1

MIMO Radar Space-Time Adaptive ProcessingUsing Prolate Spheroidal Wave Functions

Chun-Yang Chen and P. P. Vaidyanathan, Fellow, IEEE

Abstract—In the traditional transmitting beamforming radarsystem, the transmitting antennas send coherent waveformswhich form a highly focused beam. In the multiple-input multiple-output (MIMO) radar system, the transmitter sends nonco-herent (possibly orthogonal) broad (possibly omnidirectional)waveforms. These waveforms can be extracted at the receiver bya matched filterbank. The extracted signals can be used to obtainmore diversity or to improve the spatial resolution for clutter.This paper focuses on space-time adaptive processing (STAP) forMIMO radar systems which improves the spatial resolution forclutter.

With a slight modification, STAP methods developed originallyfor the single-input multiple-output (SIMO) radar (conventionalradar) can also be used in MIMO radar. However, in the MIMOradar, the rank of the jammer-and-clutter subspace becomesvery large, especially the jammer subspace. It affects both thecomplexity and the convergence of the STAP algorithm. In thispaper, the clutter space and its rank in the MIMO radar areexplored. By using the geometry of the problem rather than data,the clutter subspace can be represented using prolate spheroidalwave functions (PSWF). A new STAP algorithm is also proposed.It computes the clutter space using the PSWF and utilizesthe block diagonal property of the jammer covariance matrix.Because of fully utilizing the geometry and the structure of thecovariance matrix, the method has very good SINR performanceand low computational complexity. 1

Index Terms—Space-Time Adaptive Processing (STAP),MIMO Radar, Prolate Spheroidal Wave Function, Clutter Sub-spaces.

I. INTRODUCTION

Recently, the concept of multiple-input multiple-output(MIMO) radars has drawn considerable attention [1]-[13].MIMO radars emit orthogonal waveforms [1]-[10] or nonco-herent [11]-[13] waveforms instead of transmitting coherentwaveforms which form a focused beam in traditional trans-mitter based beamforming. In the MIMO radar receiver, amatched filterbank is used to extract the orthogonal waveformcomponents. There are two different kinds of approachesfor using the noncoherent waveforms. First, increased spa-tial diversity can be obtained [4], [5]. In this scenario, thetransmitting antenna elements are far enough from each othercompared to the distance from the target. Therefore the targetradar cross sections (RCS) are independent random variablesfor different transmitting paths. When the orthogonal compo-nents are transmitted from different antennas, each orthogonalwaveform will carry independent information about the target.This spatial diversity can be utilized to perform better detection[4], [5]. Second, a better spatial resolution for clutter can be

1Work supported in parts by the ONR grant N00014-06-1-0011, and theCalifornia Institute of Technology.

obtained. In this scenario, the distances between transmittingantennas are small enough compared to the distance betweenthe target and the radar station. Therefore the target RCSis identical for all transmitting paths. The phase differencescaused by different transmitting antennas along with the phasedifferences caused by different receiving antennas can form anew virtual array steering vector. With judiciously designedantenna positions, one can create a very long array steeringvector with a small number of antennas. Thus the spatialresolution for clutter can be dramatically increased at a smallcost [1], [2]. In this paper, we focus on this second advantage.

The adaptive techniques for processing the data from air-borne antenna arrays are called space-time adaptive processing(STAP) techniques. The basic theory of STAP for the tra-ditional single-input multiple-output (SIMO) radar has beenwell developed [32], [33]. There have been many algorithmsproposed in [27]-[33] and the references therein for improvingthe complexity and convergence of the STAP in the SIMOradar. With a slight modification, these methods can alsobe applied to the MIMO radar case. The MIMO extensionof STAP can be found in [2]. The MIMO radar STAP formultipath clutter mitigation can be found in [10]. However, inthe MIMO radar, the space-time adaptive processing (STAP)becomes even more challenging because of the extra dimen-sion created by the orthogonal waveforms. On one hand, theextra dimension increases the rank of the jammer and cluttersubspace, especially the jammer subspace. This makes theSTAP more complex. On the other hand, the extra degreesof freedom created by the MIMO radar allows us to filter outmore clutter subspace with little effect on SINR.

In this paper, we explore the clutter subspace and its rankin MIMO radar. Using the geometry of the MIMO radar andthe prolate spheroidal wave function (PSWF), a method forcomputing the clutter subspace is developed. Then we developa STAP algorithm which computes the clutter subspace usingthe geometry of the problem rather than data and utilizesthe block-diagonal structure of the jammer covariance matrix.Because of fully utilizing the geometry and the structureof the covariance matrix, our method has very good SINRperformance and significantly lower computational complexitycompared to fully adaptive methods (Section V-B).

In practice, the clutter subspace might change because ofeffects such as the internal clutter motion (ICM), velocity mis-alignment, array manifold mismatch, and channel mismatch[32]. In this paper, we consider an “ideal model”, which doesnot take these effects into account. When this model is notvalid, the performance of the algorithm will degrade. One wayto overcome this might be to estimate the clutter subspace by

2

using a combination of both the assumed geometry and thereceived data. Another way might be to develop a more robustalgorithm against the clutter subspace mismatch. These ideaswill be explored in the future.

The rest of the paper is organized as follows. In SectionII, the concept of MIMO radar will be briefly reviewed. InSection III, we formulate the STAP approach for MIMO radar.In Section IV, we explore the clutter subspace and its rankin the MIMO radar. Using prolate spheroidal wave functions(PSWF), we construct a data-independent basis for cluttersignals. In Section V, we propose a new STAP method forMIMO radar. This method utilizes the technique proposedin Section IV to find the clutter subspace and estimates thejammer-plus-noise covariance matrix separately. Finally, thebeamformer is calculated by using matrix inversion lemma. Aswe will see later, this method has very satisfactory SINR per-formance. In Section VI, we compare the SINR performanceof different STAP methods based on numerical simulations.Finally, Section VII concludes the paper.

Notations. Matrices are denoted by capital letters in bold-face (e.g., A). Vectors are denoted by lowercase letters inboldface (e.g., x). Superscript † denotes transpose conjugation.The notation diag(A,A, · · · ,A) denotes a block diagonalmatrix whose diagonal blocks are A. The notation �a� isdefined as the smallest integer larger than a.

II. REVIEW OF THE MIMO RADAR

In this section, we briefly review the MIMO radar idea.More detailed reviews can be found in [1], [2], [6]. We willfocus on using MIMO radar to increase the degrees of free-dom. Fig. 1 illustrates a MIMO radar system. The transmitting

TransmitterM antennas

ReceiverN antennas

dT2( ) 1( ) 0( )

dRMF MF…

…

Fig. 1. Illustration of a MIMO radar system with M = 3 and N = 4.

antennas emit orthogonal waveforms φk(τ). At each receivingantenna, these orthogonal waveforms can be extracted byM matched filters, where M is the number of transmittingantennas. Therefore there are a total of NM extracted signals,where N is the number of receiving antennas. The signalsreflected by the target at direction θ can be expressed as

ρtej 2π

λ (ndR sin θ+mdT sin θ), (1)

for n = 0, 1, · · · , N − 1, m = 0, 1, · · · ,M − 1. Here ρt isthe amplitude of the signal reflected by the target, dR is thespacing between the receiving antennas, and dT is the spacingbetween the transmit antennas. The phase differences arecreated by both transmitting and receiving antenna locations.

Define fs � (dR/λ) sin θ and γ � dT /dR. Equation (1) canbe further simplified as

ρtej2πfs(n+γm).

If we choose γ = N , the set {n+γm} = {0, 1, · · · , NM−1}.Thus the NM signals in (1) can be viewed as the signalsreceived by a virtual array with NM elements [2] as shownin Fig. 2. It is as if we have a receiving array of NM

Virtual array

Fig. 2. The virtual array corresponding to the MIMO radar in Fig. 1.

elements. Thus NM degrees of freedom can be obtainedwith only N +M physical array elements. One can view theantenna array as a way to sample the electromagnetic wave inthe spatial domain. The MIMO radar idea allows “sampling”in both transmitter and receiver and creates a total of NM“samples”. Taking advantage of these extra samples in spatialdomain, a better spatial resolution can be obtained.

III. STAP IN MIMO RADAR

In this section, we formulate the STAP problem in MIMOradar. The MIMO extension for STAP first appeared in [2]. Wewill focus on the idea of using the extra degrees of freedomto increase the spatial resolution for clutter.

A. Signal Model

Fig. 3 shows the geometry of the MIMO radar STAP withuniform linear arrays (ULA), where

1) dT is the spacing of the transmitting antennas,2) dR is the spacing of the receiver antennas,3) M is the number of transmitting antennas,4) N is the number of the receiving antennas,5) T is the radar pulse period,6) l indicates the index of radar pulse (slow time),7) τ represents the time within the pulse (fast time),8) vt is the target speed toward the radar station, and9) v is the speed of the radar station.

Notice that the model assumes the two antenna arrays arelinear and parallel. The transmitter and the receiver are closeenough so that they share the same angle variable θ. Theradar station movement is assumed to be parallel to the linearantenna array. This assumption has been made in most ofthe airborne ground moving target indicator (GMTI) systems.Each array is composed of omnidirectional elements. Thetransmitted signals of the mth antenna can be expressed as

xm(lT + τ) =√Eφm(τ)ej2πf(lT+τ),

for m = 1, 2, · · · ,M − 1, where φm(τ) is the baseband pulsewaveform, f is the carrier frequency, and E is the transmitted

3

…dT

dTsin

ej(2 ft-x

2dTsin

dT

x0(lT+ )…

v

target

vt

clutter

…

xM-2(lT+ ) xM-1(lT+ )

vsin …dR

dRsin

(N-1)dRsin

ej(2 ft-x

2dRsin

dR…

target

vt

clutter

…

Matchedfilterbank…

Matchedfilterbank…

Matchedfilterbank…

v

vsin

Transmitter Receiver

Fig. 3. This figure illustrates a MIMO radar system with M transmitting antennas and N receiving antennas. The radar station is moving with speed v.

energy for the pulse. The demodulated received signal of thenth antenna can be expressed as

yn(lT + τ +2rc

) ≈M−1∑m=0

ρtφm(τ)ej2πλ (sin θt(2vT l+dRn+dTm)+2vtTl)

+Nc−1∑i=0

M−1∑m=0

ρiφm(τ)ej2πλ (sin θi(2vT l+dRn+dTm))

+y(J)n (lT + τ +

2rc

) + y(w)n (lT + τ +

2rc

), (2)

where

1) r is the distance of the range bin of interest,2) c is the speed of light,3) ρt is the amplitude of the signal reflected by the target,4) ρi is the amplitude of the signal reflected by the ith

clutter,5) θt is the looking direction of the target,6) θi is the looking direction of the ith clutter,7) Nc is the number of clutter signals,8) y

(J)n is the jammer signal in the nth antenna output, and

9) y(w)n is the white noise in the nth antenna output.

For convenience, all of the parameters used in the signal modelare summarized in Table I. The first term in (2) representsthe signal reflected by the target. The second term is thesignal reflected by the clutter. The last two terms representthe jammer signal and white noise. We assume there is nointernal clutter motion (ICM) or antenna array misalignment[32]. The phase differences in the reflected signals are causedby the Doppler shift, the differences of the receiving antennalocations, and the differences of the transmitting antennalocations. In the MIMO radar, the transmitting waveformsφm(τ) satisfy orthogonality:∫

φm(τ)φ∗k(τ)dτ = δmk. (3)

The sufficient statistics can be extracted by a bank of matchedfilters as shown in Fig. 3. The extracted signals can be

TABLE ILIST OF THE PARAMETERS USED IN THE SIGNAL MODEL

dT spacing of the transmitting antennasdR spacing of the receiving antennasM number of the transmitting antennasN number of the receiving antennasT radar pulse periodl index of radar pulse (slow time)τ time within the pulse (fast time)vt target speed toward the radar stationxm transmitted signal in the mth antennaφm baseband pulse waveformsyn demodulated received signal in the nth antennavt target speed toward the radar stationv speed of the radar stationr distance of the range bin of interestc speed of lightρt amplitude of the signal reflected by the targetρi amplitude of the signal reflected by the ith clutterθt looking direction of the targetθi looking direction of the ith clutterNc number of clutter signals

y(J)n jammer signal in the nth antenna output

y(w)n white noise in the nth antenna output

expressed as

yn,m,l �∫yn(lT + τ +

2rc

)φ∗m(τ)dτ =

ρtej 2π

λ (sin θt(2vT l+dRn+dTm)+2vtTl) + (4)Nc−1∑i=0

ρiej 2π

λ (sin θi(2vT l+dRn+dTm)) + y(J)n,m,l + y

(w)n,m,l,

for n = 0, 1, · · · , N − 1, m = 0, 1, · · · ,M − 1, andl = 0, 1, · · · , L−1, where y(J)

n,m,l is the corresponding jammer

signal, y(w)n,m,l is the corresponding white noise, and L is

the number of the pulses in a coherent processing interval(CPI). To simplify the above equation, we define the followingnormalized spatial and Doppler frequencies:

fs � dRλ

sin θt, fs,i � dRλ

sin θi

fD � 2(v sin θt + vt)λ

T. (5)

4

One can observe that the normalized Doppler frequency ofthe target is a function of both target looking direction andspeed. Throughout this paper we shall make the assumptiondR = λ/2 so that spatial aliasing is avoided. Using the abovedefinition we can rewrite the extracted signal in (4) as

yn,m,l = ρtej2πfs(n+γm)ej2πfDl + (6)

Nc−1∑i=0

ρiej2πfs,i(n+γm+βl) + y

(J)n,m,l + y

(w)n,m,l,

for n = 0, 1, · · · , N − 1, m = 0, 1, · · · ,M − 1, and l =0, 1, · · · , L− 1, where

γ � dT /dR, and β � 2vT/dR. (7)

B. Fully adaptive MIMO-STAP

The goal of space-time adaptive processing (STAP) is to finda linear combination of the extracted signals so that the SINRcan be maximized. Thus the target signal can be extracted fromthe interferences, clutter, and noise to perform the detection.Stacking the MIMO STAP signals in (6), we obtain the NMLvector

y =(y0,0,0 y1,0,0 · · · yN−1,M−1,L−1

)T. (8)

Then the linear combination can be expressed as w†y, wherew is the weight vector for the linear combination. The SINRmaximization can be obtained by minimizing the total varianceunder the constraint that the target response is unity. It can beexpressed as the following optimization problem:

minw

w†Rw

subject to w†s(fs, fD) = 1, (9)

where R � E[yy†], and s(fs, fD) is the size-NML MIMOspace-time steering vector which consists of the elements

ej2πfs(n+γm)ej2πfDl, (10)

for n = 0, 1, · · · , N − 1, m = 0, 1, · · · ,M − 1, andl = 0, 1, · · · , L−1. This w is called minimum variance distor-tionless response (MVDR) beamformer [20]. The covariancematrix R can be estimated by using the neighboring range bincells. In practice, in order to prevent self-nulling, a target-freecovariance matrix can be estimated by using guard cells [32].The well-known solution to the above problem is [20]

w =R−1s(fs, fD)

s(fs, fD)†R−1s(fs, fD). (11)

However, the covariance matrix R is NML × NML. It ismuch larger than in the SIMO case because of the extradimension. The complexity of the inversion of such a largematrix is high. The estimation of such a large covariancematrix also converges slowly. To overcome these problems,partially adaptive techniques can be applied. The methodsdescribed in Section VI are examples of such partially adaptivetechniques. In SIMO radar literature such partially adaptivemethods are commonly used [32], [33].

C. Comparison with SIMO system

In the traditional transmit beamforming, or single-input-multiple-output (SIMO) radar, the transmitted waveforms arecoherent and can be expressed as

φm(τ) = φ(τ)wTm

for m = 1, 2, · · · ,M − 1, where {wTm} are the transmitbeamforming weights. The sufficient statistics can be extractedby a single matched filter for every receiving antenna. Theextracted signal can be expressed as

yn,l �∫yn(lT + τ +

2rc

)φ∗(τ)dτ =

ρtej2πfsnej2πfDl

M−1∑m=0

wTmej2πfsγm + (12)

Nc−1∑i=0

ρiej2πfs,i(n+βl)

M−1∑m=0

wTmej2πfs,iγm + yJn,l + y

(w)n,l ,

for n = 0, 1, · · · , N − 1, and l = 0, 1, · · · , L − 1, wherey(J)n,l is the corresponding jammer signal, and y

(w)n,l is the

corresponding white noise. Comparing the MIMO signals in(6) and the SIMO signals in (12), one can see that a linearcombination with respect to m has been performed on theSIMO signal in the target term and the clutter term. TheMIMO radar, however, leaves all degrees of freedom to thereceiver. Note that in the receiver, one can perform the samelinear combination with respect to m on the MIMO signal in(6) to create the SIMO signal in (12). The only differenceis that the transmitting power for the SIMO signal is lessbecause of the focused beam used in the transmitter. Forthe SIMO radar, the number of degrees of freedom is M inthe transmitter and NL in the receiver. The total number ofdegrees of freedom is M+NL. However, for the MIMO radar,the number of degrees of freedom is NML which is muchlarger than M +NL. These extra degrees of freedom can beused to obtain a better spatial resolution for clutter.

The MIMO radar transmits omnidirectional orthogonalwaveforms from each antenna element. Therefore it illumi-nates all angles. The benefit of SIMO radar is that it transmitsfocused beams which saves transmitting power. Therefore,for a particular angle of interest, the SIMO radar enjoys aprocessing gain of M compared to the MIMO radar. However,for some applications like scanning or imaging, it is necessaryto illuminate all angles. In this case, the benefit of a focusedbeam no longer exists because both systems need to consumethe same energy for illuminating all angles. The SIMO systemwill need to steer the focused transmit beam to illuminate allangles.

A second point is that for the computation of the MIMObeamformer in (11), the matrix inversion R−1 needs to becomputed only once and it can be applied for all angles.The transmitting array in a MIMO radar does not have afocused beam. So, all the ground points within a range binare uniformly illuminated. The clutter covariance seen by thereceiving antenna array is, therefore, the same for all angles.In the SIMO case, the matrix inversions need to be computed

5

for different angles because the clutter signal changes as thebeam is steered through all angles.

D. Virtual Array

Observing the MIMO space-time steering vector defined in(10), one can view the first term ej2πfs(n+γm) as a sampledversion of the sinusoidal function ej2πfsx. Recall that γ isdefined in (6) as the ratio of the antenna spacing of thetransmitter and receiver. To obtain a good spatial frequencyresolution, these signals should be critically sampled and havelong enough duration. One can choose γ = N becauseit maximizes the time duration while maintaining criticalsampling [2] as shown in Fig. 2. Sorting the sample pointsn+ γm for n = 0, 1, · · · , N − 1, and m = 0, 1, · · · ,M − 1,we obtain the sorted sample points k = 0, 1, · · · , NM − 1.Thus the target response in (10) can be rewritten as

ej2πfskej2πfDl

for k = 0, 1, · · · , NM − 1, and l = 0, 1, · · · , L− 1. It is as ifwe have a virtual receiving array with NM antennas. How-ever, the resolution is actually obtained by only M antennas inthe transmitter and N antennas in the receiver. Fig. 4 comparesthe SINR performance of the MIMO system and the SIMOsystem in the array looking direction of zero degree, that is,fs = 0. The optimal space-time beamformer described in (11)is used. The parameter L equals 16, and β equals 1.5 in this

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−7

−6

−5

−4

−3

−2

−1

0

1

Normalized Doppler

SIN

R (

dB)

SIMO: N = 10, M = 5, γ=1SIMO: N = 50, M = 1

MIMO: N = 10, M = 5, γ=1

MIMO: N = 10, M = 5, γ=10

Fig. 4. The SINR at looking direction zero as a function of the Dopplerfrequencies for different SIMO and MIMO systems.

example. In all plots it is assumed that the energy transmittedby any single antenna element to illuminate all angles is fixed.The SINR drops near zero Doppler frequency because it isnot easy to distinguish the slowly moving target from the stillground clutter. The MIMO system with γ = 1 has a slightlybetter performance than the SIMO system with the sameantenna structure. For the virtual array structure where γ = N ,the MIMO system has a very good SINR performance and itis close to the performance of the SIMO system with NMantennas because they have the same resolution for the targetsignal and the clutter signals. The small difference comes from

the fact that the SIMO system with NM antennas has a betterspatial resolution for the jammer signals. This example showsthat the choice of γ is very crucial in the MIMO radar. Withthe choice γ = 10 = N , the MIMO radar with only 15 antennaelements has about the same performance as the SIMO radarwith 51 array elements. This example also shows that theMIMO radar system has a much better spatial resolution forclutter compared to the traditional SIMO system with samenumber of physical antenna elements.

IV. CLUTTER SUBSPACE IN MIMO RADAR

In this section, we explore the clutter subspace and its rankin the MIMO radar system. The covariance matrix R in (9) canbe expressed as R = Rt + Rc + RJ + σ2I, where Rt is thecovariance matrix of the target signal, Rc is the covariancematrix of the clutter, RJ is the covariance matrix of thejammer, and σ2 is the variance of the white noise. The cluttersubspace is defined as the range space of Rc and the clutterrank is defined as the rank of Rc. In the space-time adaptiveprocessing (STAP) literature, it is a well-known fact that theclutter subspace usually has a small rank. It was first pointedout by Klemm in [18], that the clutter rank is approximatelyN+L, where N is the number of receiving antennas and L isthe number of pulses in a coherent processing interval (CPI).In [16] and [17], a rule for estimating the clutter rank wasproposed. The estimated rank is approximately

N + β(L− 1), (13)

where β = 2vT/dR. This is called Brennan’s rule. In [15],this rule has been extended to the case with arbitrary arrays.Taking advantage of the low rank property, the STAP can beperformed in a lower dimensional space so that the complexityand the convergence can be significantly improved [26]-[33].This result will now be extended to the MIMO radar. Thesetechniques are often called partially adaptive methods orsubspace methods.

A. Clutter rank in MIMO radar

We first study the clutter term in (6) which is expressed as

y(c)n,m,l =

Nc−1∑i=0

ρiej2πfs,i(n+γm+βl),

for n = 0, 1, · · · , N − 1, m = 0, 1, · · · ,M − 1, and l =0, 1, · · · , L − 1. Note that −0.5 < fs,i < 0.5 because dR =λ/2. Define ci,n,m,l = ej2πfs,i(n+γm+βl) and

ci =(ci,0,0,0, ci,1,0,0, · · · , ci,N−1,M−1,L−1

)T. (14)

By stacking the signals {y(c)n,m,l} into a vector, one can obtain

y(c) =Nc−1∑i=0

ρici.

Assume that ρi are zero-mean independent random variableswith variance σ2

c,i. The clutter covariance matrix can beexpressed as

Rc = E[y(c)y(c)†] =Nc−1∑i=0

σ2c,icic

†i .

6

Therefore, span(Rc) = span(C), where

C �(

c0, c1, · · · , cNc−1

).

The vector ci consists of the samples of ej2πfs,ix at points{n+ γm+ βl}, where γ and β are defined in (7). In general,ci is a nonuniformly sampled version of the bandlimitedsinusoidal waveform ej2πfs,ix. If γ and β are both integers,the sampled points {n+ γm+ βl} can only be integers in

{0, 1, · · · , N + γ(M − 1) + β(L− 1)}.If N+γ(M−1)+β(L−1) ≤ NML, there will be repetitionsin the sample points. In other words, some of the row vectorsin C will be exactly the same and there will be at most N +γ(M −1)+β(L−1) distinct row vectors in C. Therefore therank of C is less than N + γ(M − 1) + β(L− 1). So is therank of Rc. We summarize this fact as the following theorem:

Theorem 1: If γ and β are both integers, then rank(Rc) ≤min(N + γ(M − 1) + β(L− 1), Nc, NML).���

Usually Nc and NML are much larger than N +γ(M −1)+β(L − 1). Therefore N + γ(M − 1) + β(L − 1) is a goodestimation of the clutter rank. This result can be viewed as anextension of Brennan’s rule [16], given in (13), to the MIMOradar case.

Now we focus on the general case where γ and β arereal numbers. The vector ci in (14) can be viewed as anonuniformly sampled version of the truncated sinusoidalfunction

c(x; fs,i) �{ej2πfs,ix, 0 ≤ x ≤ X0, otherwise,

(15)

where X � N − 1 + γ(M − 1) + β(L − 1). Furthermore,−0.5 ≤ fs,i ≤ 0.5 because dR is often selected as λ/2 in(5) to avoid aliasing. Therefore, the energy of these signalsis mostly confined to a certain time-frequency region. Fig.5 shows an example of such a signal. Such signals can be

−50 0 50 100 150−1

−0.5

0

0.5

1

x

Re{

c(f s,

i,x)}

−1 −0.5 0 0.5 10

20

40

60

80

100

fs

|C(f

s,i,f s)|

(b)

Fig. 5. Example of the signal c(x; fs,i). (a) Real part. (b) Magnituderesponse of Fourier transform.

well approximated by linear combinations of �2WX + 1�orthogonal functions [19], where W is the one sided band-width and X is the duration of the time-limited functions.In the next section, more details on this will be discussed

using prolate spheroidal wave functions (PSWF). In this case,we have W = 0.5 and 2WX + 1 = N + γ(M − 1) +β(L−1). The vectors ci can be also approximated by a linearcombination of the nonuniformly sampled versions of these�N + γ(M − 1) + β(L − 1)� orthogonal functions. Thus, inthe case where γ and β are nonintegers, we can conclude thatonly �N + γ(M − 1) + β(L− 1)� eigenvalues of the matrixRc are significant. In other words,

rank(Rc) ≈ �N + γ(M − 1) + β(L− 1)�. (16)

Note that the definition of this approximate rank is actually thenumber of the dominant eigenvalues. This notation has beenwidely used in the STAP literature [32], [33]. In the SIMOradar case, using Brennan’s rule, the ratio of the clutter rankand the total dimension of the space-time steering vector canbe approximated as

N + β(L− 1)NL

=1L

+β(L− 1)NL

.

In the MIMO radar case with γ = N , the corresponding ratiobecomes

N +N(M − 1) + β(L− 1)NML

=1L

+β(L− 1)NML

.

One can observe that the clutter rank now becomes a smallerportion of the total dimension because of the extra dimensionintroduced by the MIMO radar. Thus the MIMO radar receivercan null out the clutter subspace with little effect on SINR.Therefore a better spatial resolution for clutter can be obtained.

The result can be further generalized for the arraywith arbitrary linear antenna deployment. Let xT,m,m =0, 1, · · · ,M − 1 be the transmitting antenna locations,xR,n, n = 0, 1, · · · , N − 1 be the receiving antenna locations,and v be the speed of the radar station. Without loss ofgenerality, we set xT,0 = 0 and xR,0 = 0. Then the cluttersignals can be expressed as

y(c)n,m,l =

Nc−1∑i=0

ρiej 2π

λ sin θi((xR,n+xT,m+2vT l)),

for n = 0, 1, · · · , N − 1, m = 0, 1, · · · ,M − 1, and l =0, 1, · · · , L − 1, where θi is the looking-direction of the ithclutter. The term

ej2πλ sin θi(xR,n+xT,m+2vT l)

can also be viewed as a nonuniform sampled version of thefunction ej

2πλ sin θix. Using the same argument we have made

in the uniform linear array (ULA) case, one can obtain

rank(Rc) ≈ �1 +2λ

(xR,N−1 + xT,M−1 + 2vT (L− 1))�.

The quantity xR,N−1 +xT,M−1 +2vT (L−1) can be regardedas the total aperture of the space-time virtual array. One cansee that the number of dominant eigenvalues is proportionalto the ratio of the total aperture of the space-time virtual arrayand the wavelength.

7

B. Data independent estimation of the clutter subspace withPSWF

The clutter rank can be estimated by using (16) and theparameters N , M , L, β and γ. However, the clutter subspaceis often estimated by using data samples instead of using theseparameters [26]-[33]. In this section, we propose a methodwhich estimates the clutter subspace using the geometry of theproblem rather than the received signal. The main advantageof this method is that it is data independent. The cluttersubspace obtained by this method can be used to improvethe convergence of the STAP. Experiments also show that theestimated subspace is very accurate in the ideal case (withoutICM and array misalignment).

In Fig. 5, one can see that the signal in (15) is time-limitedand most of its energy is concentrated on −0.5 ≤ fs ≤ 0.5.To approximate the subspace which contains such signals, wefind the basis functions which are time-limited and concentratetheir energy on the corresponding bandwidth. Such basisfunctions are the solutions of the following integral equation[19]

μψ(x) =∫ X

0

sinc(2W (x− ζ))ψ(ζ)dζ,

where sinc(x) � sinπxπx and μ is a scalar to be solved. This

integral equation has infinite number of solutions ψi(x) andμi for i = 0, 1, · · · ,∞. The solution ψi(x) is called prolatespheroidal wave function (PSWF). By the maximum principle[36], the solution satisfies

ψ0(x) = arg max‖ψ‖=1

∫ X

0

∫ X

0

ψ∗(x)sinc(2W (x− ζ))ψ(ζ)dζdx

ψi(x) = arg max‖ψ‖=1

∫ X

0

∫ X

0

ψ∗(x)sinc(2W (x− ζ))ψ(ζ)dζdx

subject to∫ X

0

ψ(x)ψ∗k(x)dx = 0, for k = 0, 1, · · · , i− 1,

for i = 1, 2, · · · ,∞. The function ψi(x) is orthogonal to theprevious basis components ψk(x), for k < i while concentrat-ing most of its energy on the bandwidth [−W,W ]. Moreover,only the first �2WX + 1� eigenvalues μi are significant [19].Therefore, the time-band-limited function c(x; fs,i) in (15)can be well approximated by linear combinations of ψi(x)for i = 0, 1, · · · , �2WX + 1�. In this case, W = 0.5and 2WX + 1 = N + γ(M − 1) + β(L − 1). Thus thenonuniformly sampled versions of c(x; fs,i), namely ci,n,m,l,can be approximated by the linear combination:

ci,n,m,l � ej2πfs,i(n+γm+βl) ≈rc−1∑k=0

αi,kψk(n+ γm+ βl),

for some {αi,k} where

rc � �N + γ(M − 1) + β(L− 1)�. (17)

Stacking the above elements into vectors, we have

ci ≈rc−1∑k=0

αi,kuk,

where uk is a vector which consists of the elements ψk(n +γm+ βl). Finally, we have

span(Rc) = span(C) ≈ span(Uc), (18)

where Uc �(

u0 u1 · · · urc−1

). Note that although

the functions {ψk(x)} are orthogonal, the vectors {uk} arein general not orthogonal. This is because of the fact that{uk} are obtained by nonuniform sampling which destroysorthogonality. In practice, the PSWF ψi(x) can be computedoff-line and stored in the memory. When the parameterschange, one can obtain the vectors uk by resampling thePSWF ψk(n + γm + βl) to form the new clutter subspace.In this way, we can obtain the clutter subspace by using thegeometry of the problem.

Performing the Gram-Schmidt procedure on the basis {uk},we obtain the orthonormal basis {qk}. The clutter power ineach orthonormal basis element can be expressed as q†

kRcqk.Fig. 6 shows the clutter power in the orthogonalized basiselements. In this example, N = 10, M = 5, L = 16, γ = 10,

0 50 100 150 200

−200

−150

−100

−50

0

50

100

Basis element index

Clu

tter

pow

er (

dB)

Proposed subspace methodEigen decompositionEstimated rank

Fig. 6. Plot of the clutter power distributed in each of the orthogonal basiselements.

and β = 1.5. Note that there are a total of NML = 800 basiselements but we only show the first 200 on the plot. The cluttercovariance matrix Rc is generated using the model describedin [15]. The eigenvalues of Rc are also shown in Fig. 6 forcomparison. The estimated clutter rank is �N + γ(M − 1) +β(L − 1)� = 73. One can see that the subspace obtained bythe proposed method captures almost all clutter power. Theclutter power decays to less than −200 dB for the basis indexexceeding 90.

Compared to the eigen decomposition method, the subspaceobtained by our method is larger. This is because of the factthat the clutter spatial bandwidth has been overestimated inthis example. More specifically, we have assumed the worstcase situation that the clutter spatial frequencies range from−0.5 to 0.5. In actual fact however, the range is only from−0.35 to 0.35. This comes about because of the specificgeometry assumed in this example: the altitude is 9km, therange of interest is 12.728km, and a flat ground model is used.Therefore the rank of the subspace is overestimated. It mayseem that our method loses some efficiency compared to the

8

eigen decomposition. However, note that the eigen decom-position requires perfect information of the clutter covariancematrix Rc while our method requires no data. In this example,we assume the perfect Rc is known. In practice, Rc has tobe estimated from the received signals and it might not beaccurate if the number of samples is not large enough. Notethat, unlike the eigen decomposition method, the proposedmethod based on PSWF does not require the knowledge ofRc.

V. NEW STAP METHOD FOR MIMO RADAR

In this section, we introduce a new STAP method for MIMOradar which uses the clutter subspace estimation method de-scribed in the last section. Because the clutter subspace can beobtained by using the parameter information, the performanceand complexity can both be improved. Recall that the optimalMVDR beamformer (11) requires knowledge of the covariancematrix R. In practice, this has to be estimated from data. Forexample, it can be estimated as

R =1|B|

∑k∈B

yky†k, (19)

where yk is the MIMO-STAP signal vector defined in (8) forthe kth range bin, and B is a set which contains the neighborrange bin cells of the range bin of interest. However, somenearest cells around the range bin of interest are excludedfrom B in order to avoid including the target signals [32].There are two advantages of using the target-free covariancematrix R in (11). First, it is more robust to steering vectormismatch. If there is mismatch in the steering vector s(fs, fD)in (9), the target signal is no longer protected by the constraint.Therefore the target signal is suppressed as interference. Thiseffect is called self-nulling and it can be prevented by usinga target-free covariance matrix. More discussion about self-nulling and robust beamforming can be found in [21], [22] andthe references therein. Second, using the target-free covariancematrix, the beamformer in (11) converges faster than thebeamformer using the total covariance matrix. The famousrapid convergence theorem proposed by Reed et al. [25] statesthat a SINR loss of 3 dB can be obtained by using the numberof target-free snapshots equal to twice the size of the covari-ance matrix. Note that the imprecise physical model whichcauses steering vector mismatch does not just create the self-nulling problem. It also affects the clutter subspace. Thereforeit affects the accuracy of the clutter subspace estimation inSection IV-B.

A. The proposed method

The target-free covariance matrix can be expressed as R =RJ + Rc + σ2I, where RJ is the covariance matrix of thejammer signals, Rc is the covariance matrix of the cluttersignals, and σ2 is the variance of the white noise. By (18),there exists a rc×rc matrix Ac so that Rc ≈ UcAcU†

c. Thusthe covariance matrix can be approximated by

R ≈ RJ + σ2I︸ ︷︷ ︸call this Rv

+UcAcU†c. (20)

We assume the jammer signals y(J)n,m,l in (6) are statistically

independent in different pulses and different orthogonal wave-form components [32]. Therefore they satisfy

E[y(J)n,m,l · y(J)†

n′,m′,l′ ] ={rJ,n,n′ , m = m′, l = l′

0, otherwise,

for n, n′ = 0, 1, · · · , N , m,m′ = 0, 1, · · · ,M , and l, l′ =0, 1, · · · , L. Using this fact, the jammer-plus-noise covariancematrix Rv defined in (20) can be expressed as

Rv = diag(Rvs,Rvs, · · · ,Rvs), (21)

where Rvs is an N × N matrix with elements [Rvs]n,n′ =rJ,n,n′ +σ2 for n, n′ = 0, 1, · · · , N . Therefore the covariancematrix R in (20) consists of a low-rank clutter covariancematrix and a block-diagonal jammer-pulse-noise. By using thematrix inversion lemma [37], one can obtain

R−1 ≈ R−1v −R−1

v Uc(A−1c + U†

cR−1v Uc)−1U†

cR−1v . (22)

The inverse of the block-diagonal matrix R−1v is simply

R−1v = diag(R−1

vs ,R−1vs , · · · ,R−1

vs ) and the multiplication ofthe block-diagonal matrix with another matrix is simple.

B. Complexity of the new method

The complexity of directly inverting the NML × NMLcovariance matrix R is O(N3M3L3). Taking advantage ofthe block-diagonal matrix and the low rank matrix, in (22),the complexity for computing R−1

v is only O(N3) and thecomplexity for computing A−1

c and (Ac + U†cR

−1v Uc)−1 is

only O(r3c ), where rc is defined in (17). The overall complex-ity for computing (22) is thus reduced from O(N3M3L3) toO(rcN2M2L2). This is the complexity of the multiplicationof an (NML× rc) matrix by a (rc ×NML) matrix.

C. Estimation of the covariance matrices

In (22), the matrix Uc can be obtained by the nonuniformsampling of the PSWF as described in the last section. Thejammer-pulse-noise covariance matrix Rv and the matrix Ac

both require further estimation from the received signals.Because of the block-diagonal structure, one can estimate thecovariance matrix Rv by estimating its submatrix Rvs definedin (21). The matrix Rvs can be estimated when there areno clutter and target signals. For this, the radar transmitteroperates in passive mode so that the receiver can collect thesignals with only jammer signals and white noise [33]. Thesubmatrix Rvs can be estimated as

Rvs =1Kv

Kv−1∑k=0

rkr†k, (23)

where rk is an N × 1 vector which represents the target-freeand clutter-free signals received by N receiving antennas. By(20), one can express Ac as

Ac = (U†cUc)−1U†

c(R−Rv)Uc(U†cUc)−1.

Therefore, one can estimate Ac by using

Ac =1K

K−1∑k=0

xkx†k − (U†

cUc)−1U†cRvUc(U†

cUc)−1, (24)

9

where xk = (U†cUc)−1U†

cyk and yk is the NML×1 MIMO-STAP signal vector defined in (8). Substituting (23), (24) and(22) into the MIMO-STAP beamformer in (11), we obtain

w ∝(R−1

v − R−1v Uc(A−1

c + U†cR

−1v Uc)−1U†

cR−1v )s(fs, fd)

(25)D. Zero-forcing method

Instead of estimating Ac and computing the MVDR by(25), one can directly “null out” the entire clutter subspaceas described next. Assume that the clutter-to-noise ratio isvery large and therefore all of the eigenvalues of Ac approachinfinity. We obtain A−1

c ≈ 0. Substituting it into (25), one canobtain the MIMO-STAP beamformer as

w ∝ (R−1v − R−1

v Uc(U†cR

−1v Uc)−1U†

cR−1v )s(fs, fd) (26)

Thus we obtain a “zero-forcing” beamformer which nulls outthe entire clutter subspace. The advantage of this zero-forcingmethod is that it is no longer necessary to estimate Ac. Inthis method, we only need to estimate Rvs. The method isindependent of the range bin. The matrix R−1 computed bythis method can be used for all range bins. Because there arelots of extra dimensions in MIMO radars, dropping the entireclutter subspace will reduce only a small portion of the totaldimension. Therefore it will not affect the SINR performancesignificantly, as we shall demonstrate. Thus this method canbe very effective in MIMO radars.

E. Comparison with other methods

In the sample matrix inversion (SMI) method [32], thecovariance matrix is estimated to be the quantity R in (19) andR−1 is directly used in (11) to obtain the MVDR beamformer.However, some important information about the covariancematrix is unused in the SMI method. This information includesthe parameters γ and β, the structure of the clutter covariancematrix, and the block diagonal structure of the jammer covari-ance matrix.

Our method in (25) utilizes this information. We firstestimate the clutter subspace by using parameters γ and βin (18). Because the jammer matrix is block diagonal and theclutter matrix has low rank with known subspace, by usingthe matrix inversion lemma, we could break the inversion ofa large matrix R into the inversions of some smaller matri-ces. Therefore the computational complexity was significantlyreduced. Moreover, by using the structure, fewer parametersneed to be estimated. In our method, only the rc × rc matrixA and the N × N matrix Rvs need to be estimated ratherthan the the NML × NML matrix R in the SMI method.Therefore our method also converges much faster.

In subspace methods [27]-[33], the clutter and the jammersubspace are both estimated simultaneously using the STAPsignals rather than from problem geometry. Therefore theparameters γ and β and the block diagonal structure of thejammer covariance matrix are not fully utilized. In [26], thetarget-free and clutter-free covariance matrix are also estimatedusing (23). The jammer and clutter are filtered out in twoseparate stages. Therefore the block diagonal property of the

jammer covariance matrix has been used in [26]. However,the clutter subspace structure has not been fully utilized inthis method.

VI. NUMERICAL EXAMPLES

In this section, we compare the SINR performance of ourmethods and other existing methods. In the example, theparameters are M = 5, N = 10, L = 16, β = 1.5 andγ = 10. The altitude is 9km and the range of interest is12.728km. For this altitude and range, the clutter is generatedby using the model in [15]. The clutter to noise ratio (CNR) is40 dB. There are two jammers at 20◦ and −30◦ degree. Thejammer to noise ratio (JNR) for each jammer equals 50 dB.The SINR is normalized so that the maximum SINR equals0 dB. The jammers are modelled as point sources which emitindependent white Gaussian signals. The clutter is modelledusing discrete points as described in (2). The clutter points areequally spaced on the range bin and the RCS for each clutter ismodelled as identical independent Gaussian random variables.In general, the variance of ρi will vary along the ground, aswe move within one range bin. However, for simplicity weassume this variance is fixed. The number of clutter pointsNc is ten thousand. The clutter points for different range binsare also independent. The following methods are compared:

1) Sample matrix inversion (SMI) method [32]. Thismethod estimates the covariance matrix R using (19)and directly substitutes it into (11).

2) Loaded sample matrix inversion (LSMI) method [24],[23]. Before substituting R into (11), a diagonal loadingR← R+ δI is performed. In this example, δ is chosenas ten times the white noise level.

3) Principal component (PC) method. [32]. This methoduses a KLT filterbank to extract the jammer-plus-cluttersubspace. Then the space-time beamforming can beperformed in this subspace.

4) Separate jammer and clutter cancellation method [26](abbreviated as SJCC below). This method also utilizesthe jammer-plus-noise covariance matrix Rvs which canbe estimated as in (23). The covariance matrix can beused to filter out the jammer and form a spatial beam.Then the clutter can be further filtered out by space-timefiltering [26]. In this example, a diagonal loading is usedfor the space-time filtering with a loading factor whichequals ten times the white noise level.

5) The new zero-forcing (ZF) method. This method directlynulls out the clutter subspace as described in (26).

6) The new minimum variance method. This method esti-mates Rvs and Ac and uses (25). In this example, adiagonal loading is used for Ac with a loading factorwhich equals ten times the white noise level.

7) MVDR with perfectly known R. This method is unreal-izable because the perfect R is always unavailable. It isshown in the figure because it serves as an upper boundon the SINR performance.

Fig. 7 shows the comparison of the SINR for fs = 0 as afunction of the Doppler frequencies. The SINR is defined as

SINR � |w†s(fs, fD)|2w†Rw

,

10

where R is the target-free covariance matrix. To compare these

−0.5 0 0.5−16

−14

−12

−10

−8

−6

−4

−2

0

Normalized Doppler frequency

SIN

R (

dB)

SMI, K=2000LSMI, K=300PC, K=300SJCC, K

v=20, K

c=300

Our method, Kv=20, K

c=300

Our method ZF, Kv=20

MVDR, perfect Ry

Fig. 7. The SINR performance of different STAP methods at lookingdirection zero as a function of the Doppler frequency.

methods, we fix the number of samples K and the numberof jammer-plus-noise samples Kv . In all of the methodsexcept the SMI method, 300 samples and 20 jammer-plus-noise samples are used. We use 2000 samples instead of 300samples in the SMI method because the estimated covariancematrix in (19) with 300 samples is not full-rank and thereforecan not be inverted. The spatial beampatterns and space-timebeampatterns for the target at fs = 0 and fD = 0.25 for fourof these methods are shown in Fig. 8 and Fig. 9 respectively.The spatial beampattern is defined as

ML−1∑k=0

|w†(1:N)+kMLs(fs)|2,

where s(fs) is the spatial steering vector(1 ej2πfs · · · ej2πfs(N−1)

)T,

and w(1:N)+kML represents N successive elements of wstarting from kML+1. The space-time beampattern is definedas

|w†s(fs, fD)|,where s(fs, fD) is the space-time steering vector defined in(10). The spatial beampattern represents the jammer and noiserejection and the space-time beampattern represents the clutterrejection. In Fig. 8, one can see the jammer notches at thecorresponding jammer arrival angles −30◦ and 20◦. In Fig. 9,one can also observe the clutter notch in the beampatterns. InFig. 7, lacking use of the covariance matrix structure, the SMImethod requires a lot of samples to obtain good performance.It uses 2000 samples but the proposed minimum variancemethod which has a comparable performance uses only 300samples. The PC method and LSMI method utilize the factthat the jammer-plus-clutter covariance matrix has low rank.Therefore they require fewer samples than the SMI method.The performance of these two are about the same. The SJCCmethod further utilizes the fact that the jammer covariance

−0.5 0 0.5−90

−80

−70

−60

−50

−40

−30

Normalized spatial frequency fs

Spa

tial b

eam

patte

rn (

dB)

Our method ZF, Kv = 20

PC, K = 300SJCC, K

v=20, K=300

MVDR, perfect R

Fig. 8. Spatial beampatterns for four STAP methods.

matrix is block diagonal and estimates the jammer-plus-noisecovariance matrix. Therefore the SINR performance is slightlybetter than the LSMI and PC methods. Our methods notonly utilize the low rank property and the block diagonalproperty but also the geometry of the problem. Thereforeour methods have better SINR performance than the SJCCmethod. The proposed zero-forcing (ZF) method has aboutthe same performance as the minimum variance method. Itconverges to a satisfactory SINR with very few clutter-freesamples. According to (16), the clutter rank in this exampleis approximately

�N + γ(M − 1) + β(L− 1)� = 73.

Thanks to the MIMO radar, the dimension of the space-timesteering vector is MNL = 800. The clutter rank is just a smallportion of the total dimension. This is the reason why the ZFmethod, which directly nulls out the entire clutter space, worksso well.

VII. CONCLUSIONS

In this paper, we first studied the clutter subspace and itsrank in MIMO radars using the geometry of the system. Wederived an extension of Brennan’s rule for estimating the di-mension of the clutter subspace in MIMO Radar systems. Thisrule is given in (16). An algorithm for computing the cluttersubspace using nonuniform sampled PSWF was described.Then we proposed a space-time adaptive processing methodin MIMO radars. This method utilizes the knowledge of thegeometry of the problem, the structure of the clutter space,and the block diagonal structure of the jammer covariancematrix. Using the fact that the jammer matrix is block diagonaland the clutter matrix has low rank with known subspace,we showed how to break the inversion of a large matrixR into the inversions of smaller matrices using the matrixinversion lemma. Therefore the new method has much lowercomputational complexity. Moreover, we can directly null outthe entire clutter space for large clutter. In our ZF method,only the N × N jammer-plus-noise matrix Rvs needs to be

11

Normalized spatial frequency fs

Nor

mal

ized

Dop

pler

freq

uenc

y f D

(a)

−0.5 0 0.5−0.5

0

0.5

Normalized spatial frequency fs

Nor

mal

ized

Dop

pler

freq

uenc

y f D

(b)

−0.5 0 0.5−0.5

0

0.5

Normalized spatial frequency fs

Nor

mal

ized

Dop

pler

freq

uenc

y f D

(c)

−0.5 0 0.5−0.5

0

0.5

Normalized spatial frequency fs

Nor

mal

ized

Dop

pler

freq

uenc

y f D

(d)

−0.5 0 0.5−0.5

0

0.5

−70

−60

−50

−40

−30

−20

−10

Fig. 9. Space-time beampatterns for four methods: (a) The proposed zero-forcing method, (b) Principal component (PC) method [32], (c) Separate jammerand clutter cancellation method (SJCC) [26] and (d) Sample matrix inversion (SMI) method [32].

estimated instead of the the NML×NML matrix R in theSMI method, where N is the number of receiving antennas, Mis the number of transmitting antennas, and L is the numberof pulses in a coherent processing interval. Therefore, for agiven number of data samples, the new method has betterperformance. In Section VI, we provided an example wherethe number of training samples was reduced by a factor of100 with no appreciable loss in performance compared to theSMI method.

In practice, the clutter subspace might change because ofeffects such as the internal clutter motion (ICM), velocity mis-alignment, array manifold mismatch, and channel mismatch[32]. In this paper, we considered an “ideal model”, whichdoes not take these effects into account. When this model isnot valid, the performance of the algorithm will degrade. Oneway to overcome this might be to estimate the clutter subspaceby using a combination of both the assumed geometry and thereceived data. Another way might be to develop a more robustalgorithm against the clutter subspace mismatch. These ideaswill be explored in the future.

REFERENCES

[1] D. J. Rabideau and P. Parker, “Ubiquitous MIMO Multifunction DigitalArray Radar,” Proc. 37th IEEE Asilomar Conf. on Signals, Systems, andComputers, vol. 1, pp. 1057-1064, Nov. 2003.

[2] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output (MIMO)radar and imaging: degrees of freedom and resolution,” Proc. 37th IEEEAsilomar Conf. on Signals, Systems, and Computers, vol. 1, pp. 54-59,Nov. 2003.

[3] E. Fishler, A. Haimovich, R. S. Blum, D. Chizhik, L. J. Cimini, and R.A. Valenzuela, “MIMO Radar: An Idea Whose Time Has Come,” Proc.IEEE Radar Conference, pp. 71-78, April 2004.

[4] E. Fishler, A. Haimovich, R. S. Blum, L. J. Cimini, D. Chizhik, and R.A. Valenzuela, “Performance of MIMO Radar Systems: Advantages ofAngular Diversity,” Proc. 38th IEEE Asilomar Conf. on Signals, Systems,and Computers, vol. 1, pp. 305-309, Nov. 2004.

[5] E. Fishler, A. Haimovich, R. S. Blum, L. J. Cimini, D. Chizhik, andR. A. Valenzuela, “Spactial Diversity in Radars-Models and DetectionPerformance,” IEEE Trans. on Signal Processing, vol. 54, Issue 3, pp.823-837, March 2006.

[6] F. C. Robey, S. Coutts, D. Weikle, J. C. McHarg, and K. Cuomo, “MIMORadar Theory and Experimental Results,” Proc. 38th IEEE Asilomar Conf.on Signals, Systems, and Computers, vol. 1, pp. 300-304, Nov. 2004.

[7] K. W. Forsythe, D. W. Bliss, and G. S. Fawcett, “Multiple-Input Multiple-Output (MIMO) Radar Performance Issues,” Proc. 38th IEEE AsilomarConf. on Signals, Systems, and Computers, pp. 310-315, Nov. 2004.

[8] H. A. Khan, W. Q. Malik, D. J. Edwards, and C. J. Stevens, “UltraWideband Multiple-Input Multiple-Output Radar,” IEEE InternationalRadar Conference, pp. 900-904, May 2005.

[9] H. A. Khan and D. J. Edwards, “Doppler problems in orthogonal MIMOradars,” IEEE International Radar Conference, pp. 24-27, April 2006.

[10] V. F. Mecca, D. Ramakrishnan, and J. L. Krolik, “MIMO RadarSpace-Time Adaptive Processing for Multipath Clutter Mitigation” IEEEWorkshop on Sensor Array and Multichannel Signal Processing, pp. 249-253, July 2006.

[11] D. R. Fuhrmann and G. S. Antonio, “Transmit Beamforming for

12

MIMO Radar Systems Using Partial Signal Correlation,” Proc. 38th IEEEAsilomar Conf. on Signals, Systems, and Computers, pp. 295-299, Nov.2004.

[12] G. S. Antonio and D. R. Fuhrmann, “Beampattern Synthesis for Wide-band MIMO Radar Systems,” Proc. 1st. IEEE International Workshop onComputational Advances in Multi-Sensor Adaptive Processing, pp. 105-108, Dec. 2005.

[13] K. W. Forsythe and D. W. Bliss, “Waveform Correlation and Optimiza-tion Issues for MIMO Radar,” Proc. 39th IEEE Asilomar Conf. on Signals,Systems, and Computers, pp. 1306-1310, Nov. 2005.

[14] Q. Zhang and W. B. Mikhael, “Estimation of the Clutter Rank in theCase of Subarraying for Space-Time Adaptive Processing,” ElectronicsLetters, vol. 33, Issue 5, pp. 419-420, 27 Feb. 1997.

[15] N. A. Goodman and J.M. Stiles, “On Clutter Rank Observed by ArbitraryArrays,”IEEE Trans. on Signal Processing, vol. 55, Issue 1, pp. 178-186,Jan. 2007.

[16] J. Ward, “Space-Time Adaptive Processing for Airborne Radar,” Tech-nical Report 1015, Lincoln Laboratory, Dec. 1994.

[17] L. E. Brennan and F. M. Staudaher, “Subclutter Visibility Demonstra-tion,” Technical Report RL-TR-92-21, Adaptive Sensors Inc., 1992.

[18] R. Klemm, “Adaptive clutter supression for airborne phased arrayradars”, IEE Proc. F, 130, (1), pp. 125-132, 1983.

[19] D. Slepian, and H. O. Pollak, “Prolate Spheroidal Wave Functions,Fourier Analysis and Uncertainty-III: the dimension of the space ofessentially time-and-band-limited signals,” Bell Syst. Tech. J., pp. 1295-1336, July 1962.

[20] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,”Proc. IEEE, vol. 57, no. 8, pp. 1408-1418, Aug. 1969.

[21] C. Y. Chen and P. P. Vaidyanathan, “Quadratically Constrained Beam-forming Robust Against Direction-of-Arrival Mismatch,” accepted toIEEE Trans. on Signal Processing.

[22] R. G. Lorenz and S. P. Boyd, “Robust minimum variance beamforming,”IEEE Trans. on Signal Processing, vol. 53, Issue 5, pp. 1684-1696, May2005.

[23] Y. I. Abramovich, “Controlled method for adaptive optimization of filtersusing the criteriion of maximum SNR,” Radio Engineering and ElectronicPhysics, vol. 26, pp. 87-95, March 1981.

[24] B. D. Carlson, “Covariance matrix estimation errors and diagonalloading in adaptive arrays,” IEEE Trans. on Aerospace and ElectronicSystems, vol. 24, Issue 4, pp. 397-401, July 1988.

[25] J. S. Reed, J. D. Mallett, and L. E. Brennan, “Rapid Convergence Ratein Adaptive Arrays,” IEEE Trans. on Aerospace and Electronic Systems,vol. AES-10, Issue 6, pp. 853-863, Nov. 1974.

[26] R. Klemm, “Adaptive air- and spaceborne MTI under jamming condi-tions,” IEEE National Radar Conference, pp. 167-172, April 1993.

[27] J. R. Guerci, J. S. Goldstein, and I. S. Reed, “Optimal and AdaptiveReduced-Rank STAP,” Special Section on Space-Time Adaptive Process-ing, IEEE Trans. on Aerospace and Electronic Systems, vol. 36, Issue 2,pp. 647-663, April 2000.

[28] A. M. Haimovich and M. Berin, “Eigenanalysis-based space-timeadaptive radar: performance analysis,” IEEE Trans. on Aerospace andElectronic Systems, vol. 33, Issue 4, pp. 1170-1179, Oct. 1997.

[29] J. S. Goldstein, I. Si. Reed, and L. L. Scharf, “A Multistage Representa-tion of the Wiener Filter Based on Orthogonal Projections,” IEEE Trans.on Information Theory, vol. 44, Issue 7, pp. 2943-2959, Nov. 1998.

[30] B. Friedlander, “A Subspace Method for Space Time Adaptive Process-ing,” IEEE Trans. on Signal Processing, vol. 53, Issue 1, pp. 74-82, Jan.2005.

[31] X. Wen, A. Wang, L. Li, and C. Han, “Direct Data Domain Approachto Space-Time Adaptive Signal Processing,” Proc. of the ICARCV, vol.3, pp. 2070-2075, Dec. 2004.

[32] J. R. Guerci, Space-time adaptive processing, Artech House, 2003.[33] R. Klemm, Principles of Space-Time Adaptive Processing, IEE, 2002.[34] M. A. Richards, Fundamentals of Radar Signal Processing, McGraw-

Hill, 2005.[35] H. L. Van Trees, Optimum Array Processing: Part IV of Detection

Estimation and Modulation Theory, New York: Wiley Interscience, 2002.[36] J. P. Keener, Principles of Applied Mathematics, Addison-Welsley, 1988.[37] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge 1985.