Target Velocity Estimation and Antenna Placement for ... - CiteSeerX

IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010

79

Target Velocity Estimation and Antenna Placement for MIMO Radar With Widely Separated Antennas Qian He, Student Member, IEEE, Rick S. Blum, Fellow, IEEE, Hana Godrich, Member, IEEE, and Alexander M. Haimovich, Member, IEEE

Abstract—This paper studies the velocity estimation performance for multiple-input multiple-output (MIMO) radar with widely spaced antennas. We derive the Cramer–Rao bound (CRB) for velocity estimation and study the optimized system/configuration design based on CRB. General results are presented for an extended target with reflectivity varying with look angle. Then detailed analysis is provided for a simplified case, assuming an isotropic scatterer. For given transmitted signals, optimal antenna placement is analyzed in the sense of minimizing the CRB of the velocity estimation error. We show that when all antennas are located at approximately the same distance from the target, symmetrical placement is optimal and the relative position of transmitters and receivers can be arbitrary under the orthogonal received signal assumption. In this case, it is also shown that for MIMO radar with optimal placement, velocity estimation accuracy can be improved by increasing either the signal time duration or the product of the number of transmit and receive antennas. Index Terms—Antenna placement, Cramer–Rao bound (CRB), multiple-input multiple-output (MIMO) radar, maximum-likelihood (ML) estimate, velocity estimation.

I. INTRODUCTION

R

ESEARCH on multiple-input multiple-output (MIMO) radar has been drawing more and more attention from both the communication and radar communities. Most studies on MIMO radar can be classified into two categories. The first category [1]–[8] employs widely separated antennas at both the transmit and receive ends. Among these studies, both coherent and non-coherent processing has been considered [1]. Non-coherent MIMO radar requires time synchronization between the sensors. Besides time synchronization, coherent MIMO radar

Manuscript received January 26, 2009; revised July 20, 2009. Current version published nulldate. The work of Q. He and R. S. Blum were supported in part by the Air Force Research Laboratory under Agreement FA9550-09-1-0576, in part by the National Science Foundation under Grant CCF-0829958, in part by the U.S. Army Research Office under Grant W911NF-08-1-0449, and in part by the China Scholarship Council. The work of H. Godrich and A. M. Haimovich were supported in part by the U.S. Air Force Office of Scientific Research under agreement FA9550-09-1-0303. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Jian Li. Q. He is with the Electrical Engineering Department, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China, and also with Lehigh University, Bethlehem, PA 18015 USA (e-mail: qianhe@uestc. edu.cn; [email protected]). R. S. Blum is with the Electrical and Computer Engineering Department, Lehigh University, Bethlehem, PA 18015 USA (e-mail: [email protected]. edu). H. Godrich and A. M. Haimovich are with the Electrical and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTSP.2009.2038974

requires additional phase synchronization. The second category [9]–[16] uses closely spaced antennas. Widely dispersed antennas enable MIMO radar to view the target from several different angles simultaneously. This lowers the chance that all antennas measure low Doppler shifts. Recently, we have begun to study the detection of moving targets using MIMO radar [7]. We showed that MIMO radar with widely separated antennas enjoys both the diversity gain and the geometry gain. The diversity gain is similar to that uncovered in [3], so we do not focus on this in the current paper. The geometry gain is a different type of gain that results from viewing the target from different directions. While a single antenna may be blind to a target moving in a direction orthogonal to the line towards the antenna, a group of widely spaced antennas will not be. This is the essence of geometry gain. In the current paper we focus on geometry gain in our numerical results, since it is less well understand. To complete [7] and answer some remaining questions, we focus here on the ultimate velocity estimation performance obtainable through MIMO radar with widely spaced antennas by studying the Cramer–Rao bound (CRB). We show the utility of these results by using them to study proper antenna placement. Various CRB studies for MIMO radar performance have appeared in the literature. For example, the CRB for target localization in a coherent MIMO radar is presented in [4], [8]. The average CRB for bearing estimation in MIMO radar is analyzed in [2]. In [6], the outage CRB is introduced to evaluate MIMO radar direction finding performance. In [16], range compression and waveform optimization problems for MIMO radar are investigated based on the CRB matrix. None of these papers consider target velocity estimation performance, which is another important issue for radar systems [17]–[20]. MIMO radar systems optimization has also been investigated. Most existing publications concentrate on waveform optimization. For example, in [21], waveform design methods for maximizing the conditional mutual information (MI) or minimizing the mean-square error (MMSE) are discussed. The two criteria lead to the same solution under the same total power constraint. In [22], the uncertainty of the target power spectrum is considered and the MI and MMSE criteria lead to different minimax robust waveforms. Different from this work, in [23] and [24], waveform optimization for target parameter estimation based on the CRB is studied. In [25] the MIMO radar ambiguity function is used for designing good waveforms. In this paper, the CRB is used to study target velocity estimation in MIMO radar with widely dispersed antennas. We present an optimization strategy for transmit/receive antenna placement. Considering a target moving with constant velocity and a model which includes Gaussian noise and path loss, we calcu-

1932-4553/$26.00 © 2010 IEEE

80


late the CRB for velocity estimation using MIMO radar. The derived CRB provides an effective way for finding the optimal value for certain parameters. With the other system parameters fixed, we focus on the optimal antenna placement. Assuming all antennas are located equidistant from the target, it is shown that symmetrically placing the transmit and receive antennas is the best choice, and the optimal achievable performance is not affected by the relative position of the transmit and receive antennas under an orthogonal received signal assumption. This result is also extended to a more general case without the equidistance assumption. This optimal placement result justifies some of our numerical findings, which also show that larger sensor separations, corresponding to larger angular spreads, lead to better performance. We develop the maximum-likelihood (ML) estimator, compare the mean square error (MSE) of the ML estimate with the CRB result for some specific examples, and show the ML estimate and the CRB agree for sufficiently large signal-to-noise ratio (SNR). While initially, we focus on a target whose reflectivity is the same during the observation, we show how to extend our results for more general models. For completeness, the extreme sensor placements resulting in a singular Fisher information matrix, which means one cannot accurately estimate the desired quantities, are briefly discussed. The paper is organized as follows. The signal model is presented in Section II. Section III derives the ML estimate of the velocity and calculates the corresponding CRB for a case with an extended target whose reflectivity varies with angle. In Section IV, we start to concentrate on a special case, assuming an isotropic scatter, which simplifies the analysis and enables us to provide useful insight. Then, Section V describes a general optimization based on the derived CRB, with optimal antenna placement studied as a particular example. Numerical results are provided in Section VI. Finally, Section VII describes extensions to cases outside the original assumptions and also provides conclusions for this paper.

Fig. 1. Location of transmitters and receivers with respect to the target and its motion.

Fig. 1. Since the focus of this paper is target velocity estimation, we assume that through the observation interval, the target is known to be present in a resolution cell with nominal coordi. Let the transmitters and receivers be located nates , , and , , respecat tively. The low-pass equivalent of the signal received by the th transmitters and receiver due to the transmissions from all the reflection from the target is given by (1) where is the unknown complex target reflecdenotes the time delay of the th tivity for the th path2 is the Doppler frequency detected by the th path, and receiver due to the th transmitter. The observation interval is assumed long enough so that all transmitted signals can be observed, irrespective of their delay. The terms (2) in (1) represent the propagation coefficients, where

II. SIGNAL MODEL Consider a coherent MIMO radar that has transmit and receive antennas. The elements at both the transmit and receive ends are widely dispersed, see [1], [7] for motivation. The set of transmitted signals expressed in low-pass equivalent , , where is the form is given by is nortotal transmitted energy. Each of the waveforms , where is the obmalized and satisfies servation interval. Assume a narrowband signal such that the common complex envelope notation can be applied to the trans. Each of these signals mitted signal,1, i.e., is emitted by an individual transmit antenna. Note that the sig, are expressed in a completely general nals form, other than normalization, so that for example, either single pulse or multiple pulse waveforms can be assumed. In addition, it is assumed that the target returns do not fluctuate during the observation interval. Consider a target moving with an unknown constant velocity , where and are velocity components in a two-dimensional Cartesian coordinate system, as shown in 1Note

that the actual transmitted signal is the real part, but we use complex envelope description so we deal with complex signals.

, with the path length from the

representing th transmitter to the target, and

representing the path length from the target to the th receiver. Note that the path loss is . We also have assumed to be inversely proportional to , where is the speed of light. The phase shift induced by the carrier frequency is captured in the exponent of . in (1). The The noise at the th receiver is denoted by , at different receivers are noise components is a zero-mean spatially independent, and temporally each white complex Gaussian process with power spectral density , (PSD) . If the noise is colored3 with PSD is needed at each rethen a whitening filter ceiver to remove the correlation of the noise. The positions of the transmit and receive antennas with respect to the target motion are shown in Fig. 1. The transmitter 2Here we assume the target reflectivity is different for each transmit-to-receive antenna path. If this is not the case we can reduce the number of unknown parameters as discussed later.

G!

3For simplicity, ( ) is assumed known. In practice it may have to be estimated. For further details, the reader is referred to the extensive literature on space-time adaptive processing (STAP).

HE et al.: TARGET VELOCITY ESTIMATION AND ANTENNA PLACEMENT FOR MIMO RADAR

and receiver

view the target from different angles and , and . The Doppler shift where observed at the th receiver due to the signal transmitted by the th transmitter is given by

81

A. Cramer–Rao Bound for Velocity Estimation In this subsection, we compute the CRB for the model introduced previously. The variance of the th element of any unbiased estimator for an unknown parameter vector satisfies [28], [30]

where

is the Fisher information matrix (FIM) defined as

(3) where

denotes the carrier wavelength and

(8) and

the

gradient

operation

is

(4) and , define two position matrices, and . These matrices are defor termined by the placement of the transmit and receive antennas with respect to the target of interest. The signals received from all the receivers are collected in . The joint probability a vector4 density function (pdf) of the observations (time samples at multiple receive antennas) parameterized by the unknown parameter vector

. The CRB matrix is the inverse of the Fisher . Note that here the information matrix, are assumed to be deterministic reflection coefficients unknown. Using the approach described in [6], the CRBs for the reflection coefficients modeled as random variables are provided in Appendix I. is a function of the Doppler frequency shifts Since , , , we introduce the parameter vector

(9)

(5)

which is a column vector with elements containing the observed Doppler shifts from every transmit–receive path as well as the real and imaginary parts of the target complex reflectivities. Using the chain rule, the FIM can be expressed as

is given by [26]

(10) (6)

The term

can be calculated from (3) and (9) as

Taking the logarithm of the probability density function, we obtain the log-likelihood function where

(7) where is some constant independent of the parameters collected in .

and and are the all zero matrix and the identity matrix, respectively, with dimensions given in the corresponding subscripts. , which is a matrix, We now calculate using (8) with replacing . Considering the structure of has to do with the order of the elements arranged in , it is convenient to divide it into four block submatrices

III. CRB AND ML ESTIMATE FOR TARGET VELOCITY ESTIMATION In this section, the CRB and the ML estimate of the target velocity are discussed for MIMO radar. 4The

superscript y denotes transposition.

The upper left submatrix involves derivatives of the th components of such Doppler frequencies. Consider the

82

that corresponds to . Then


, as per (9), and assume the th component , and the th component corresponds to

for where5

(11) and . When , where the integral term in (11) represents the square of the effective pulse width of the th transmitted signal plus the square of the pulse center of the th received signal (see (34) for more detail), scaled by the squared magnitude of the propagation coefficient for the th path. Note that in this calculation, the received signals are delayed by a certain amount of time (the delay for the path from the th transmit antenna to the th receive antenna), and the observation interval is chosen so that the received sig, then the integral nals are visible with this delay. When term in (11) is related to the squared average of the overlap time between the signals transmitted from the th and th transmit antennas when received at the th receive antenna. Note that due to the complex signals involved, a conjugate enters into the calculation, as found normally in cross-correlation of complex sigfor different , into an nals. For each , we can arrange matrix , the th entry of which is . The lower right submatrix involves derivatives of the target complex reflectivities. Consider the th components of such , as per (9), and assume the th that or , and the th component component corresponds to or . Then we have corresponds to

where

(12) The integral term in (12) represents the cross-correlation between two signals as received at the th receive antenna when transmitted from the th and th transmit antennas , or the self-correlation when both transmitted signals come . Note that the received from the th transmit antenna Doppler shifts and propagation coefficients enter into these calculations as expected. involves derivatives related to The upper right submatrix both the Doppler frequency and the target complex reflectivities. th components of such that , Consider the , as per (9), and assume the th compo, and the th component corresponds to nent corresponds to or . Then we compute

and

where

(13) The integral term in (13) is similar to that in (12), except for an additional in the integrand. This makes it implement an effective center of mass type calculation for the cross-correlated signals to estimate the time at which the center of this cross-cor, the integral term represents related signal resides. When the time center of the th transmitted signal when received at the th receiver. and in (10), the FIM is obtained Plugging

,

and

5The superscript 3 denotes conjugation, and < and = denotes the real and imaginary operators.

(14) In order to get the CRB matrix, we need to invert the FIM. and are invertible, and one can apply the Usually, matrix inversion lemma [33] to (14) to calculate the inverse of the FIM as in (15), shown at the bottom of the page. Here, we are and , whose variances interested only in the estimates of are bounded by CRB

(16)

(15)


CRB

(17)

Thus, it is sufficient to calculate the upper-left 2 of , called

CRB CRB

2 submatrix

(18)

The two diagonal elements of determine the CRBs and . They are dependent on several of the estimates of parameters, such as the wavelength, the energy used per transmitter, the PSD of the noise, the target reflectivity, the range cell , and related to the transof interest, the terms mitted signals, and the number and placement of the radar antennas implied in the position matrices and .

83

IV. VELOCITY ESTIMATION ANALYSIS FOR AN ISOTROPIC SCATTERER To simplify the analysis and provide useful insight, we consider the isotropic scatter, which gives the same reflectivity for are the same for all paths, i.e., all directions, and hence , for all . Note that the discussions in the rest of the paper are based on this simplification. A. Cramer–Rao Bound For an isotropic scatter, the log-likelihood function can be written as

(20) The dimensions of the unknown parameter vectors are greatly reduced

B. ML Estimate of Velocities ML estimation is an asymptotically optimal method [28]. For signal-in-noise problems, the ML estimator achieves the CRB for sufficiently large SNRs. In our problem, the ML estimate is given by

and

which yields

and

where . unknown parameters that need Note that contains and are to be estimated simultaneously, although only of interest. Since the closed-form solution for the ML estimate seems not available, numerical evaluation is needed. The com-dimensional search is very putational complexity for a . A practical approach is to use high, especially for large iterative optimization procedures such as the space alternating generalized expectation–maximization (SAGE) algorithm [29].

where the submatrices and are the same as previously, but and have much fewer elements than their former versions, and . In the submatrix , involving the th components we have of for

C. Optimized System/Configuration Design Based on CRB The results of Section III-A show that the CRBs of the esand are dependent on several parameters. We timates of require the optimized system/configuration design with respect to minimize a metric to a parameter vector and . formed by the sum of the CRBs of the estimates of The value that optimizes this metric is given by CRB

CRB

(19) denotes the objective function dependent on paramwhere denotes the feasible set of . Numerical methods eter , and are usually resorted to for solving this problem.

and

where

(21)

84


In the submatrix ,

, involving the

th components of we find

for

and

and

where

(27)

B. ML Estimate of Velocities (22) and in (10), the FIM matrix is obtained Plugging and given in (23), shown at bottom the of the page, where the and , are matrix is defined after (11), and the th row of the position matrices and . The upper-left 2 2 submatrix of related to the velocity estimates is

CRB CRB where ,

and

(24)

of

Based on the log-likelihood function in (20), the ML estimate is

For any by letting

, the likelihood function is maximized

are defined as

(25)

and

(26)

(23)


Then the ML estimate of the target complex reflectivity is obtained

85

A. Optimal Antenna Placement With Orthogonal Signals and an Isotropic Scatterer In the rest of this section, we consider the case of orthogonal signals, formulated in Assumption 1, which follows. Assumption 1 (Orthogonal Signals): We assume that the transmitted signals are orthogonal if if

where is defined in (21). Substituting back into the loglikelihood function, the ML estimate of the target velocities can be derived

and maintain orthogonality for a variety of allowed time delays , and Doppler frequency shifts , , i.e., if if

(28) As further simplification seems not available for the ML estimator, needs to be determined numerically. In the numerical study, we adopt a two-dimensional grid search followed by an iterative optimization to obtain the maximum-likelihood estimate. Assuming the received signals satisfy an orthogonality criterms in are zero, and teria, we show later that the hence is a constant independent of velocity, which simplifies (28). Then the optimal estimator selects the velocity vector in the phase term that maximizes the output of a matched filter, which is matched to a shifted version of the transmitted signal. A bank of matched filters that covers the allowable velocity region can be used to produce an approximation to this optimal estimator. However, if the orthogonal received signal assumpterms enter in . Note that the tion does not hold, then the terms are affected by the velocity vector. values of the In this case, a more complicated optimal estimator is required. The more complicated estimate still employs a matched filter, but the quantity maximized is the matched filter divided by a term that characterizes the impact of the interference one gets between signals originating from different transmit antennas, as might be expected with non-orthogonal signals. V. OPTIMIZED SYSTEM/CONFIGURATION DESIGN BASED ON CRB AND AN ISOTROPIC SCATTERER Using the CRB submatrix (24), for fixed wavelength and SNR is fixed), the CRBs can be expressed as a (so that function of the quantities (25), (26), and (27). Plugging CRB and CRB into (19), the objective function becomes (29) and thus the optimal parameter should satisfy (30)

.

See Appendix II for further discussion. Given Assumption 1, we can simplify (see Appendix III) the results in (11), (21), and (22) as (31) , (32) (33)

where (34) , is the effective pulse width of the represents the time transmitted signal, and center of the transmitted signal [35]. Plugging , , and back into (24)–(27), the CRB submatrix is obtained. Before discussing the optimal antenna placement, we define symmetrical antenna placement. Definition 1 (Symmetrical Antenna Placement): Symmetrical antenna placement satisfies

, (35) where and are arbitrary constant angles. Next, we introduce an equidistance assumption. Assumption 2 (Equidistance): We assume all antennas are placed around the area being monitored, such that they have roughly the same distance from the target. In this case, the antenna placement variable is an vector consisting of the transmit and receive antenna directions with respect , where to the target. The feasible set is . (without Under Assumptions 1–2, in the loss of generality, we normalize this term to , and are the same for sequel for simplicity),

86

all and ( , , , (25)–(27) can be simplified to


), and hence the terms

Proof of Lemma 3: Under the assumptions given in Lemma 3, according to (103) in Appendix VII, at all symmetrical placements, the quantities , , (36)–(38) are given by

(36)

(41)

(37)

The value of the objective function is the same for all symmetric antenna placements (42)

and (38) It can be proved that for a MIMO radar system with widely separated transmitters and widely separated , , and . See Appendix IV. receivers, In this case, the optimal solution in (30) can be determined by the necessary condition that it makes the gradient of the objective function equal to zero

(39) or more precisely

(40) , the elements of the gradient Plugging , , (36)–(38) in can be calculated. See Appendix V. The following theorem indicates that there is only one kind of solution to (40), which therefore provides the solution to the optimization problem (30) and leads to the best achievable velocity estimation performance. Theorem 1: Assume a MIMO radar system with widely separated transmitters and widely separated receivers satisfies Assumptions 1–2, then for waveforms with and from (34), only symmetrical placement can arbitrary be optimum. The proof of Theorem 1 is obvious based on the following two lemmas. Lemma 1: Assume a MIMO radar system with widely separated transmitters and widely separated receivers satisfies Assumptions 1–2, then no solution other than a symmetric antenna placement will solve the zero gradient condition (40) for waveforms with arbitrary and from (34). Proof of Lemma 1: See Appendix VI. Lemma 2: Assume a MIMO radar system with widely separated transmitters and widely separated receivers satisfies Assumptions 1–2, then any symmetrical antenna placement will solve the zero gradient condition (40). Proof of Lemma 2: See Appendix VII. Lemma 3: Assume a MIMO radar system with widely separated transmitters and widely separated receivers satisfies Assumptions 1–2, then all symmetric antenna placements give exactly the same performance.

Consequently, all symmetric antenna placements give exactly the same performance. Theorem 2: Assume a MIMO radar system with widely separated transmitters and widely separated receivers satisfies Assumptions 1–2, then for waveforms with arbitrary and from (34), any symmetric antenna placement is optimal, assuming an optimum placement exists. The relative positions between the transmit and receive antennas have no impact on the performance. Proof of Theorem 2: According to Theorem 1, no solutions other than symmetric placements can be optimal, and according to Lemma 3, all symmetric placements give exactly the same performance. Therefore, any symmetric placement must be optimal. If you substitute the placements of the form given in (35), and . one finds the resulting expression is independent of Therefore, the relative positions between the transmit and receive antennas have no impact on the performance. We further conclude from (42) that for a MIMO radar system with symmetric (optimal) placement, the velocity estimation acor the product of the curacy can be improved by increasing . number of the transmit and receive antennas Singular FIM Cases: When an FIM is singular, one cannot accurately estimate the desired quantities without additional constrains. To understand the problems that can result, consider a single pair of collocated transmit and receive antennas. Since they can measure, at most, only a single Doppler frequency, they cannot possibly measure a two-dimensional velocity. Further, if the target is moving in a direction perpendicular to the vector between the antennas and the target, then no velocity measurement can be made. Improper antenna placements could result in singular FIMs, from (14) a linear comif they make any row or column of bination of the others. To simplify matters, we focus on the case of orthogonal signals. Investigations for other cases follow similar steps. Plugging (31)–(33) into (23), under Assumptions 1–2, a singular FIM appears when the first two rows/columns of the FIM are proportional to each other, that is

(43)

For example, when a radar system has one transmitter and one receiver


87

Fig. 3. MSE of ML estimate and CRB for a 3 2 3 MIMO radar with f = 9 ; = 110 ; = 351 g, f = 20 ; = 70 ; = 205 g, fR = Fig. 2. MSE of ML estimate and CRB for a 3 2 3 MIMO radar with f = 4000; R = 8290; R = 6000g m, fR = 8000; R = 7300; R = 5000g 50 ; = 81 ; = 230 g, f = 167 ; = 65 ; = 80 g, fR = m. 4000; R = 8290; R = 6000g m, fR = 8000; R = 7300; R = 5000g m.

thus (43) is satisfied, resulting in a singular FIM. For another example that leads to singular FIM, consider a radar system with multiple transmit and receive antennas, but all the transmitters and and/or receivers are co-located, then for all and . Hence, (43) is achieved by

Intuitively, these two systems provide only one propagation can be obpath, from which only one Doppler frequency served. This provides essentially a single observation, which is certainly not enough to estimate the two-dimensional quantity . Fig. 4. MSE of ML estimate and CRB for a 2 2 2 MIMO radar with f = 50 ; = 81 g, f = 65 ; = 80 g, fR = 4000; R = 8290g m, fR = 7300; R = 5000g m.

VI. NUMERICAL RESULTS

In this section, the CRBs are computed for some example MIMO radar systems. In Figs. 2–4, the empirical MSE of the ML estimate is compared to the CRB. The figures are distinguished by different antenna placements and different numbers of antennas. We use Monte Carlo simulations with 2000 iterations per SNR value to obtain the MSE. We vary in (1), and label our plots using a scaled version of . In particular, we use , where is the maximum distance between the target and radar antennas. Assume a moving target with velocity m/s is in the resolution

cell of interest. Suppose the signal time duration6 is s and the carrier frequency is GHz. A set of pulsed sinusoidal signals , with frequency increment Hz are employed for transmission (see Appendix II). The sampling frequency used is KHz. In Figs. 2 and 3, an MIMO radar is considered with two different antenna placements. In Fig. 2, the transmitters are located at while the receivers are located at . In Fig. 3, the transmitters are located at and the receivers are 6The signal time duration is set to a large number to shorten the simulation time for getting the MSE curve of the ML estimates.

88


Fig. 5. Antenna positions corresponding to Figs. 2–4. Triangles represent transmitters while circles represent receivers. (a) Antenna positions in Fig. 2. (b) Antenna positions in Fig. 3. (c) Antenna positions in Fig. 4.

located at . The distance between the transmit antennas and the target are assumed to be m, while the distance between the receive antennas and the target are m. In Fig. 4, a 2 2 MIMO radar, with fewer transmit and receive antennas, is considered. The angles defining the transmit and and receive antenna positions are , respectively. The transmitters and receivers are located at m and m away from the target. The simulation results show that the MSE of the ML estimates approach the CRBs as SNR becomes large. This is consistent with the theoretical asymptotic efficiency of ML estimates, and also supports the correctness of the CRB derived in the previous section. It can be observed that the CRBs in Figs. 2 and 3 are smaller than those in Fig. 4. This indicates that MIMO radars with more antennas can achieve better performance. It is also found that the estimated error bounds (variance) for are larger than those for in Figs. 2 and 4, while the estimated error bounds (variance) for are larger than that for in Fig. 3. These results show that the estimation accuracy for and are quite different and depend on the chosen antenna positions. The corresponding antenna positions in these examples are illustrated in Fig. 5, where triangles represent transmitters and circles represent receivers. Projecting the antenna locations to the and axes we find that in (a) and (c) the antennas are more widely spread along the -dimension. In this case, the antennas are more sensitive to the target motion in the -direction, thus the estimation accuracy of is more accurate than that for in Figs. 2 and 4. On the contrary, in (b), the antennas are more widely spread along the -dimension, for the same reason the estimation accuracy for is more accurate than that for in Fig. 3. Thus, the results in Figs. 2–4 show that, generally, the more we spread out the antennas, the better the performance. Such findings are partially explained by our optimum placement results from Section V, which also suggest it is best to spread out the antennas as much as possible. A. Cramer–Rao Bound Analysis In this subsection, we provide some further analysis using the CRB result in (24). For simplicity, we assume all antennas are located at approximately the same distance from the target, and let m for all and .

Fig. 6. CRB for a 4

2 3 MIMO radar system with non/symmetric placement.

R = R = 6500 m; Nonsymmetric placement 1 (nonsym 1) f = 10 ; = 120 ; = 34 ; = 80 g, f = 199 ; = 245 ; = 300 g. Nonsymmetric placement 2 (nonsym 2) f = 140 ; = 150 ; = 51 ; = 115 g, f = 325 ; = 60 ; = 300 g. Symmetric placement (sym) [ ; ; ; ] = [0 ; 90 ; 180 ; 270 ] + , [ ; ; ] = [0 ; 120 ; 240 ] + . Assume the velocity of the target is m/s. The transmitted signals employ pulsed sinusoidal signals , with frequency increment for their low-pass equivalent, time duration ms. The carrier frequency is GHz and the sampling frequency is KHz. In Fig. 6, we plot the CRBs for a 4 3 MIMO radar system with different placements. Let KHz. We first allocate the antennas non-symmetrically. The directions of the transmit and receive antennas in case 1 and are . In case 2, they are located at and . Then we consider symmetric placement, where the transmit antennas are at and the receive antennas are at , where and are arbitrary. Fig. 6 shows that case 1 has the highest CRB. When the antennas are symmetrically placed, the CRB achieves the lowest value.7 In Fig. 7, we consider the CRBs for a 4 4 MIMO radar system with optimally (symmetrically) placed antennas transmitting orthogonal signals. In this example, we also use pulsed sinusoidal signals , . The frequency increment is chosen as KHz, such that the signals are approximately orthogonal. All antennas have the same distance to the target m, , . We vary the signal time duration from to , and to , where ms. The sampling frequency is KHz. As 7Note that with antenna positions fixed, one can compute the CRBs for different target positions. The results seem not very sensitive to the target position in cases we looked at.


Fig. 7. Signal time duration effect on velocity estimation (using optimal placement). Signal time duration varies from T to 2T , and to 5T . T = 6:4 ms.

shown in the figure, the estimation accuracy is improved by increasing the signal time duration. It is found that for all cases in this example, the CRB curves for and estimates fall on top of each other. This is not surprising if we take a look at (41), which implies that the accuracies of the velocity estimation in the and axes are equal when the antennas are optimally placed. Fig. 8 compares the CRBs for MIMO radar systems using a different number of transmit and receive antennas . Assuming orthogonal signals and optimal placement are adopted, the parameter settings in this example are the same as in Fig. 7, while the signal time duration is fixed at . It is seen that the larger the product of , the lower the CRB values. For example, the 8 8 system with has the highest value of the four, hence the lowest CRB curve. It is noticed that the 5 5 system performs better than the 7 3 system, even though both cases have a total of ten antennas. Also, the 8 8 system having a total of 16 antennas outperforms the 3 16 system with 19 antennas in all. Realizing the fact that it is the product of transmitter and receiver numbers that determines the estimation accuracy is significant. It enables a MIMO radar system to achieve better performance with lower complexity (fewer antennas). Finally, in Fig. 9, we give an example to show that our results are also valid for multiple pulse waveforms. The transmitted waveforms are given by , , where is the pulse width, is the pulse repetition time, is the number of pulses during one coherent processing interval (CPI), is the time duration of one CPI, denotes the duty cycle, and is the rectangular window function. Here, we choose s, ms, , ms, MHz, and GHz. Consider a 4 3 MIMO radar system. Suppose all antennas have the same distance to the target m, , . The CRB curves are plotted for both symmetrical (S) and nonsymmetrical (N) placement. In the nonsymmet-

89

Fig. 8. Effect of transmitter and receiver numbers (M mation (using optimal placement).

2 N ) on velocity esti-

2

Fig. 9. CRB for a 4 3 MIMO radar applying multiple pulse signals with T = 2 s, T = 0:2 ms, K = 32, T = 6:4 ms, 1f = 5 MHz, f = 1 GHz; R = R = 6500 m. Antennas angles are either nonsymmetrical (N) = 140 ; = 150 ; = 51 ; = 115 , = 325 ; = 60 ; = 300 or symmetrical (S) [ ; ; ; ] = [0 ; 90 ; 180 ; 270 ] + , [ ; ; ] = [0 ; 120 ; 240 ]+ , where and can take on any values.

g

gf

f

rical cases, the angles defining the transmit and receive antenna positions are and , respectively. The figure shows similar results as we already discussed in the previous pulsed sinusoidal signal cases. VII. DISCUSSIONS AND CONCLUSION In the signal model considered, the unknown target reflecare assumed to be constant during the observation tivities interval. As an extension, one may assume that the target reflectivity fluctuates during the observation interval (from pulse varies with pulse number, which to pulse), such that each , where , and denotes the number gives of pulses in the observation interval. In this case, the parameter

90


vector

is a

column vector given by . The dimension of the resulting FIM is then . The extension involves similar calculations to those given in this paper for simpler cases, whereas for large values of the computation complexity increases greatly. As another extension to the analysis in Section V-A, suppose we no longer constrain all antennas to be the same distance from the target. Suppose the possible region for us to place radar antennas is the outside of a circle with diameter , which surrounds the target. Since we often cannot get too close to a target, this is a reasonable scenario. In order to use minimum power, it is easy to see we should place all the antennas at the same disfrom the target. If Assumptions 1 and 2 are also satistance fied, then the results in Section V apply for this scenario if we replace with . In this paper, the CRB for the target velocity estimation has been derived when a MIMO radar is deployed with widely dispersed antennas. A signal model was formulated. The equations specifying the ML estimates of target velocity were developed, and the mean square error of the estimates was compared to the CRB for some examples. These comparisons substantiate the CRB results. Optimization strategies were discussed for MIMO radar velocity estimation. The conditions for optimal antenna placement were derived. For MIMO radar transmitting orthogonal signals, symmetrically placing all the transmitters and all the receivers gives the best achievable performance. The relative position between the transmitters and receivers does not affect the results under orthogonal signals. The symmetric placement result shows that larger sensor separations, corresponding to angular spreads, lead to better performance, which also justifies some of our numerical findings. It was shown by both analysis and simulation that for any MIMO radar system with optimal placement, it is possible to improve the velocity estimation accuracy by increasing the effective signal time duration or the product of the number of transmit and receive antennas. APPENDIX I CAMERA-RAO BOUND FOR RANDOM REFLECTIVITY USING THE APPROACH FROM [6] In this section, we derive the CRB for velocity estimation when the reflectivity coefficients are asand sumed as random variables. Denote , then in . The FIM conditioned on is (5) can be written as

(44) where derivative with respect to

. Plugging (3) into (7) and taking and we find

(45)

where

and

The CRB conditioned on reflectivity

is

CRB CRB CRB

(46)

Further, we introduce several approaches to get rid of the dependence on the nuisance parameter from the CRBs. Following the discussion in [6], we can compute the average CRB (ACRB) by averaging (46) with respect to ACRB ACRB

CRB CRB

(47) (48)

where the expectation is taken with respect to the pdf . We can also compute the outage CRB (OCRB) [6] for a given probability based on (46), considering the CRBs in (46) depend on the realization of OCRB OCRB

CRB CRB

(49) (50)

Another approach is to compute the modified CRB (MCRB) [31] based on (45) MCRB

(51)

MCRB

(52)

Note that since (according to Jensen’s inequality or [32]), the ACRB is tighter than the MCRB. The true CRB (TCRB) is tighter than the above CRBs discussed, although it is very hard to evaluate TCRB

(53)

TCRB

(54)

Note that the inner expectation in (53) or (54) is taken with re, and the outer expectation is taken with spect to the pdf . respect to the pdf


91

time delay difference. Note that when is greater than or approximates , the overlap part of these two signals vanishes, which implies (56) is close to 0 and the signals can be considered as orthogonal. Therefore, it is possible for the two delayed , which makes signals to be nonorthogonal only if the following discussion necessary. . Thus, We denote the overlap interval as

Fig. 10. Frequency spectrum for transmitted signals (solid curves) and the effective parts of their time delayed versions (dotted curves).

APPENDIX II FREQUENCY SPREAD SIGNALS In this appendix, we describe one class of signals which could be employed. Under appropriate assumptions, these signals approximately satisfy the conditions of Assumption 1 for orthogonal signals. We discuss these assumptions. These assumptions essentially ignore small sidelobes in the Fourier transform of these signals to approximate them with truncated (in frequency domain) signals. After discussing the assumptions, we compute the CRB using the untruncated signals and also by assuming ideal orthogonal signals and show the two CRBs are extremely close in cases of practical interest. Consider a set of pulsed sinusoidal signals8 otherwise

where and represent the overlapping parts of and , the time duration of which are both . More specifically, suppose , then we have

The spectrum of and , i.e., and , are plotted by dotted curves in Fig. 10. In a similar way, for the time delayed signals, the orthogonality is approximately maintained if

(55)

, denotes the signal time duration, and is the frequency increment between and . The approximate bandwidth of each signal is . The spectrum of and , and , are shown in Fig. 10. Note that in this section, it is always assumed . According to Parseval’s theorem that where

(56)

By observing the spectrum, we see that when , the main lobes of and are sufficiently sepand (56) approximates 0, which arated, so that and approximately orthogonal. Here, we are makes ignoring contributions from small sidelobes which is similar to , truncating the signals in the frequency domain. As if the frequency increment satisfies

(58) Since

, the inequality is maintained by requiring . As , the condition be, which is the same as in (57). Hence, the comes requirement for (57) guarantees (58). That is, if (57) is met, the time delayed signals are orthogonal. Next, we further discuss the Doppler shift restriction for orthogonality. The effective time delayed and Doppler and shifted signals are given by . The frequency spectrum of the efand , i.e., and fective time delayed signals , are replotted in Fig. 11 using solid curves, the center fre. After Doppler frequency shift, quency of which are and the spectrum of the effective time delayed and Doppler shifted and . signals are They are shown in Fig. 11 using dashed curves. Thus, the is , while center frequency of the effective signal becomes . Clearly, the center frequency of , that is provided

(57) then any two signals from the signal set are approximately orthogonal. and are time delayed by and Assume that . The nonzero part of the delayed signal occupies , and the nonzero part of occupies . Hence, the time duration of the overlap be, where denotes the tween them is 8Windowing can be applied in practice but we omit discussion of this well studied topic.

or (59) the overlapping part between and can be ignored. Thus, the orthogonality is well approximated. Note that since (58) is satisfied, we have . Hence, the term

92


Fig. 11. Frequency spectrum for the time-delayed equivalent signals (solid curves) and their Doppler shifted versions (dotted curves).

Fig. 13. CRBs for frequency spread signals (FS) and ideal orthogonal approx , (b) T : s , (c) s imation (orth), where (a) T T : s s < . , and (d) T

= 6 67 =

Fig. 12. CRBs for frequency spread signals (FS) and ideal orthogonal approx, (b) f = T , (c) f imation (orth), where (a) f =T , (d) f =T , (e) f =T , and (f) f =T .

1 =2

1 =0 1 =12 1 = 10 1 = 50

1 =1

in (59) can be ignored, resulting in or . Numerical Investigations: We now examine the performance of a system using frequency spread signals by comparing its exact CRB results with the CRBs obtained using ideal orthogonal signals as an approximation. We assume the GHz, following system parameters: carrier frequency target velocity (7.75,13.42) m/s (in the resolution cell of interest), distances between transmit antennas and the target are m, distances between receive antennas and the target are m, directions from the transmit and receive antenna positions to the target are and . Suppose the signal time ms. In Fig. 12, the CRB results for the duration is frequency spread signals (FS) are plotted to compare with those obtained using ideal orthogonal signals (orth). In this example, is true for every . In (a), the frequency increment , which means all the transmitted signals are the same. As shown in the figure, the resulting two CRB curves are far . apart for this identical signals case. In (b), we set Condition (57) is still not met and the transmitted signals are nonorthogonal. It can be observed that the CRB curves are not close in this case either. In (c) to (f), the frequency increment to , , and . We find that varies from

= 500 =5

= 26 7 =

when , the CRB curves are close to each other for the scales shown in Fig. 12 and thus the orthogonality is increases, (57) is more closely approximately achieved. As approximated; therefore, the CRB curves get closer. Results similar to Fig. 12 have been observed in other cases with different antenna placement and a different number of antennas. While the remaining parameters are unchanged, we next consider a new transmit antenna placement with m. Thus the maximum and minimum time-delay differences s and s, respecfor this scenario are such that (57) is met for each case. In tively. Let s, which is much greater than the Fig. 13(a), we set and hence all . In this maximum time delay difference case, we need to check (58) and (59) for the orthogonality of the time delayed and Doppler shifted signals. Since (57) is met, (58) and (59) are guaranteed. The results show that the two CRB curves are very close. The time durations for (b) to (d) are s , s , and s . approach or In these cases, we observe that though some are greater than , those two CRB curves are close to each other in every figure. This is due to (57) being satisfied, which ensures the orthogonality being well approximated.

SIMPLIFICATION OF

APPENDIX III , , FOR ORTHOGONAL SIGNALS

The integral in (11) can be written as

Considering the support of is independent of the function

(or ), and is differen-


tiable with respect to (or ), we can interchange the order of integration and differentiation [34] to get

93

summation terms contained in the minuend and the subtrahend. ,9 then Assume

and

(60)

According to Jensen’s Inequality, we have (61) When the orthogonal signals assumption (Assumption 1) holds, , and therefore the above integral equals zero for for For

where the equality holds if and only if Plugging (60) into (61) we obtain

, it is straightforward to get

.

(62) where the equality holds if and only if

Observing (21), is composed of a bunch of weighted inteare zero under Assumption 1. grals terms. The one for Thus,

(63) Since

, (63) is equivalent to

and The integral in (22) can be written as

(64)

For a MIMO radar system with widely separated widely separated receivers, transmitters and and , where , thus (64) cannot be met. Therefore, the equality condition in (62) cannot be satisfied and the equal sign is eliminated. Further considering the square of any real number is non-negative, we obtain (65)

Interchanging the order of integration and differentiation gives

Now we consider the other parts of (36). As discussed in (34) with (positive pulse Section V-A, (from their definitions) so width ) and (66) From (65) and (66) we find which equals zero under Assumption 1 for . Therefore, all summands on the right-hand side of (22) vanish except that . Therefore, for and therefore (67) THE PROOF OF

APPENDIX IV , , AND

In this section, we prove some inequalities related to (36), (37), and (38) in Section V. by comparing the size of In the first step, we prove the minuend and the subtrahend of (36). We first examine the

Similarly, we can prove

(68) 9The

symbol vect(1) denotes the matrix vectorization operator.

94


Finally, we prove . Looking back at (24), we find is a positive multiple of a variance , that which is always non-negative, so (76) (69) and According to (67) and (69), we obtain (70) APPENDIX V ELEMENTS OF THE GRADIENT Under the assumptions given in Section V, we calculate the . elements of the gradient Focusing first on the th transmit antenna yields

(77) (71)

Substituting (72)–(74) into (71) and after some manipulation, we obtain

where the terms , , and are computed and as follows. Plugging in , , (36)–(38) and taking derivative with respect to , we can obtain

(78) where

(72)

(79) In the same way, we can calculate the elements of the gradient with respect to the th receive antenna

(73) and where

(74) where

(75)


and

95

is equivalent to

Substitute (81) into (80), thus

(84) , Note that for a certain placement, the coefficients , , and in (84) do not vary with . Next, we will show by contradiction that the zero gradient condition requires all these coefficients equal to zero. Let us assume that the gradient is equal to zero but the coefficients in (84) are not all equal to zero. For example, we assume , , , and . In this case, (84) is equivalent to (85) Consequently, every element of

can be obtained.

APPENDIX VI PROOF OF LEMMA 1

Zero gradient means that holds for . Equivalently, it implies (85) holds for all . That is

In this section, we show that under the assumptions given in Lemma 1, symmetric placement is the necessary condition for the gradient being equal to zero, i.e., no other placement will solve the zero gradient condition. requires The gradient being zero for and for . We first investigate the derivatives with respect to transmit antennas. The derivative with respect to the th transmit antenna in is given in (78). Since it has been shown that is equivalent to (70),

Note that as and do not vary with , for a certain placement, they are fixed for all the equations in (86). As is assumed in Lemma 1, the transmit antennas are widely separated, thus . The range of each is . Hence, (86) implies that the following equation:

(80)

(87)

can be

with respect to has at least distinct solutions in . This leads to a contradiction, since with fixed nonzero coefficients and , the trigonometric equation (87) has only [36]. Therefore, for any , the two solutions in , , , and placement with cannot have zero gradient. Similarly, we can show that other cases, when any one or , , , and are not equal to some of the terms zero, will lead to a contradiction and thus fail to solve the zero . gradient condition for any The analysis for the derivatives with respect to receive antennas is similar. transmit anTherefore, for MIMO radar with any receive antennas, the placement that entennas and any sures zero gradient must simultaneously satisfy

Plugging written as

,

, and

from (75)–(77) into (79),

(81) where

(82)

(86)

and and

(88) (89) (90)

Applying (88)–(90) to (82) and (83), we have

(91) (83)

96


where and are tunable parameters for radar waveforms. To make sure (91) holds for arbitrary and , following the similar analysis as discussed above, it is required that

and

(92)

Hence, the placement with zero gradient must satisfy all the equations in (92). It can be shown that for odd and , only or , either symmetric placement satisfy (92); for even symmetric placement or nonsymmetric placement with angle pairs that differ by (i.e., or ) will satisfy (92). To see this, we can imagine drawing pictures in the complex (take the transmit antennas for explane. Consider ample) as a vector on the unit circle. Using Euler’s formula [37], the sum of these vectors being equal to zero (i.e., ) ensures and . Next we will show that the possible nonsymmetric placement with angle pairs that differ by can be ruled out, thus only symmetric placement can solve the zero gradient condition. Plug, and (92) ging into , , (36)–(38), we have

Since ditions and

(67) and (68) (See Appendix IV), the con(88) and (89) give . That is

(93)

According to summation formulas (see Appendix VIII for a proof)

for

(94)

for

(95)

where is an arbitrary constant, symmetric placement ensures (93). However, the nonsymmetric placement with angle pairs and that differ by does not satisfy (93) for any should be ruled out. , the placement with As a consequence, for any zero gradient must satisfy the equations in both (92) and (93), and thus symmetric placement (35) is the only solution. APPENDIX VII PROOF OF LEMMA 2 In this section, we prove that under the assumptions given in Lemma 2, symmetric placement is a sufficient condition for the gradient being equal to zero, i.e., symmetric placement will solve the zero gradient condition.


Assume symmetric placement is used such that , and where the transmit and receive antenna positions are given in (35). We first observe the derivative with respect to the th transmit (78). Based on the expression in (78), we antenna by showing that , are going to prove , and for symmetric placement. , , and in In the first step, we calculate the terms (79). Use the summation formula (75)–(77) contained in (see Appendix VIII for a proof)

for where and

97

and

(96)

is an arbitrary constant, then for in (35) satisfy

, (102)

(97) ,

Thus, the corresponding terms in equal to zero, and we obtain

,

For symmetric placement , where and , the summation terms in (100)–(102) can be treated by using the summation formulas (94)–(96). It can be derived that, for

(75)–(77) are

(103) and hence

and

(98)

Substituting (98) into (79), we obtain (99) Next we compute and we obtain

and . Plugging into , , and

(104) (99), Thus, we have proved that (104), and (103). Plugging them in (78), we find the derivative with respect to any transmit antenna is equal to zero for symmetric placement

in (36)–(38), for

(105)

Similarly, we can show that the derivative with respect to any receive antenna is equal to zero for symmetric placement for

(100)

(106)

The gradient of the objective function at the symmetric placement, which is made up of these terms in (105) and (106), is therefore equal to zero

(107) (101)

Consequently, the symmetric placement does make the gradient equal to zero.

98


APPENDIX VIII SUMMATION FORMULAS In this section, we give proofs for some useful summation formulas. First, we introduce two basic formulas derived from [38]

and Let , , and According to (108), we have

(111)

(108) , where integer

(112)

. and

and Therefore, for arbitrary

(109) and

(113)

REFERENCES

(110) , , and Let According to (108), we have

and Therefore, for arbitrary

, where integer

.

[1] A. M. Haimovich, R. S. Blum, and L. Cimini, “MIMO radar with widely separated antennas,” IEEE Signal Process. Mag., vol. 25, no. 1, pp. 116–129, Jan. 2008. [2] E. Fishler, A. M. Haimovich, R. S. Blum, D. Chizhik, L. Cimini, and R. Valenzuela, “MIMO radar: An idea whose time has come,” in Proc. IEEE Radar Conf., Apr. 2004, pp. 71–78. [3] E. Fishler, A. M. Haimovich, R. S. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Spatial diversity in radars—Models and detection performance,” IEEE Trans. Signal Process., vol. 54, no. 3, pp. 823–838, Mar. 2006. [4] N. Lehmann, A. M. Haimovich, R. S. Blum, and L. Cimini, “High resolution capabilities of MIMO radar,” in Proc. 40th Asilomar Conf. Signal, Syst. Comput., Nov. 2006, pp. 25–30. [5] A. De Maio and M. Lops, “Design principles of MIMO radar detectors,” IEEE Trans. Aerosp. Electron. Syst., vol. 43, no. 3, pp. 886–898, Jul. 2007. [6] N. Lehmann, E. Fishler, A. M. Haimovich, R. S. Blum, L. Cimini, and R. Valenzuela, “Evaluation of transmit diversity in MIMO radar direction finding,” IEEE Trans. Signal Process., vol. 55, no. 5, pp. 2215–2225, May 2007. [7] Q. He, N. Lehmann, R. S. Blum, and A. M. Haimovich, “MIMO radar moving target detection in homogeneous clutter,” IEEE Trans. Aerosp. Electron. Syst., 2010, to be published. [8] H. Godrich, A. M. Haimovich, and R. S. Blum, “Concepts and Applications of a MIMO Radar System With Widely Separated Antennas,” in MIMO Radar Signal Processing, J. Li and P. Stoica, Eds. New York: Wiley, 2008. [9] J. Li and P. Stoica, “MIMO radar with colocated antennas,” IEEE Signal Process. Mag., vol. 24, no. 5, pp. 106–114, Sep. 2007. [10] C.-Y. Chen and P. P. Vaidyanathan, “MIMO radar space-time adaptive processing using prolate spheroidal wave functions,” IEEE Trans. Signal Process., vol. 56, no. 2, pp. 623–635, Feb. 2008. [11] A. Leshem, O. Naparstek, and A. Nehorai, “Information theoretic adaptive radar waveform design for multiple extended targets,” IEEE J. Sel. Topics Signal Process., vol. 1, no. 1, pp. 42–55, Jun. 2007.


[12] T. Aittomaki and V. Koivunen, “Low-complexity method for transmit beamforming in MIMO radars,” in IEEE Int. Conf. Acoust., Speech, Signal Process., Apr. 2007, vol. 2, pp. 15–20. [13] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output (MIMO) radar and imaging: Degrees of freedom and resolution,” in Proc. 37th Asilomar Conf. Signal, Syst., Comput., Nov. 2003, pp. 54–59. [14] D. J. Rabideau and P. Parker, “Ubiquitous MIMO digital array radar,” in Proc. 37th Asilomar Conf. Signal, Syst., Comput., Nov. 2003, pp. 1057–1064. [15] F. C. Robey, S. Coutts, D. Werikle, C. McHarg, and K. Cuomo, “MIMO radar theory and experimental results,” in Proc. 38th Asilomar Conf. Signal, Syst., Comput., Nov. 2004, pp. 300–304. [16] J. Li, L. Xu, P. Stoica, K. W. Forsythe, and D. W. Bliss, “Range compression and waveform optimization for MIMO radar: A CRB based study,” IEEE Trans. Signal Process., vol. 56, no. 1, pp. 218–232, Jan. 2008. [17] J. Ward, “Cramer–Rao bounds for target angle and Doppler estimation with space-time adaptive processing radar,” in Proc. 29th Asilomar Conf. Signal, Syst., Comput., Nov. 1995, pp. 1198–1202. [18] S. R. Dooley and A. K. Nandi, “Adaptive time delay and Doppler shift estimation for narrowband signals,” IEE Proc. Radar, Sonar Navig., vol. 146, no. 5, pp. 243–250, Oct. 1999. [19] A. Dogandzic and A. Nehorai, “Estimating range, velocity, and direction with a radar array,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), Mar. 1999, pp. 2773–2776. [20] A. Dogandzic and A. Nehorai, “Cramer–Rao bounds for estimating range, velocity, and direction with an active array,” IEEE Trans. Signal Process., vol. 49, no. 6, pp. 1122–1137, Jun. 2001. [21] Y. Yang and R. S. Blum, “MIMO radar waveform design based on mutual information and minimum mean-square error estimation,” IEEE Trans. Aerosp. Electron. Syst., vol. 43, no. 1, pp. 330–343, Jan. 2007. [22] Y. Yang and R. S. Blum, “Minimax robust MIMO radar waveform design,” IEEE J. Sel. Topics Signal Process., vol. 1, pp. 147–155, Jun. 2007. [23] L. Xu, J. Li, P. Stoica, K. W. Forsythe, and D. W. Bliss, “Waveform optimization for MIMO radar: A Cramer–Rao bound based study,” in Proc. 2007 IEEE Int. Conf. Acoust., Speech, Signal Process., Apr. 2007, pp. 917–920. [24] K. W. Forsythe and D. W. Bliss, “Waveform correlation and optimization issues for MIMO radar,” in Proc. 39th Asilomar Conf. Signal, Syst., Comput., Nov. 2005, pp. 1306–1310. [25] G. San Antonio, D. R. Fuhrmann, and F. C. Robey, “MIMO radar ambiguity functions,” IEEE J. Sel. Topics Signal Process., vol. 1, no. 1, pp. 167–177, Jun. 2007. [26] H. V. Poor, An Introduction to Signal Detection and Estimation, 2nd ed. New York: Dowden & Culver, 1994. [27] S. Coutts, K. Cuomo, J. McHarg, F. Robey, and D. Weikle, “Distributed coherent aperture measurements for next generation BMD radar,” in Proc. IEEE Workshop Sens. Array Multichannel Signal Process., 2006, pp. 390–393. [28] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, 1st ed. Englewood Cliffs, NJ: Prentice-Hall, 1993. [29] J. A. Fessler and A. O. Hero, “Space-alternating generalized expectation-maximization algorithm,” IEEE Trans. Signal Process., vol. 42, no. 10, pp. 2664–2677, Oct. 1994. [30] H. L. Van Trees, Optimum Array Processing. New York: Wiley, 2002. [31] A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer–Rao bound and its application to synchronization problems,” IEEE Trans. Commun., vol. 42, no. 2, pp. 1391–1399, Feb. 1994. [32] B. Z. Bobrovsky, E. Mayer-Wolf, and M. Zakai, “Some classes of global Cramer–Rao bounds,” Ann. Statist., vol. 15, no. 4, pp. 1421–1438, 1987. [33] W. J. Duncan, “Some devices for the solution of large sets of simultaneous linear equations,” London, Edinburgh, Dublin Philosoph. Mag. J. Sci., vol. 7, no. 35, pp. 660–670, 1944. [34] E. W. Weisstein, CRC Concise Encyclopedia of Mathematics. Boca Raton, FL: CRC, 2003. [35] H. L. Van Trees, Detection, Estimation and Modulation Theory: Part III. New York: Wiley, 1971. [36] W. R. Ransom, “Solutions of a trigonometric equation,” Amer. Math. Monthly, vol. 56, no. 6, pp. 402–403, Jun. 1949.

99

[37] M. A. Moskowitz, DA Course in Complex Analysis in One Variable. Singapore: World Scientific, 2002. [38] K. S. Kunz, Numerical Analysis. New York: McGraw-Hill, 1957.

Qian He (S’09) received the B.Eng. degree in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2004. She is currently working towards the Ph.D. degree in the Electronic Engineering Department, UESTC. From 2007 to 2009, she was a Visiting Scholar in the Electrical and Computer Engineering Department, Lehigh University, Bethlehem, PA. Her current research interests include statistical signal processing, array signal processing, and their applications in radar and communication systems. Ms. He is a member of Sigma Xi. She was awarded the Distinguished Graduate of UESTC Award in 2004. She was awarded a scholarship under the State Scholarship Fund from 2007 to 2009.

Rick S. Blum (S’83–M’84–SM’94–F’05) received the B.S. degree in electrical engineering from the Pennsylvania State University, State College, in 1984 and the M.S. and Ph.D. degrees in electrical engineering from the University of Pennsylvania, Philadelphia, in 1987 and 1991. From 1984 to 1991, he was a Member of Technical Staff at General Electric (GE) Aerospace, Valley Forge, PA, and he graduated from the GE’s Advanced Course in Engineering. Since 1991, he has been with the Electrical and Computer Engineering Department at Lehigh University, Bethlehem, PA, where he is currently a Professor and holds the Robert W. Wieseman Chaired Professorship in Electrical Engineering. His research interests include communications, sensor networking, sensor processing, and related topics in the areas of signal processing and communications. Dr. Blum is currently an Associate Editor for the IEEE COMMUNICATIONS LETTERS and he is on editorial board for the Journal of Advances in Information Fusion of the International Society of Information Fusion. He was an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and edited special issues for this journal and for JSAC. He was a member of the Signal Processing for Communications Technical Committee of the IEEE Signal Processing Society and is a member of the Communications Theory Technical Committee of the IEEE Communication Society. He is also on the awards Committee of the IEEE Communication Society. He is an IEEE Third Millennium Medal winner, a member of Eta Kappa Nu and Sigma Xi, and holds several patents. He was awarded an ONR Young Investigator Award in 1997 and an NSF Research Initiation Award in 1992. His IEEE Fellow Citation “for scientific contributions to detection, data fusion, and signal processing with multiple sensors” acknowledges some early contributions to the field of sensor networking.

Hana Godrich (S’91–M’04) received the B.Sc. degree in electrical engineering from The Technion–Israeli Institute of Technology, Haifa, in 1987, and the M.Sc. degree in electrical engineering from Ben-Gurion University, Beer-Sheva, Israel, in 1993. She is currently working towards the Ph.D. degree in electrical engineering at the New Jersey Institute of Technology (NJIT), Newark. From 1987 to 1990, she was a Research Engineer with the IAF. From 1993 to 1996, she served as the ATE Manager of Scitex, Herzelia, Israel. From 1997 to 2003, she was a partner and consultant at Enetpower, Israel, focusing on mission-critical facilities design. Her research interests include multiple-input multiple-output systems, statistical signal processing, communication, and information theory.

100


Alexander M. Haimovich (S’82–M’87) received the B.Sc. degree in electrical engineering from The Technion–Israeli Institute of Technology, Haifa, in 1977, the M.Sc. degree in electrical engineering from Drexel University, Philadelphia, in 1983, and the Ph.D. degree in systems from the University of Pennsylvania, Philadelphia, in 1989. He is a Professor of Electrical and Computer Engineering at the New Jersey Institute of Technology, Newark. His research interests include MIMO radar and communication systems, wireless networks, and source localization. Prof. Haimovich is currently a Guest Editor for the Special Issue on MIMO Radar of the IEEE JOURNAL FOR SPECIAL TOPICS IN SIGNAL PROCESSING.