARAMETER IDENTIFIABILITYParameter identifiability is basically a consistency aspect; wewant to establish the uniqueness of the solution to the param-eter estimation problem as either the signal-to-interference-plus-noise ratio (SINR) goes to infinity or the snapshotnumber N goes to infinity [19]. It is clear that in either case,assuming that the interference-plus-noise term ε(n) is uncor-related with x(n), the identifiability property of the first termin (6) is not affected by the second term. In particular, it fol-lows that asymptotically, we can handle any number of inter-ferences. Of course, for a finite snapshot number N and afinite SINR, the accuracy will degrade as the number of inter-ferences increases, but that is a different issue—the parame-ter identifiability is not affected.
Using results from [24] and [25], we can show that when theMt transmitted waveforms are linearly independent of eachother, a sufficient and generically (for almost every vector β)necessary condition for parameter identifiability is
Kmax ∈[
Mt + Mr − 22
,MtMr + 1
2
), (7)
depending on the array geometry and how many antennas areshared between the transmit and receive arrays [19].
Furthermore, generically, (i.e., for almost any vector β), theidentifiability can be ensured under the following condition[19], [24]:
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Kmax ∈[
2(Mt + Mr) − 53
,2MtMr
3
), (8)
which typically yields a larger number Kmax than the one givenin (7).
For a phased-array radar, the condition similar to (7) is
Kmax =⌈
Mr − 12
⌉, (9)
and that similar to (8) is
Kmax =⌈
2Mr − 33
⌉, (10)
where �·� denotes the smallest integer greater than or equal to agiven number.
Hence, the maximum number of targets that can beuniquely identified by a MIMO radar can be up to Mt timesthat of its phased-array counterpart. To illustrate the extremecases, note that when a filled (i.e., 0.5-wavelength interele-ment spacing) uniform linear array (ULA) is used for bothtransmitting and receiving, which appears to be the worstMIMO radar scenario from the parameter identifiabilitystandpoint, the maximum number of targets that can beidentified by the MIMO radar is about twice that of itsphased-array counterpart. This is because the virtual aper-ture bc(θ) ⊗ ac(θ) of the MIMO radar system has onlyMt + Mr − 1 distinct elements. On the other hand, when thereceive array is a filled ULA with Mr elements and the trans-mit array is a sparse ULA comprising Mt elements with Mr/2-wavelength interelement spacing, the virtual aperture of theMIMO radar system is a filled-element (MtMr) ULA, i.e., thevirtual aperture length is Mt times that of the receive array[4], [19]. This increased virtual aperture size for this caseleads to the result that the maximum number of targets thatcan be uniquely identified by the MIMO radar is Mt timesthat of its phased-array counterpart.
We present several numerical examples to demonstrate theparameter identifiability of MIMO radar, as compared to itsphased-array counterpart. First, consider a MIMO radar systemwhere a ULA with M = Mt = Mr = 10 antennas and half-wave-length spacing between adjacent antennas is used both fortransmitting and receiving. The transmitted waveforms areorthogonal to each other. Consider a scenario in which K tar-gets are located at θ1 = 0◦ , θ2 = 10◦ , θ3 = −10◦ , θ4 = 20◦ ,θ5 = −20◦ , θ6 = 30◦ , θ7 = −30◦ , · · · , with identical complexamplitudes β1 = · · · = βK = 1. The number of snapshots isN = 256. The received signal is corrupted by a spatially andtemporally white circularly symmetric complex Gaussian noisewith mean zero and variance 0.01 [i.e., signal-to-noise ratio(SNR) = 20 dB] and by a jammer located at 45◦ with anunknown waveform (uncorrelated with the waveforms transmit-ted by the radar) with a variance equal to 1 [i.e., interference-to-noise ratio (INR) = 20 dB].
Consider the CRB of {θk}, which gives the best performanceof an unbiased estimator. By assuming that {ε(n)}N
n=1 in (6)
are independently and identically distributed (i.i.d.) circularlysymmetric complex Gaussian random vectors with mean zeroand unknown covariance Q, the CRB for {θk} can be obtainedusing the Slepian-Bangs formula [23]. Figure 2(a) shows theCRB of θ1 for the MIMO radar as a function of K. For compari-son purposes, we also provide the CRB of its phased-arraycounterpart, for which all the parameters are the same as forthe MIMO radar except that Mt = 1 and that the amplitude ofthe transmitted waveform is adjusted so that the total trans-mission power does not change. Note that the phased-arrayCRB increases rapidly as K increases from 1 to 6. The corre-sponding MIMO CRB, however, is almost constant when K isvaried from 1 to 12 (but becomes unbounded for K > 12).Both results are consistent with the parameter identifiabilityanalysis (see [19] for details): Kmax ≤ 6 for the phased-arrayradar and Kmax ≤ 12 for the MIMO radar.
[FIG2] Performance of a MIMO radar system where a ULA withM = 10 antennas and 0.5-wavelength interelement spacing isused for both transmitting and receiving. (a) CRB of θ1 versus Kand (b) LS spatial spectrum when K = 12.
2 4 6 8 10 12
10−4
10−3
10−2
10−1
100
101
K
CR
B
Phased−ArrayMIMO Radar
(a)
−80 −60 −40 −20 0 20 40 60 800
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ulus
of C
ompl
ex A
mpl
itude
(b)
We next consider a simple semi-parametric least-squares (LS)method [14] for MIMO radar param-eter estimation. Figure 2(b) showsthe LS spatial spectrum as a func-tion θ , when K = 12. Note that all12 target locations can be approxi-mately determined from the peaklocations of the LS spatial spectrum.
Consider now a MIMO radar sys-tem with Mt = Mr = 5 antennas. The distance between adja-cent antennas is 0.5-wavelength for the receiving ULA and2.5-wavelength for the transmitting ULA. We retain all thesimulation parameters corresponding to Figure 2 except thatthe targets are located at θ1 = 0◦ , θ2 = 8◦ , θ3 = −8◦ ,θ4 = 16◦ , θ5 = −16◦ , θ6 = 24◦ , θ7 = −24◦ , · · · in this exam-
ple. Figure 3(a) shows the CRB of θ1,for both the MIMO radar and thephased-array counterpart, as a func-tion of K. Again, the MIMO CRB ismuch lower than the phased-arrayCRB. The behavior of both CRBs isconsistent with the parameter identi-fiability analysis: Kmax ≤ 3 for thephased-array radar and Kmax ≤ 16 forthe MIMO radar. Moreover, the
parameters of all K = 16 targets can be approximately deter-mined with the simple LS method, as shown in Figure 3(b).
DIRECT APPLICATION OF ADAPTIVETECHNIQUES FOR PARAMETER ESTIMATIONAs shown recently in [14], MIMO radar makes it possible to useadaptive localization and detection techniques directly, unlikea phased-array radar. This is another significant advantage of aMIMO radar system since adaptive techniques are known tohave much better resolution and much better interferencerejection capability than their data-independent counterparts.For example, in space-time adaptive processing applications,secondary range bins are needed to be able to use adaptivetechniques [26]–[28]; however, selecting quality secondaryrange bins is in itself a major challenge [29]. Since the MIMOprobing signals reflected back by the targets are actually lin-early independent of each other, the direct application of adap-tive techniques is made possible for a MIMO radar systemwithout the need for secondary range bins or even for rangecompression [20].
Let
A = [β∗1 a(θ1) β∗
2 a(θ2) · · · β∗Ka(θK)]. (11)
Then the sample covariance matrix of the target reflected wave-forms is A∗RxxA , where Rxx = (1/N)
∑Nn=1 x(n)x∗(n) . For
example, when orthogonal waveforms are used for MIMO prob-ing and N ≥ Mt, Rxx is a scaled identity matrix. Then A∗RxxAhas full rank, i.e., the target reflected waveforms are not com-pletely correlated with each other (or coherent), if the columnsof A are linearly independent of each other, which requires thatK ≤ Mt. The fact that the target reflected waveforms are nonco-herent allows the direct application of many adaptive techniquesfor target localization [14].
We demonstrate the performance of the Capon method for tar-get localization. Consider the scenario of a MIMO radar with a ULAcomprising M = Mt = Mr = 10 antennas with half-wavelengthspacing between adjacent antennas. This array is used both fortransmitting and receiving. Without loss of generality, the totaltransmit power is set to 1. Assume that K = 3 targets are locatedat θ1 = −40◦, θ2 = 0◦, and θ3 = 40◦ with complex amplitudesequal to β1 = β2 = β3 = 1. There is a strong jammer at 25◦ withan unknown waveform (uncorrelated with the transmitted MIMOradar waveforms) with a power equal to 106 (60 dB). Eachtransmitted signal pulse has N = 256 samples. The received signal
IEEE SIGNAL PROCESSING MAGAZINE [110] SEPTEMBER 2007
A MULTI-INPUT MULTI-OUTPUTRADAR SYSTEM
CAN TRANSMIT MULTIPLEPROBING SIGNALS THAT
MAY BE CORRELATEDOR UNCORRELATED WITH
EACH OTHER.
[FIG3] Performance of a MIMO radar system with Mt = Mr = 5antennas, and with half-wavelength interelement spacing forthe receive ULA and 2.5-wavelength interelement spacing for thetransmit ULA. (a) CRB of θ1 versus K and (b) LS spatial spectrumwhen K = 16.
2 4 6 8 10 12 14 16
K
(a)
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ompl
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itude
10−4
10−3
10−2
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100
101
CR
B
−80 −60 −40 −20 0 20 40 60 80
Angle (°)
(b)
Phased−ArrayMIMO Radar
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is also corrupted by a zero-mean circu-larly symmetric spatially and temporallywhite Gaussian noise with variance σ 2.
Since we do not assume any priorknowledge about the target locations,orthogonal waveforms are used forMIMO probing. (We refer to this asinitial probing, since after we get thetarget location estimates with thisprobing, we can optimize the transmitted beampattern toimprove the estimation accuracy, as shown in the followingsection.) Using the data collected as a result of this initialprobing, we can obtain the Capon spatial spectrum and thegeneralized likelihood ratio test (GLRT) function [14]. Anexample of the Capon spectrum for σ 2 = −10 dB is shown inFigure 4(a), where very narrow peaks occur around the targetlocations. Note that in Figure 4(a), a false peak occurs aroundθ = 25◦ due to the presence of the very strong jammer. Thecorresponding GLRT pseudo-spectrum as a function of θ isshown in Figure 4(b). Note that the GLRT is close to one atthe target locations and close to zero at any other locationsincluding the jammer location. Therefore, the GLRT can beused to reject the jammer peak in the Capon spectrum. Theremaining peak locations in the Capon spectrum are the esti-mated target locations.
PROBING SIGNAL DESIGNThe probing signal vector transmitted by a MIMO radar systemcan be designed to approximate a desired transmit beampatternand also to minimize the cross-correlation of the signalsbounced from various targets of interest—an operation that,like the direct application of adaptive techniques, would behardly possible for a phased-array radar [7], [15]–[17].
The power of the probing signal at a generic focal point withlocation θ is given by [see (1)]
P(θ) = a∗(θ)Ra(θ), (12)
where R is the covariance matrix of x(n), i.e.,
R = E {x(n)x∗(n)}. (13)
The spatial spectrum in (12), as a function of θ , will be calledthe transmit beampattern.
The first problem we will consider in this section consists ofchoosing R under a uniform elemental power constraint,
Rmm = cM
, m = 1, · · · , M; with c given , (14)
where M is a short notation for Mt, Rmm denotes the (m, m)thelement of R, to achieve the following goals:
■ Control the spatial power at a number of given target loca-tions by matching (or approximating) a desired transmitbeampattern.■ Minimize the cross-correlation between the probing signals
at a number of given target locations;note from (1) that the cross-correlationbetween the probing signals at loca-tions θ and θ is given by a∗(θ)Ra(θ).
Let φ(θ) denote a desired transmitbeampattern, and let {μl}L
l =1 be a finegrid of points that cover the locationsectors of interest. We assume thatthe said grid contains points which
are good approximations of the locations {θk}Kk=1 of the targets
of interest, and that we dispose of (initial) estimates {θk}Kk=1 of
{θk}Kk=1, where K denotes the number of targets of interest
that we wish to probe further. We can obtain φ(θ) and {θk}Kk=1
using the Capon and GLRT approaches presented in the previ-ous section.
As stated above, our goal is to choose R such that the trans-mit beampattern, a∗(θ)Ra(θ), matches or rather approximates[in a least squares (LS) sense] the desired transmit beampattern,φ(θ), over the sectors of interest, and also such that the cross-correlation (beam)pattern, a∗(θ)Ra(θ) (for θ �= θ ), is minimized(once again, in an LS sense) over the set {θk}K
k=1 .Mathematically, we want to solve the following problem
[FIG4] (a) The Capon spatial spectrum and (b) the GLRT pseudo-spectrum as functions of θ, for the initial omnidirectionalprobing.
0
0.5
1
1.5
2
0
0.5
1
1.5
2
Cap
on S
pect
rum
GLR
T
−50 0 50Angle (°)
(a)
−50 0 50Angle (°)
(b)
THE MAXIMUM NUMBER OFTARGETS THAT CAN BE
UNIQUELY IDENTIFIED BY AMIMO RADAR CAN BE UP TO
Mt TIMES THAT OF ITS PHASED-ARRAY COUNTERPART.
minα,R
{1L
L∑l =1
wl [αφ(μl) − a∗(μl)Ra(μl)]2
+ 2wc
K2 − K
K −1∑k=1
K∑p=k+1
∣∣∣a∗(θk)Ra(θp)
∣∣∣2⎫⎬⎭
s.t. Rmm = cM
, m = 1, · · · , M
R ≥ 0, (15)
where α is a scaling factor, wl ≥ 0, l = 1, · · · , L, is the weightfor the l th grid point and wc ≥ 0 is the weight for the cross-correlation term. The reason for introducing α in the designproblem is that typically φ(θ) is given in a normalized form(e.g., satisfying φ(θ) ≤ 1,∀θ ), and our interest lies in approxi-mating an appropriately scaled version of φ(θ), not φ(θ) itself.The value of wl should be larger than that of wk if the beam-pattern matching at μl is considered to be more importantthan the matching at μk. Note that by choosing maxl wl > wc
we can give more weight to the first term in the design criteri-on above, and viceversa for maxl wl < wc. We show in [15],[17] that this design problem can be efficiently solved in poly-nomial time as a SQP.
To illustrate the beampattern matching design, considerthe example considered in Figure 4. The initial target loca-tion estimates obtained using Capon or GLRT can be used toderive a desired beampattern. In the following numericalexamples, we form the desired beampattern by using thedominant peak locations of the GLRT pseudo-spectrum,denoted as θ1, · · · , θK , as follows (with K being the resultingestimate of K, and K = K ):
φ(θ) ={
1, θ ∈ [θk − �, θk + �], k = 1, · · · , K,
0, otherwise,(16)
[FIG6] MSEs of (a) the location estimates and (b) the complexamplitude estimates for the first target as functions of−10 log10 σ 2, obtained with initial omnidirectional probing andprobing using the beampattern matching design for � = 5◦,wc = 1, and c = 1.
[FIG5] MIMO beampattern matching designs for � = 5◦ andc = 1. The beampatterns are obtained using wc = 0 or wc = 1.
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mpa
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wc=1
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where 2� is the chosen beamwidth for each target (� shouldbe greater than the expected error in {θk}). Figure 5 isobtained using (16) with � = 5◦ in the beampattern match-ing design in (15) along with a mesh grid size of 0.1◦, wl = 1,l = 1, · · · , L, and either wc = 0 or wc = 1. Note that thedesigns obtained with wc = 1 and with wc = 0 are similar toone another. However, the cross-correlation behavior of theformer is much better than that of the latter in that thereflected signal waveforms corresponding to using wc = 1 arealmost uncorrelated with each other.
Next, we examine the mean-squared errors (MSEs) of thelocation estimates obtained by Capon and of the complexamplitude estimates obtained by AML [22]. In particular, wecompare the MSEs obtained using the initial omnidirection-al probing with those obtained using the optimal beampat-tern matching design shown in Figure 5 with � = 5◦ and
IEEE SIGNAL PROCESSING MAGAZINE [113] SEPTEMBER 2007
wc = 1. Figure 6(a) and (b) shows the MSE curves of thelocation and complex amplitude estimates obtained for thetarget at −40◦ from 1,000 Monte-Carlo trials (the results forthe other targets are similar). The estimates obtained usingthe optimal beampattern matching design are much better:the SNR gain over the omnidirec-tional design is larger than 10 dB.
Another beampattern design prob-lem we consider consists of choosing Runder the uniform elemental powerconstraint in (14) to achieve the fol-lowing goals:
■ Minimize the sidelobe level in aprescribed region.■ Achieve a predetermined 3 dB main-beam width.
This problem can be formulated as follows:
mint,R
− t
s.t. a∗(θ0)Ra(θ0) − a∗(μl)Ra(μl) ≥ t, ∀μl ∈ �
a∗(θ1)Ra(θ1) = 0.5a∗(θ0)Ra(θ0)
a∗(θ2)Ra(θ2) = 0.5a∗(θ0)Ra(θ0)
R ≥ 0
Rmm = cM
, m = 1, · · · , M, (17)
where θ2 − θ1 (with θ2 > θ0 and θ1 < θ0) determines the 3 dBmain-beam width and � denotes the sidelobe region of interest.As shown in [15] and [17], this minimum sidelobe beampatterndesign problem can be efficiently solved in polynomial time as asemidefinite program (SDP).
Finally, consider the conventional phased-array beampatterndesign problem in which only the array weight vector can beadjusted and therefore all antennas transmit the same different-ly scaled waveform. We can readily modify the previouslydescribed beampattern matching or minimum sidelobe beam-pattern designs for the case of phased-arrays by adding the con-straint rank(R) = 1. However, due to the rank-one constraint,both these originally convex optimization problems becomenon-convex. The lack of convexity makes the rank-one con-strained problems much harder to solve than the original con-vex optimization problems [30]. Semidefinite relaxation (SDR)is often used to obtain approximate solutions to such rank-constrained optimization problems [31]. Typically, the SDR isobtained by omitting the rank constraint. Hence, interestingly,the MIMO beampattern design problems are SDRs of the corre-sponding phased-array beampattern design problems.
In the numerical examples below, we have used the Newton-like algorithm presented in [30] to solve the rank-one con-strained design problems for phased-arrays. This algorithm usesSDR to obtain an initial solution, which is the exact solution tothe corresponding MIMO beampattern design problem.Although the convergence of the said Newton-like algorithm isnot guaranteed [30], we did not encounter any apparent prob-lem in our numerical simulations.
Consider the minimum sidelobe beampattern design prob-lem in (17), with the main beam centered at θ0 = 0◦, with a 3dB width equal to 20◦ (θ2 = −θ1 = 10◦), and with c = 1, forthe same MIMO radar scenario as the one considered in Figure4. The sidelobe region is � = [−90◦,−20◦] ∪ [20◦, 90◦]. The
MIMO minimum-sidelobe beampatterndesign is shown in Figure 7(a). Notethat the peak sidelobe level achieved bythe MIMO design is approximately 18dB below the mainlobe peak level.Figure 7(b) shows the correspondingphased-array beampattern obtained byusing the additional constraint
rank(R) = 1. The phased-array design fails to provide a propermainlobe (it suffers from peak splitting) and its peak sidelobelevel is much higher than that of its MIMO counterpart. We note
[FIG7] Minimum sidelobe beampattern designs under theuniform elemental power constraint when the 3 dB main beamwidth is 20◦. (a) MIMO and (b) phased-array.
−50 0 50−30
−20
−10
0
10
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)
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ttern
(dB
)
MIMO RADAR MAKES ITPOSSIBLE TO USE ADAPTIVE
LOCALIZATION AND DETECTIONTECHNIQUES DIRECTLY.
that, under the elemental power constraint, the number of DoFof the phased-array that can be used for beampattern design isequal to only M − 1 (real-valued parameters); consequently, it isdifficult for the phased-array to synthesize a proper beampat-tern. The MIMO design, on the other hand, can be used toachieve a much better beampattern due to its much larger num-ber of DoF, i.e., M2 − M.
CONCLUSIONSWe have provided a review of some recent results on the emerg-ing technology of MIMO radar with colocated antennas. We haveshown that the waveform diversity offered by such a MIMO radarsystem enables significant superiority over its phased-arraycounterpart, including much improved parameter identifiability,direct applicability of adaptive techniques for parameter estima-tion, as well as superior flexibility of transmit beampatterndesigns. We hope that this overview of our recent results on theMIMO radar, along with the related results obtained by our col-leagues, will stimulate the interest deserved by this topic in bothacademia and government agencies as well as industry.
ACKNOWLEDGMENTThis work was supported in part by the National ScienceFoundation under Grant No. CCF-0634786 and the SwedishResearch Council (VR). Opinions, interpretations, conclusions,and recommendations are those of the authors and are not nec-essarily endorsed by the United States Government.
AUTHORSJian Li ([email protected]) received the M.Sc. and Ph.D. degrees inelectrical engineering from The Ohio State University,Columbus in 1987 and 1991, respectively. She is a professorwith the Department of Electrical and Computer Engineering,University of Florida, Gainesville. Her current research interestsinclude spectral estimation, statistical and array signal process-ing, and their applications. She is a Fellow of IEEE and IET. Sheis presently a member of two of the IEEE Signal ProcessingSociety technical committees: Signal Processing Theory andMethods (SPTM) Technical Committee and Sensor Array andMultichannel (SAM) Technical Committee.
Petre Stoica is a professor of system modeling at UppsalaUniversity, Uppsala, Sweden. Additional information is availableat http://user.it.uu.se/ps/ps.html.
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IEEE SIGNAL PROCESSING MAGAZINE [114] SEPTEMBER 2007
