Adaptive MIMO Radar

Luzhou Xu 1, Jian Li 1, Petre Stoica 2

Abstract

We investigate several adaptive techniques for a multiple-input multiple-output (MIMO)

radar system (that is, a radar with multiple transmitting and receiving antennas). By

transmitting independent waveforms via different antennas, the echoes due to targets at

different locations are linearly independent of each other, which allows the direct appli-

cation of many adaptive techniques to achieve high resolution and excellent interference

rejection capability. In the absence of array steering vector errors, we discuss the applica-

tion of several existing adaptive algorithms including Capon, APES (amplitude and phase

estimation) and CAPES (combined Capon and APES), and then propose an alternative

estimation procedure, referred to as the combined Capon and approximate maximum like-

lihood (CAML) method. Via several numerical examples, we show that the proposed

CAML method can provide excellent estimation accuracy of both target locations and

target amplitudes. In the presence of array steering vector errors, we apply the robust

Capon beamformer (RCB) and doubly constrained robust Capon beamformer (DCRCB)

approaches to the MIMO radar system to achieve accurate parameter estimation and

superior interference and jamming suppression performance.

Index Terms – Adaptive arrays, MIMO Radar, direction of arrival (DOA), Capon, ampli-

tude and phase estimation (APES), robust Capon beamformer (RCB), doubly constrained

robust Capon beamformer (DCRCB).

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS 3

Submitted in April 2006

1Luzhou Xu and Jian Li are with the Department of Electrical and Computer Engineering, University of Florida,

Gainesville, USA.2Petre Stoica is with the Department of Information Technology, Uppsala University, Uppsala, Sweden.3Please address all correspondences to: Dr. Jian Li, the Department of Electrical and Computer Engineering,

P. O. Box 116130, University of Florida, Gainesville, USA. Phone: (352) 392-2642, Fax: (352) 392-0044, Email:

li@dsp.ufl.edu.

I. Introduction

A MIMO (multiple-input multiple-output) radar uses multiple antennas to simultane-

ously transmit several (possibly linearly independent) waveforms and it also uses multiple

antennas to receive the reflected signals [1] [2] [3] [4] [5] [6] [7] [8]. The statistical MIMO

radar, proposed in [2], [3] and [4], is different from the conventional radar systems in that

the signal “fading” effect due to the target’s scintillations, which can cause severe perfor-

mance degradations, is mitigated via the use of a new radar architecture and the spacing

between the adjacent elements of the array is large to achieve the diversity gain. In [5],

a transmitting beamforming approach, using partial signal correlation, is proposed and it

is shown that a desired transmitting beampattern can be achieved or approximated via

choosing the correlation matrix of the waveforms. In [6], a MIMO radar technology is

suggested to improve the radar resolution. The idea is to transmit N (N > 1) orthogo-

nal coded waveforms by N antennas and to receive the reflected signals by M (M > 1)

antennas. At each receiving antenna output, the signal is matched-filtered using each of

the transmitted waveforms to obtain NM channels, to which the data-adaptive Capon

beamformer [9] is applied. It is proved in [6] that the beampattern of the proposed MIMO

radar is obtained by the multiplication of the transmitting and receiving beampatterns,

which gives high resolution. However, the derivation of the proposed method in [6] and its

theoretical performance analysis are valid only in the single target case. When there are

multiple targets, the problem in [6] turns out to be a coherent-source problem, in which

case Capon and many other adaptive methods fail to work properly. Specifically, the

equation of the signal covariance (see Equation (7) in [6]), on which the Capon algorithm

relies, is no longer valid. Moreover, the distributive law of the Khatri-Rao product used

1

in the performance analysis in [6] (see Equations (9) and (10) in [6]) is no longer valid in

the presence of multiple targets.

We consider herein a new MIMO radar scheme that can deal with multiple targets. In

this paper, we focus on the narrow-band MIMO radar. Similar to some of the aforemen-

tioned MIMO radar approaches, linearly independent waveforms are transmitted simul-

taneously via multiple antennas. Due to the different phase shifts associated with the

different propagation paths from the transmitting antennas to targets, these independent

waveforms are linearly combined at the targets with different phase factors. As a result,

the signal waveforms reflected from different targets are linearly independent of each other,

which allows for the application of Capon and of other adaptive array algorithms. We con-

sider applying two well-known adaptive approaches, i.e., the Capon and APES (Amplitude

and Phase Estimation) [10] algorithms, to the proposed MIMO radar system to estimate

the target locations and the reflected signal amplitudes. We show via numerical examples

that Capon can provide accurate estimates of the target locations, while its amplitude

estimates are biased downward. APES gives accurate amplitude estimates at the cost of

poorer resolution and hence poorer estimation accuracy of target locations. To exploit the

merits of both Capon and APES, we consider a combined method, referred to as CAPES

[11], which first estimates the target locations using the Capon method and then refines

the amplitude estimates at the estimated locations using the APES estimator. To further

improve the amplitude estimation accuracy, we combine Capon and an approximate max-

imum likelihood (AML) method recently introduced in [12]. We refer to the so-obtained

method as CAML. In CAML, the AML estimator, which is devised in [12] based on a

diagonal growth curve (DGC) model, is used to replace APES in CAPES. As shown in

2

Section V, this new method can provide higher amplitude estimation accuracy than that

of the existing methods.

We also investigate robust adaptive methods in the presence of array calibration errors.

It is well-known that the performance of Capon degrades severely in the presence of steering

vector errors. This problem also affects APES and other adaptive methods, but to a lesser

extent. In the presence of array calibration errors, we suggest the use of the recently

proposed robust Capon beamformer (RCB) (see [13], [14] and references therein) and the

doubly constrained robust Capon beamformer (DCRCB) [15] to process the MIMO radar

data. As shown in the numerical examples, when array calibration errors are present, RCB

and DCRCB significantly outperform the aforementioned (non-robust) adaptive methods.

The remainder of this paper is organized as follows. In Section II, we describe our

MIMO radar scheme and the associated data model. Several adaptive methods and two

robust adaptive methods are presented in Sections III and IV, respectively. We provide

several numerical examples in Section V. Finally, Section VI contains our conclusions.

II. Signal Model

Consider a MIMO narrowband radar system with N arbitrarily located transmitting

antennas and M arbitrarily located receiving antennas as shown in Fig. 1 4. The system

simultaneously transmits N linearly independent waveforms, denoted by sn ∈ CL×1 (n =

1, 2, · · ·N) with L being the snapshot number. Let θ be the location parameter of a

generic target, for example, the direction of arrival (DOA) when the targets are in the far

4Note that in Fig. 1, we have separated the transmitting array from the receiving array for the sake of generality.

However, in practical applications, the two sub-arrays may have many common elements. Also, the targets in Fig.

1 are in the near field for illustration purposes only. In practice, the targets can be in the far field.

3

field of the arrays 5, and let at(θ) be the corresponding steering vector for the transmitting

antenna array. Then the waveform vector of the reflected signals from the target at θ is

aTt (θ)S with (·)T denoting the transpose and S = [s1 s2 · · · sN ]T . Note that aT

t (θ)S is a

function of the location parameter θ. Hence, the signals reflected from targets at different

locations are linearly independent of each other. By assuming that the targets are located

in the same range bin, the reflected signals arrive at the receiving array at about the same

time and the arrival time is known.

The signal matrix at the output of the receiving array has the form:

X = ar(θ)β(θ)aTt (θ)S + Z, (1)

where the columns of X ∈ CM×L are the received data snapshots, ar(θ) ∈ CM×1 is the

steering vector of the receiving antenna array and β(θ) ∈ C denotes the complex amplitude

of the reflected signal from θ, i.e., the “reflection coefficient” of the focal point θ. The

matrix Z ∈ CM×L denotes the residual term, which includes the unmodelled noise, inter-

ferences from targets at locations other than θ, and intentional or unintentional jamming.

For notional simplicity, we will not show explicitly the dependence of Z on θ.

The problem is to estimate β(θ) for each θ of interest from the observed data matrix

X. The estimates of {β(θ)} can be used to form a spatial spectrum in the 1D case or a

radar image in the 2D case. We can then estimate the locations of the targets and their

“reflection coefficients” by searching for the peaks in the so-obtained spectrum (or image).

5Note that when the transmitting and receiving arrays are different, the DOAs for the two arrays are different.

Since we know the relative positions of the two arrays, we can use the same DOA variable for notational convenience.

4

III. Adaptive Approaches in the Absence of Array Calibration Errors

A simple way to estimate β(θ) in (1) is via the Least Squares (LS) method:

βLS(θ) =aH

r (θ)XSH a∗t (θ)

L ‖ ar(θ) ‖2 [aTt (θ)RSSa∗t (θ)]

(2)

where (·)H and (·)∗ denote the conjugate transpose and the complex conjugate, respec-

tively, ‖ · ‖ denotes the Euclidean norm, and

RSS =1

LSSH (3)

is the correlation matrix of the waveforms. However, as any other data-independent

beamforming-type method, the LS method suffers from high sidelobes and low resolu-

tion. In the presence of strong interference and jamming, this method completely fails to

work.

We present below several adaptive beamforming-based methods, which, as shown via nu-

merical examples in Section V, have much higher resolution and much better interference-

rejection capability than the LS method.

A. Capon

The Capon estimator consists of two main steps. The first is the Capon beamforming

step [9] [16]. The second is a LS estimation step, which involves basically a matched

filtering.

The Capon beamformer can be formulated as follows:

minw

wHRw subject to wHar(θ) = 1, (4)

where w ∈ CM×1 is the weight vector used to achieve noise, interference and jamming

suppression while keeping the desired signal undistorted, and R is the sample covariance

5

of the observed snapshots, i.e.,

R =1

LXXH . (5)

Following [9] and [16], we can readily obtain the solution to (4) as:

wcapon =R−1ar(θ)

aHr (θ)R−1ar(θ)

, (6)

The output of the Capon beamformer is thus given by

aHr (θ)R−1X

aHr (θ)R−1ar(θ)

. (7)

Substituting (1) into (7) yields

aHr (θ)R−1X

aHr (θ)R−1ar(θ)

= β(θ)aTt (θ)S +

aHr (θ)R−1Z

aHr (θ)R−1ar(θ)

. (8)

Applying the LS method to (8), we get the Capon estimate of β(θ) as follows:

βCapon(θ) =aH

r (θ) R−1XSH a∗t (θ)

L [aHr (θ)R−1ar(θ)][aT

t (θ)RSSa∗t (θ)], (9)

where RSS is defined in (3).

B. APES

The Amplitude and Phase Estimation (APES) approach [10] [17] [18] is a non-parametric

spectral analysis method with superior estimation accuracy [19] [20]. We apply this method

to the proposed MIMO radar system to achieve better amplitude estimation accuracy.

By following [17], the APES method can be formulated as:

minw,β

‖ wHX− β(θ)aTt (θ)S ‖2 subject to wHar(θ) = 1, (10)

where w ∈ CM×1 is the weight vector. Intuitively, the goal of (10) is to find a beamformer

whose output is as close as possible to a signal with the waveform given by aTt (θ)S.

6

Minimizing the cost function in (10) with respect to β(θ) yields

βAPES(θ) =wHXSHa∗t (θ)

L aTt (θ)RSSa∗t (θ)

. (11)

Then the optimization problem in (10) reduces to

minw

wHQw subject to wHar(θ) = 1 (12)

with

Q = R− XSH a∗t (θ)aTt (θ)SXH

L2 aTt (θ)RSSa∗t (θ)

. (13)

Solving the optimization problem in (12) gives the APES beamformer weight vector [17]:

wAPES =Q−1ar(θ)

aHr (θ)Q−1ar(θ)

. (14)

By inserting (14) into (11), the APES estimate of β(θ) is readily obtained as:

βAPES(θ) =aH

r (θ)Q−1XSH a∗t (θ)

L [aHr (θ)Q−1ar(θ)] [aT

t (θ)RSSa∗t (θ)]. (15)

Note that the only difference between the Capon estimator and the APES estimator is

that the sample covariance matrix R in (9) is replaced in (15) by the residual covariance

estimate Q. However, this seemingly minor difference makes these two methods behave

quite differently (see, e.g., Section V).

C. CAPES and CAML

As discussed in [10] (see also Section V), the above two adaptive methods behave dif-

ferently. Capon has high resolution, and hence can provide accurate estimates of the

target locations. However, the Capon amplitude estimates are significantly biased down-

ward. The APES estimator gives much more accurate amplitude estimates at the target

locations, but at the cost of lower resolution.

7

To reap the benefits of both Capon and APES, an alternative estimation procedure,

referred to as CAPES, is proposed in [11]. CAPES first estimates the peak locations using

the Capon estimator and then refines the amplitude estimates at these locations using the

APES estimator. CAPES can be directly applied to the MIMO radar target localization

and amplitude estimation problem considered herein.

Inspired by the CAPES method, we propose below a new approach, referred to as

CAML, which combines Capon and the recently proposed approximate maximum likeli-

hood (AML) estimator based on a diagonal growth curve (DGC) model [12]. In CAML,

AML, instead of APES, is used to estimate the target amplitudes at the target locations

estimated by Capon. Since, unlike APES, AML estimates the amplitudes of all targets at

given locations jointly rather than one at a time, the latter can provide better estimation

accuracy than the former.

Let θk (k = 1, 2, · · ·K) denote the estimated target locations, i.e., the peak locations of

the spatial spectrum estimated by Capon. Here K is assumed known. If K is unknown, it

can be determined accurately using a generalized likelihood ratio test (GLRT), the details

of which are given in the Appendix. Let

Ar = [ar(θ1) ar(θ2) · · · ar(θK)], (16)

At = [at(θ1) at(θ2) · · · at(θK)], (17)

and

β = [β(θ1) β(θ2) · · · β(θK)]T . (18)

Then the data model in (1) can be re-formulated as a DGC model [12]:

X = ArB(ATt S) + Z with B = diag(β), (19)

8

where diag(β) denotes a diagonal matrix whose diagonal elements are equal to the elements

of β, and Z is the residual term. Since the maximum likelihood (ML) estimate of β in

(19) cannot be produced in closed-form, we use the AML estimator in [12] instead, which

is asymptotically equivalent to the ML for a large snapshot number L.

By using the AML method in Equation (18) of [12], the estimate of β can be readily

obtained as:

βAML =1

L[(AH

r T−1Ar)¯ (AHt R∗

SSAt)]−1 vecd(AH

r T−1XSHA∗t ), (20)

where ¯ denotes the Hadamard product [21], vecd(·) denotes a column vector formed by

the diagonal elements of a matrix, and

T = L R− 1

LXSHA∗

t (ATt RSSA

∗t )−1AT

t SXH . (21)

IV. Adaptive Approaches in the Presence of Array Calibration Errors

The previous adaptive methods assume that the transmitting and receiving arrays are

perfectly calibrated, i.e., at(θ) and ar(θ) are accurately known as functions of θ. However,

in practice, array calibration errors are often inevitable. The presence of array calibration

errors and the related small snapshot number problem [22] can degrade significantly the

performance of the adaptive methods discussed so far.

A. RCB

We consider the application of the robust Capon beamformer (RCB) ( see [13], [14] and

references therein) to a MIMO radar system that suffers from calibration errors. RCB

allows ar(θ) to lie in an uncertainty set. Without loss of generality, we assume that ar(θ)

9

belongs to an uncertainty sphere:

‖ ar(θ)− ar(θ) ‖2≤ ε (22)

with both ar(θ), the nominal receiving array steering vector, and ε being given. (For the

more general case of ellipsoidal uncertainty sets, see [13], [14] and references therein.) Note

that the calibration errors in at(θ) will also degrade the accuracy of the estimate of β(θ)

(see (9)). However, the LS (or matched filtering) approach of (9) is quite robust against

calibration errors in at(θ).

The RCB method is based on the following covariance fitting formulation [13] [14] [23]:

maxσ2(θ),ar(θ)

σ2(θ) subject to R− σ2(θ)ar(θ)aHr (θ) ≥ 0

‖ ar(θ)− ar(θ) ‖2≤ ε,

(23)

where σ2(θ) denotes the power of the signal of interest and P ≥ 0 means that P is

Hermitian positive semi-definite. The optimization problem in (23) can be simplified as

[13]:

minar(θ)

aHr (θ)R−1ar(θ) subject to ‖ ar(θ)− ar(θ) ‖2≤ ε. (24)

By using the Lagrange multiplier methodology [13], the solution to (24) is found to be

ar(θ) = ar(θ)− [I + λ(θ)R]−1ar(θ), (25)

where I denotes the identity matrix. The Lagrange multiplier λ(θ) ≥ 0 in (25) is obtained

as the solution to the constraint equation

‖ [I + λ(θ)R]−1 ar(θ) ‖2= ε, (26)

which can be solved efficiently by using the Newton method since the left side of (26) is a

monotonically decreasing function of λ(θ) (see [13] for more details). Once the Lagrange

10

multiplier λ(θ) is determined, ar(θ) is obtained from (25). To eliminate a scaling ambiguity

(see [13]), we scale ar(θ) such that ‖ ar(θ) ‖2= M . Replacing ar(θ) in (9) by the scaled

steering vector ar(θ) yields the RCB estimate of β(θ).

B. DCRCB

A doubly constrained robust Capon beamformer (DCRCB) is proposed in [15]. In

the DCRCB, in addition to the spherical constraint in (22), the steering vector ar(θ) is

constrained to have a constant norm, i.e.,

‖ ar(θ) ‖2= M. (27)

Then the covariance fitting formulation (see (23)) becomes:

maxσ2(θ),ar(θ)

σ2(θ) subject to R− σ2(θ)ar(θ)aHr (θ) ≥ 0

‖ ar(θ)− ar(θ) ‖2≤ ε

‖ ar(θ) ‖2= M.

(28)

The optimization problem in (28) can be simplified to [15]:

minar(θ)

aHr (θ)R−1ar(θ) subject to Re[aH

r (θ)ar(θ)] ≥ M − ε

2

‖ ar(θ) ‖2= M,

(29)

where Re(·) denotes the real part of a complex number. Let u1 be the principal eigenvector

of R, and let

ar(θ) = M12 u1 exp

{j arg

[uH

1 ar(θ)]}

(30)

with arg(·) denoting the phase of a complex number. If ar(θ) in (30) satisfies

Re[ aHr (θ)ar(θ) ] ≥ M − ε

2, (31)

11

then, obviously, ˆar(θ) = ar(θ) is the optimal solution of (29). Otherwise, by using the

Lagrange multiplier methodology [15], the optimal solution to (29) is

ˆar(θ) =(M − ε

2

) [R−1 + λ(θ)I]−1 ar(θ)

aHr (θ) [R−1 + λ(θ)I]−1 ar(θ)

. (32)

The Lagrange multiplier λ(θ) in (32), which can be negative, is obtained as the solution

to the constraint equation

aHr (θ) [R−1 + λ(θ)I]−2 ar(θ){

aHr (θ) [R−1 + λ(θ)I]−1 ar(θ)

}2 =M

(M − ε2)2

. (33)

Like in the case of RCB, the constraint equation (33) can be solved efficiently by using

the Newton method since the left side of (33) is a monotonically decreasing function of

λ(θ) (see [15] for more details).

We summarize the DCRCB estimator of β(θ) as follows:

1. Compute ar(θ) by (30).

2. If ar(θ) satisfies (31), estimate β(θ) by replacing ar(θ) in (9) by ˆar(θ) = ar(θ), and

stop.

3. If (31) is not satisfied, solve (33) for λ(θ). Calculate ˆar(θ) by (32) using the so-obtained

λ(θ).

4. Compute the DCRCB estimate of β(θ) by replacing ar(θ) in (9) by ˆar(θ), and stop.

V. Numerical Examples

Consider a MIMO radar system where a uniform linear array with N = M = 10 antennas

and half-wavelength spacing between adjacent antennas is used both for transmitting and

for receiving. The transmitted waveforms sn (n = 1, 2, · · · N) are orthogonal quadrature

phase shift keyed (QPSK) sequences, and hence we have RSS = I (this choice of Rss is

12

optimal in a maximum-SNR sense, see [24]).

Consider a scenario in which K = 3 targets are located at θ1 = −40◦, θ2 = −25◦, and

θ3 = −10◦ with “reflection coefficients” β1 = 4, β2 = 4, and β3 = 1, respectively. There is

a strong jammer at 0◦ with an unknown waveform and with amplitude 1000, i.e., 60 dB

above β3. The received signal has L = 256 snapshots and is corrupted by a zero-mean

spatially colored Gaussian noise with an unknown covariance matrix. The (p, q)th element

of the unknown noise covariance matrix is 1

SNR 0.9|p−q|ej(p−q)π

2 .

We first consider the case of no array calibration errors. Let SNR = 10 dB. The moduli

of the spatial spectral estimates of β(θ), versus θ, obtained by using LS, Capon, APES,

CAPES and CAML are given in Figs. 2(a), (b), (c), (e) and (f), respectively. For com-

parison purpose, we also show the true spatial spectrum via dashed lines in these figures.

As seen in Fig. 2(a), the LS method suffers from high sidelobes and low resolution. Due

to the presence of the strong jamming signal, the LS estimator fails to work properly.

Capon and APES possess great interference and jamming suppression capabilities. The

Capon method gives very narrow peaks around the target locations. However, the Capon

amplitude estimates are biased downward. The APES method gives more accurate am-

plitude estimates around the target locations but its resolution is slightly worse than that

of Capon. Note that in Figs. 2(a)-2(c) a false peak occurs at θ = 0◦ due to the presence

of the strong jammer. Despite the fact that the jammer waveform is statistically indepen-

dent of the waveforms transmitted by the MIMO radar, a false peak still exists since the

jammer is 60 dB stronger than the weakest target and the snapshot number is finite.

To reject the false peak, we use a generalized likelihood ratio test (GLRT) for each angle

13

of interest; the GLR is given by (see the Appendix for details):

ρ(θ) = 1− aHr (θ)R−1ar(θ)

aHr (θ)Q−1ar(θ)

, (34)

where R and Q are given in (5) and (13), respectively. Fig. 2 (d) gives the GLR as a

function of the target location parameter θ. As expected, we get high GLRs at the target

locations and low GLRs at other locations including the jammer location. By comparing

the GLR with a threshold, the false peak due to the strong jammer can be readily detected

and rejected, and a correct estimate of the number of the targets can be obtained. This

information can be passed to CAPES and CAML to obtain Figs. 2(e) and 2(f). As seen

from these figures, both CAPES and CAML give accurate estimates of the target locations

and amplitudes.

Next we consider a more challenging example where θ3 is −21◦ while all the other

simulation parameters are the same as before. As shown in Fig. 3(c), in this example the

APES method fails to resolve the two closely spaced targets at θ2 = −25◦ and θ3 = −21◦.

On the other hand, Capon gives well-resolved peaks around the target locations, but its

amplitude estimates are significantly biased downward. Based on the GLR in Fig. 3(d),

the false peak due to the strong jammer can be readily rejected. Once again CAPES

and CAML outperform the other methods and provide quite accurate target parameter

estimates.

To illustrate the improvement of CAML over CAPES for target amplitude estimation,

we consider the amplitude estimation performance of Capon, CAPES, and CAML as a

function of SNR. All the other simulation parameters are the same as in Fig. 3. Figs.

4(a) and 4(b) show, respectively, the bias and mean-squared error (MSE) of the amplitude

14

estimate for the target at θ3. The results are obtained using 1000 Monte-Carlo simulations.

As seen from these figures, the Capon estimate is biased downward significantly. Due to

this bias, an error floor occurs in the MSE of the Capon estimate. CAPES is also biased

downward at very low SNR. The CAML estimator is unbiased and outperforms CAPES

for the entire range of the SNR considered, especially at low SNR.

We next consider the case where array calibration errors are present. To simulate the

array calibration error, each element of the steering vector at(θ) = ar(θ), for each incident

signal, is perturbed with a zero-mean circularly symmetric complex Gaussian random

variable with variance 0.005 and then scaled to have norm√

M , so that the squared

Euclidean norm of the difference between the true array steering vector and the presumed

one is about 0.05. We let SNR = 30 dB. The other simulation parameters are the same

as those in Fig. 2. For the sake of description convenience, the moduli of the amplitude

estimates, i.e., the y-axes of Figs. 5(a)-5(d), are on a dB scale. As shown in Fig. 5(a),

the Capon method fails to work properly in the presence of array calibration errors, as

expected: its amplitude estimates at the target locations are severely biased downward (by

more than 60 dB for some of them). Although APES gives much better performance than

Capon, its amplitude estimates at the target locations are about 10 dB lower than the true

amplitudes. On the other hand, both RCB and DCRCB provide accurate estimates of the

target amplitudes as well as target locations, but their peaks are wider (and hence their

resolution is poorer) compared to what is shown in Fig. 2(b), as expected (robustness to

array calibration errors inherently reduces the resolution).

Figs. 6(a)-6(b) show the GLRs corresponding to RCB and DCRCB, as functions of θ,

which are obtained by replacing ar(θ) in (34) by ar(θ) and ˆar(θ) from Subsections IV-A

15

and IV-B, respectively. As we can see, both RCB and DCRCB give high GLRs at the

target locations and a low GLR at the jammer location. Based on the GLRs, we can again

readily and correctly estimate the number of targets to be 3. Plotting the spatial spectral

estimates in Figs. 5(c) and 5(d) only for the angles at which the corresponding GLRs are

above a given threshold (say 0.8), we obtain the refined spatial spectral estimates in Figs.

6(c) and 6(d). These refined spatial spectral estimates obtained by RCB and DCRCB

provide an accurate description of the target scenario.

VI. Conclusions

We have considered several adaptive techniques for a MIMO radar system, where mul-

tiple antennas transmit linearly independent (perferrably uncorrelated) waveforms and

multiple antennas receive the reflected signals. Due to the different phase shifts caused

by the different propagation paths from the transmitting antennas to the targets, these

linearly independent waveforms are combined with different phase factors at the targets.

The result is that the signals reflected from different targets are linearly independent of

each other, which enables the application of the adaptive techniques to achieve high reso-

lution and superior interference and jamming rejection capability. In the absence of array

calibration errors, we have considered Capon, APES and CAPES. Capon provides high

resolution and APES gives accurate amplitude estimates at the target locations, while

CAPES combines the merits of both Capon and APES. We have also introduced a new

approach, referred to as CAML, which can further improve the amplitude estimation ac-

curacy of (C)APES. In the presence of array calibration errors, we have shown that the

RCB and DCRCB approaches can provide accurate estimates of both target locations and

16

target amplitudes. Finally, we have proposed a generalized likelihood ratio test that can

be used to determine the number of targets by separating the jammers from targets.

Appendix: Generalized Likelihood Ratio Test

We assume that the columns of the residual term Z in (1) are independently and identi-

cally distributed circularly symmetric complex Gaussian random vectors with zero-mean

and unknown covariance matrix Q. We derive the GLRT for each θ of interest. For

notational brevity, we omit the argument θ of ρ, ar, at and β.

Following [25] (see also [26]), we define the GLR as follows:

ρ = 1−[maxQ f(X|β = 0,Q)

maxβ,Q f(X|β,Q)

] 1L

, (35)

where

f(X|β,Q) = π−LM |Q|−L exp{−tr

[Q−1(X− arβaT

t S)(X− arβaTt S)H

]}(36)

is the probability density function of the observed data matrix X given the parameters

β and Q, and tr(·) and | · | denote the trace and determinant of a matrix, respectively.

From (35), we note that the value of the GLR, ρ, lies between 0 and 1. If there is a target

at a θ of interest, we usually have maxβ,Q f(X|β,Q) À maxQ f(X|β = 0,Q), i.e., ρ ≈ 1;

otherwise ρ ≈ 0.

Solving the optimization problems in (35) with respect to Q yields

maxQ

f(X|β = 0,Q) = (πe)−LM |R|−L, (37)

and

maxβ,Q

f(X|β,Q) = (πe)−LM

{min

β

∣∣∣∣1

L(X− arβaT

t S)(X− arβaTt S)H

∣∣∣∣}−L

. (38)

17

Applying the technique used in [10] and [27], we obtain

∣∣∣∣1

L(X− arβaT

t S)(X− arβaTt S)H

∣∣∣∣

=

∣∣∣∣R− XSHa∗t β∗aH

r

L− arβaT

t SXH

L+ |β|2ara

Tt RSSa

∗ta

Hr

∣∣∣∣

=

∣∣∣∣∣∣Q + (aT

t RSSa∗t )

(arβ − XSHa∗t

L(aTt RSSa∗t )

)(arβ − XSHa∗t

L(aTt RSSa∗t )

)H∣∣∣∣∣∣

=|Q|∣∣∣∣∣∣I + (aT

t RSSa∗t ) Q

−1

(arβ − XSHa∗t

L(aTt RSSa∗t )

)(arβ − XSHa∗t

L(aTt RSSa∗t )

)H∣∣∣∣∣∣

=|Q|1 + (aT

t RSSa∗t )

(arβ − XSHa∗t

L(aTt RSSa∗t )

)H

Q−1

(arβ − XSHa∗t

L(aTt RSSa∗t )

)

≥|Q|[1 +

aTt SXH

L2(aTt RSSa∗t )

Q−1

(I− ara

Hr Q−1

aHr Q−1ar

)XSHa∗t

],

(39)

where we have used the fact that |I + AB| = |I + BA| [16], and the equality holds when

β = βAPES. Note that

|Q|[1 +

aTt SXH

L2(aTt RSSa∗t )

Q−1

(I− ara

Hr Q−1

aHr Q−1ar

)XSHa∗t

]

=|Q|∣∣∣∣∣I + Q−1

(I− ara

Hr Q−1

aHr Q−1ar

)XSHa∗ta

Tt SXH

L2(aTt RSSa∗t )

∣∣∣∣∣

=

∣∣∣∣∣Q +

(I− ara

Hr Q−1

aHr Q−1ar

)XSHa∗ta

Tt SXH

L2(aTt RSSa∗t )

∣∣∣∣∣

=

∣∣∣∣∣R− araHr Q−1XSHa∗ta

Tt SXH

L2(aHr Q−1ar)(aT

t RSSa∗t )

∣∣∣∣∣

=|R|∣∣∣∣∣I−

R−1araHr Q−1XSHa∗ta

Tt SXH

L2(aHr Q−1ar)(aT

t RSSa∗t )

∣∣∣∣∣

=|R|[1− aH

r Q−1(R− Q)R−1ar

aHr Q−1ar

]

=|R| aHr R−1ar

aHr Q−1ar

,

(40)

where again we have used the fact that |I + AB| = |I + BA|.

18

From (39) and (40), it follows that

minβ

∣∣∣∣1

L(X− arβaT

t S)(X− arβaTt S)H

∣∣∣∣ = |R| aHr R−1ar

aHr Q−1ar

. (41)

By using (37), (38), and (41) in (35), the GLR in (34) follows immediately.

References

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Nov. 2004.

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Nov. 2004.

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19

[11] A. Jakobsson and P. Stoica, “Combining Capon and APES for estimation of spectral lines,” Circuits, Systems

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20

azine, pp. 39–54, September 2003.

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IEEE Transactions on Signal Processing, vol. 52, pp. 23–35, January 2004.

21

s 1

s 2

s N

x 1 x

2 x

M ... ...

...

Targets

Transmitting array Receiving array

Fig. 1. A MIMO radar system.

22

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(a) (b)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

−60 −40 −20 0 20 40 600

0.2

0.4

0.6

0.8

1

DOA (deg)

GLR

(c) (d)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(e) (f)

Fig. 2. Spatial spectral estimates and GLR in the absence of array calibration errors, when θ1 = −40◦,

θ2 = −25◦ and θ3 = −10◦. (a) LS, (b) Capon, (c) APES, (d) GLR, (e) CAPES, and (f) CAML.

23

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(a) (b)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

−60 −40 −20 0 20 40 600

0.2

0.4

0.6

0.8

1

DOA (deg)

GLR

(c) (d)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(e) (f)

Fig. 3. Spatial spectral estimates and GLR in the absence of array calibration errors, when θ1 = −40◦,

θ2 = −25◦ and θ3 = −21◦. (a) LS, (b) Capon, (c) APES, (d) GLR, (e) CAPES, and (f) CAML.

24

−40 −30 −20 −10 0 10 20 30−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

SNR (dB)

Bia

s CaponCAPESCAML

−40 −30 −20 −10 0 10 20 3010

−10

10−8

10−6

10−4

10−2

100

102

SNR (dB)

MS

E

CaponCAPESCAML

(a) (b)

Fig. 4. Empirical Bias and MSE versus SNR in the absence of array calibration errors, when θ1 = −40◦,

θ2 = −25◦, and θ3 = −21◦. (a) Bias and (b) MSE.

−60 −40 −20 0 20 40 60−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(dB

)

−60 −40 −20 0 20 40 60−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(dB

)

(a) (b)

−60 −40 −20 0 20 40 60−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(dB

)

−60 −40 −20 0 20 40 60−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(dB

)

(c) (d)

Fig. 5. Spatial spectral estimates in the presence of array calibration errors, when θ1 = −40◦, θ2 = −25◦

and θ3 = −10◦. (a) Capon, (b) APES, (c) RCB with ε = 0.1 , and (d) DCRCB with ε = 0.1.

25

−60 −40 −20 0 20 40 600

0.2

0.4

0.6

0.8

1

DOA (deg)

GLR

−60 −40 −20 0 20 40 600

0.2

0.4

0.6

0.8

1

DOA (deg)

GLR

(a) (b)

−60 −40 −20 0 20 40 60−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(dB

)

−60 −40 −20 0 20 40 60−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

DOA (deg)

Mod

ulus

of C

ompl

ex A

mpl

itude

(dB

)

(c) (d)

Fig. 6. GLRs and refined spatial spectral estimate in the presence of array calibration errors, when

θ1 = −40◦, θ2 = −25◦ and θ3 = −10◦. (a) GLR of RCB, (b) GLR of DCRCB, (c) refined spatial spectral

estimate of RCB, and (d) refined spatial spectral estimate of DCRCB.

26