Clutter Rank Of MIMO Radar With A Speical Class of Waveforms

Guohua WANG, Yilong LU School of Electrical and Electronic Engineering

Nanyang Technological University Singapore

{Wang0330, eylu}@ntu.edu.sg

Abstract—In this paper, the rank of clutter covariance matrix (CCM) is studied for space-time adaptive processing (STAP) implemented on the multiple-input multiple-output (MIMO) radar with waveform diversity. It is found that the clutter rank is dependent on the rank and structure of the covariance matrix of the transmitting waveforms, and that clutter rank of orthogonal waveforms acts as the upper bound for all kinds of waveforms. More specifically, three simple rules are proposed to estimate the clutter rank for one special class of waveforms. This class of waveforms, though we call it special, indeed can include coherent waveforms, orthogonal waveforms, as well as part of partially correlated waveforms. In this way, the method of estimating the clutter rank for both coherent and orthogonal waveforms are herein unified. The application of these rules only requires the computation of the waveform covariance matrix (WCM) and a polynomial based on the WCM, thus it is convenient and computationally efficient. The proposed rules are well verified by simulation results and also general enough to be suitable for both conventional space time radar systems and the MIMO radars with orthogonal waveforms.

I. INTRODUCTION MIMO radar has recently drawn considerable attention

[1]–[4] and the related study has been extended to space-time adaptive processing (STAP) systems [1], [5], [6]. Through MIMO configuration, the STAP system will benefit from more degrees of freedom (DOFs) to perform effective cancellation of the clutter due to the waveform diversity in MIMO radar. However, the severity of clutter will also increase due to the same reason, which is determined by the rank of the clutter covariance matrix (CCM). The knowledge of clutter rank is important for reducing the computation complexity while improving the efficiency of signal processing. Therefore, estimation of the clutter rank is crucial for predicting the performance of MIMO STAP systems.

So far, the clutter rank of the CCM has been intensively studied for conventional adaptive radars with uniform linear arrays aligned along the radar platform’s velocity and pulse repetition frequency matched to the antenna element spacing. Several rules [7]–[8] have been developed for these STAP systems under ideal conditions, where effects such as the

internal clutter motion, velocity misalignment, array manifold mismatch, and channel mismatch are not taken into account. Specifically, paper [7] introduced the well known Brennan’s rule and proved a special case of it. Paper [8] extended the Brennan’s rule to estimate the clutter rank in the case of subarraying. Paper [9] generalized these rules for analyzing the rank and the eigenspectrum of the CCM observed by space-time radar systems with arbitrarily configured arrays and varying look geometry. Based on abovementioned rules, paper [6] has derived the rule for estimating the clutter rank of MIMO radar with orthogonal waveforms. It should be mentioned that all above rules are considering conventional transmitting waveforms, i.e., coherent or orthogonal waveforms. However, as illustrated in [10]–[12], MIMO radar performance could be optimized via waveform optimization. And the optimal waveforms may be partially correlated other than orthogonal or coherent with each other. In these cases, the conventional rules for clutter rank estimation will fail. Hence, it is worthy of study what the severity of clutter subspace would be when using different optimal waveforms, in other words, what the relationship would be between the clutter rank and the transmitting waveforms.

In this paper, we extend the current rule to estimate the rank of the CCM observed by MIMO radar that is operating with arbitrary optimal waveforms under ideal condition. Firstly, the model of echoes due to clutter is constructed based on both the waveforms and the configuration of the MIMO STAP system. Then the clutter rank is analyzed. Specifically, we here focus on estimation of the clutter rank due to a special class of waveforms by modifying the method in [8] and three estimation rules are proposed. These rules indicate that the clutter rank of the MIMO adaptive radar with waveform diversity depends on the structure of waveforms, or more precisely, the rank and structure of the covariance matrix of the transmitting waveforms. Based on this relationship the waveform performance can be further examined. The proposed rules can be viewed as an extension of Brennan’s rule. Thus they are suitable for both conventional STAP systems and the MIMO radar systems

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with orthogonal waveforms. The simulated results are provided to verify the proposed rule well.

The rest of this paper is organized as follows. Section II presents the geometry configuration and signal model. Then Section III proposes the rank estimation rules. After that Section IV provides simulation results to confirm the rules. Finally, Section V concludes this paper.

II. GEOMETRY CONFIGURATION AND SIGNAL MODEL

Fig.1 MIMO-STAP geometry

The assumed MIMO radar system geometry is shown in

Fig. 1. In this paper we use the same conventions as in [7]. The radar platform travels at velocity v in the positive direction of x axis. The altitude of the array phase center is h. The surface of the earth is assumed to be flat for the interested area of observation. There are N isotropic transmitting elements uniformly spaced by a distance of dTx, and M receiving elements uniformly spaced by a distance of dRx. As illustrated in Fig.1, the transmitting antennas are collocated with the receiving antennas, meaning that this system is with monostatic MIMO radar configuration. Thus, the transmitting array and the receiving array have the same azimuth angle and elevation angle. The elevation angle φ and the azimuth angle θ are illustrated in Fig. 1. In each pulse repetition interval (PRI), the system will transmit a waveform set

[ ]1 2, ,..., T N NsN C ×= ∈S s s s (1)

where sn Є CNs, n=1,2,..,N, is the waveform emitted by the n-th transmitter; the superscript T means matrix transpose. In one coherent processing interval (CPI), a train of L repeating waveform sets will be emitted by all transmitting elements. The pulse repetition interval is Tr and the pulse duration is TP. We assume that the transmitting waveforms meet the narrowband assumption. Thus the Doppler phase shift within the pulse can be neglected. For the l-th pulse, the echoes on the m-th receiver according to all clutter patches at the range ring of interest are [5]:

( ) ( ) ( )2

2 ( 1) ( 1) 2 1

10

s tN

j f n m j f lm n

n

e e dπ

π α π

θ

ξ θ θ− + − −

==

= ∑∫y s (2)

where ym Є CNs, α=dTx/dRx, ft=sin(θ)cos(φ)(2vTr)/ λo, fs=dRx sin(θ)cos(φ)/ λo and ξ(θ) is the reflect coefficient of clutter at θ. We assume that dRx and λo have been properly chosen so

that –0.5≤fs≤0.5 can always hold in order to avoid aliasing in spatial frequency. We also define the transmitting steering vector and receiving steering vector in (3) and (4), respectively.

( ) ( )2 12[1,e , ,e ]ss j f Nj f TTx

π απ αθ −=V 1NC ×∈ , (3)

( ) ( )2 121,e , ,e ssTj f Mj f

Rxππθ −⎡ ⎤= ⎣ ⎦V 1MC ×∈ , (4)

Define the Doppler vector as ( )2 12 1( )= 1, , , tt

Tj L fj f Ltf e e Cππ − ×⎡ ⎤ ∈⎣ ⎦U . (5)

Then for the l-th pulse, the clutter echoes on the receiving elements should be in the form

( ) ( ) ( ) ( )2

2 1

0

tj l f Tl Rx Txe d

ππ

θ

ξ θ θ θ θ−

=

= ∫Y V V S . (6)

To get the sufficient statistics for STAP signal processing, we can employ SH, where the superscript H means the conjugate transpose, as the filter bank in the receivers. Thus the clutter echoes are compressed by SH. Then we can stack the filtered clutter data of the l-th pulse in a vector form:

( )

( ) ( ) ( ) ( )2

2 1

0

t

Hl l

j l f TRx Tx S

stack

stack e dπ

π

θ

ξ θ θ θ θ−

=

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∫

X YS

V V R, (7)

where RS=SSH. If the iso-range ring is divided in the cross-range

dimension into NC (NC >>NML) clutter patches, then the discrete form of the clutter data from the l-th pulse is

( ) ( ) ( ) ( ),2 1

1

Ct i

Nj l f T

l i Rx i Tx i Si

stack e πξ θ θ θ−

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑X V V R (8)

with ft,i=sin(θi)cos(φ)(2vTr)/ λo, and fs,i=dRx sin(θi)cos(φ)/ λo. Generally, ξ(θi) can be modeled as a zero-mean independent complex Gaussian random variable with the variance of σ2

i.Now we can define the total clutter snapshot as 1 2=[ ; ; ;X ]C Lχ X X . (9)

For the clutter patch at azimuth of θi, the snapshot should be ( ) ( )( )

( )( ) ( )( ),

,

= ( ) stack

= ( )

TC i i t,i Rx i Tx i S

Ti t,i S Rx i Tx i

i C i

f

f

ξ θ θ

ξ θ θ

ξ

⊗

⊗ ⊗

χ U V V R

U R V V

V

, (10)

where ⊗ means Kronecker product and VC,iЄ CNML is the space-time steering vector. Lastly we define

,1 ,2 ,, , ,CC C C C N⎡ ⎤= ⎣ ⎦V V V V , (11–1)

( )2 2 21 2=diag , , ,

CNσ σ σΞ . (11–2) The clutter covariance matrix could be written as

E H HC C C C C⎡ ⎤= =⎣ ⎦R χ χ V ΞV , (13)

where E[▪] is the expected value operator.

III. CLUTTER RANK ESTIMATION In this section we will estimate the rank of the CCM.

Because Ξ is positive-definite matrix, the rank of RC is equal to the rank of VC [7]. Therefore, we can just study the rank of VC. Furthermore, because each column of VC is a function of

2009 International WD&D Conference 109 9781-4244-2971-4/09/$25.00©2009 IEEE

RS, the rank of VC would depend on the rank of RS, let us say, r (r≤N). For simplification, we set Zi=exp[j2πdRxsin(θi)cos(φ)/ λo], β=2vTr/dRx, and (RS)m,n=Rm,n. Then we can decompose VC,i for each pair of m and l as

( ) ( )1 1, , ,

m l TC i,m l i S Tx iz β− + −=V R V , (14–1)

where VTx,i=VTx(θi). We rewrite VC as

( )

, ,2 ,

, 2 ,2 2 , 2

, , ,2 , , ,

C

C

C

C 1,1,1 C ,1,1 C N ,1,1

C 1, ,1 C , ,1 C N , ,1C

C 1,M L C ,M L C N ,M LT orthogonalS M L C

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

= ⊗

V V VV V VV

V V V

R I V

, (14–2)

where VC,i,m,l Є CN, and orthogonalCV is the counterpart matrix

for orthogonal waveforms. From (14-2), it can be said that CV comes from conducting nonlinear row operations on

orthogonalCV , which will change the rank of orthogonal

CV . As rank(AB)≤min(rank(A), rank(B)), the rank of CCM from orthogonal waveform behaves as an upper bound for CCM from general waveforms. In the following sections, we will show how to precisely estimate the rank of CCM from a special class of RS. The property of this special class of RS will be also specified.

Because RS is normal and positive semidefinite it can be decomposed as RS=UΛUH, where U is unitary matrix and Λ=diga(λ1, λ2, …, λr, 0,…, 0) is diagonal matrix with only r positive diagonal elements. We also denote U= [u1, u2,…, uN]. Thus we can get RSU=UΛ, which indicates that we can get (15) by row operations:

( )( )

( )

1

2

i

iT T T

S Tx Txr i

f zf z

f z

⎡ ⎤⎢ ⎥⎢ ⎥

= Λ = ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

U R V U V

0

. (15)

where, fq(zi)= λq(uq)TVTx, q=1,…,r, is the polynomials function of argument zi. If we can express fq (zi) as fq (zi)= aq zi^(bq), where aq and bq are non-zero real values, the rank estimation problem will be perfectly resolved as discussed in the following. Then we can define:

( ) ( )

( )( )

( )( ) ( )

1

21 1

2 21 1 1 1

, ,

0

r

bi i

bi i

m l m lC i,m l i i b

r i r i

f z a zf z a z

z zf z a z

β β− + − − + −

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

V

0

(16–1)

Then the same row operation can be conducted on Vc and leads to a new clutter matrix

, ,2 ,

, 2 ,2 2 , 2

, , ,2 , , ,

C

C

C

C 1,1,1 C ,1,1 C N ,1,1

C 1, ,1 C , ,1 C N , ,1C

C 1,M L C ,M L C N ,M L

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

V V VV V VV

V V V

(16–2)

Because there are some all-zero rows, we can rearrange the rows of CV and get

, , ,

, 2 , ,

, , ,0 0 0

0 0 0

C

C

C

C 1,1 C 2,1 C N ,1

C 1, C 2,2 C N ,2

C C 1,r C 2,r C N ,r

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

V V VV V V

V V V V (16-3)

with ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

00

11

1 1 1 1,

1 1 1 1

q

q

q

q

biib

i i

m l b m lC i,q q i qi i

m l b m li i

zzz z

f z az z

z zβ β

+

+

− + − + − + −

− + − + − + −

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

V ,

where ,C i,qV is get by putting together matrix elements with same b q for all pairs of m and l(see (16-1)). Without loss of generality, we assume b1 ≤… ≤ bq ≤… ≤ br for convenience. Because the above row operation will not change the rank of matrix, thus the rank of new clutter matrix CV is the same as that of original clutter matrix Vc. Then we can just estimate the clutter rank by investigating CV in stead of original clutter matrix.

As pointed out in [6], the column vector ,C iV in CV can be viewed as a nonuniform sampled version of the truncated sinusoidal function:

( ),2

1,

, [ , ],

0 , otherwise

s i qr

j f xq q q q

qs i n

a e x b b XV f x

π

=

⎧ ∈ +⎪= ⎨⎪⎩

∑ (17)

where X=M-1+β(L-1). We also define the bandwidth of sinusoidal function as W=0.5 based on –0.5≤fs,i≤0.5. Because the coefficient aq has no effect on the dimension of the sinusoidal function, the dimension of function in (17) will be the same as that in (18):

( ) { },21 1

,[ , ], [ , ],,

otherwise0,

s ij f xr r

s ix b b X b b XeV f x

π⎧ ∈ + +⎪= ⎨⎪⎩

. (18)

From (18), we can see that the time domain of this sinusoidal function may consist of q sub-domains. Thus, we can no more apply the Lemma in [8] directly to this function without considering the function’s time-frequency property. Before proceeding, we should assume that β≤M to make sure each sub-domain is suitable to this Lemma and has k’=X+1 eigenvalues, which is not pointed out in [6]. Then, applying the Lemma in [8] to the sinusoidal function with different time-frequency property, the rank of CCM can be approximately given by the following three rules. Rule 1: If bq+X≥ bq+1, q=1,…,r-1, which means that the

succeeding sub-domain will overlap the preceding sub-domain and the total domain is one continuous interval, then the rank of CCM is

( ) ( ) 11C C rr rank L M b bβ= ≈ − + + −⎢ ⎥⎣ ⎦V , (19–1)

where the brackets ⎢ ⎥⎣ ⎦ indicate rounding to the nearest integer.

2009 International WD&D Conference 110 9781-4244-2971-4/09/$25.00©2009 IEEE

Rule 2: If bq+X≤ bq+1, q=1,…,r-1, which means that the succeeding sub-domain will not overlap the preceding sub-domain and the total domain has r independent continuous intervals, then the rank of CCM is

( ) ( )( )1

1r

C Cq

r rank L Mβ=

⎢ ⎥= ≈ − +⎢ ⎥

⎣ ⎦∑V . (19–2)

Rule 3: Otherwise, the rank of CCM should be computed by applying Rule 1 for overlapping sub-domain and Rule 2 for independent sub-domain.

Though these three rules are derived from a special class of Rs, they are applicable to many kind of waveforms. For example, for orthogonal waveforms and coherent waveforms, bq, q=1,…,r is constant over i. Thus by simplifying Rule 1, the rank of the clutter covariance matrix with waveform diversity is

( ) ( ) ( )1 1C Cr rank M L rβ α= = + − + −⎢ ⎥⎣ ⎦V (20)

IV. NUMERICAL STUDIES In this section we verify the correctness of the extended

rules by simulated clutter data. The clutter simulation assumptions are given as those in [7]. The clutter to noise ratio is 50 dB. The basic parameters used in the simulations are listed in the Table 1. We take three kinds of waveforms extensively used in the related literatures for discussion. They are coherent waveforms, orthogonal waveforms and waveforms designed by the method of maximum power criteria for known target locations in [10]. Here we only need to illustrate Rule 2.

TABLE 1． SATP SIMULATION PARAMETERS Parameter Symbol Value

Number of transmitting elements N 10 Number of receiving elements M 10 Number of pulses L 16 Radar platform velocity v 200 m/s Altitude h 10 km Operating frequency fo 450 MHz Space between transmitting elements dRx 0.3333 m Elevation angle φ 0 degree

Fig. 3 shows eigenspectrum of CCM to verify Rule 2 of this paper. Here we set α=10. The waveforms come from the method of maximum power criteria for known target location [10]. The focus region is set to be [-20 20] degree and 3 dB point is -12 and 12 degree. The rank of the waveform covariance matrix is 2. The pattern of this waveform is illustrated in Fig.2. Through matrix operation, we can get UHRS=UΛ=[0,…,0; …; 0,…,0; 0,…,0,1; 3,…3,0]. Then, Δ=bi,2–bi,1=(N-1)α-(N-2)α/2=90-40=50>X, for β=1, 2.4. Thus, Rule 2 is suitable to this case. When β=1, rC=r(M+β(L–1))=50; when β=2.4, rC=r(M+β(L–1))=92. From Fig.2 we can see that Rule 2 well predicted the clutter rank of the simulated data. However, if the rule of synthetic aperture-bandwidth is applied the rank will be heavily overestimated. As pointed above, the rank from orthogonal waveforms will be the upper bound as illustrated by the red solid line in Fig.3.

Furthermore, by comparison of the clutter ranks from different waveforms in the same geometry configuration, we can see that the radar measurements from waveforms with lower rank covariance matrix may have less severe clutter subspace. In other words, there are more dimensions available for moving target detection. From this aspect, the orthogonal waveform may be not the optimal waveform for the STAP radar system in some cases.

-80 -60 -40 -20 0 20 40 60 80-30

-25

-20

-15

-10

-5

0

Angle (degree)

Bea

mp

atte

rn (d

B)

Fig.2. Pattern of correlated waveforms

0 50 100 150 200 250 300-200

-150

-100

-50

0

50

100

150

Eigenvalue Number

Re

lati

ve

Po

wer

(dB

)

Eigen decompositionRule 2 of this paperRule 1 of this paper fororthogonal waveforms

=2.4

=2.4

=1 =1

Fig. 3. Illustration of Rule 2

V. CONCLUSION In this paper we extended the Brannan’s rule for

estimation of the rank of the CCM in MIMO radar with waveform diversity. It is found that the clutter rank is dependent on the rank and structure of the covariance matrix of the transmitting waveforms, and that clutter rank of orthogonal waveforms acts as the upper bound for all kinds of waveforms. More specifically, three simple rules are proposed to estimate the clutter rank for one special class of waveforms. This class of waveforms, though we call it special, indeed can include coherent waveforms, orthogonal waveforms, as well as part of partially correlated waveforms. In this way, the method of estimating the clutter rank for both coherent and orthogonal waveforms are herein unified. Hence, there would be another useful aspect based on the

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relationship between the clutter rank and the waveform for optimal waveform selection. The simulation result verified the proposed rules.

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[3] E. Fisher, A. Haimovich, R. Blum, L. Cimini, , D. Chizhik, R. Valenzuela, “Performance of MIMO radar systems advantages of angular diversity,” in Proc. of the 38th Asilomar Conf. on Signal, Systems and Computers, California, U.S.A, November 2004, pp.305-309.

[4] J. Li, “MIMO radar: diversity means superiority,” in Proc. of the ASAP- 2006, Jun. 2006, pp.305–309.

[5] C. Y. Chen and P. P. Vaidyanathan, “A subspace method for MIMO radar space-time adaptive processing,” in Proc. of the ICASSP-2007, April, 2007, pp.925–928.

[6] C. Y. Chen and P. P. Vaidyanathan, “ Beamforming issues in modern MIMO radars with doppler,” in Proc. of the 40th Asilomar Conf. on Signal, Systems and Computers,, Oct, 2006, pp.41–45.

[7] J. Ward, Space-Time Adaptive Processing for Airborne Radar, MIT Lincoln Laboratory, Lexington, MA, Tech. Rep. 1015, DEC. 1994.

[8] Q. Zhang and W. B. Mikhael, “Estimation of the clutter rank in the case of subarraying for space-time adaptive processing,” Electron. Lett., vol. 33, no. 5, pp. 419-420, Feb. 27, 1997.

[9] N. A. Goodman and J. M. Stiles, “ On clutter rank observed by arbitrary Arrays,” IEEE Trans. on Signal Processing, Vol. 55 , no. 1, pp. 178–186, Jan. 2007.

[10] S. Peter, J. Li, and Y. Xie, “On probing single design for MIMO radar,” in IEEE Trans. on Signal Processing, Vol. 55 , no. 8, pp. 4151–4161, Aug. 2007.

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