Noncoherent MIMO Radar for Location and Velocity ... - CiteSeerX

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 7, JULY 2010

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Noncoherent MIMO Radar for Location and Velocity Estimation: More Antennas Means Better Performance Qian He, Student Member, IEEE, Rick S. Blum, Fellow, IEEE, and Alexander M. Haimovich, Member, IEEE

Abstract—This paper presents an analysis of the joint estimation of target location and velocity using a multiple-input multiple-output (MIMO) radar employing noncoherent processing for a complex Gaussian extended target. A MIMO radar transmit and receive antennas is considered. To provide with insight, we focus on a simplified case first, assuming orthogonal waveforms, temporally and spatially white noise-plus-clutter, and independent reflection coefficients. Under these simplifying assumptions, the maximum-likelihood (ML) estimate is analyzed, and a theorem demonstrating the asymptotic consistency, large , of the ML estimate is provided. Numerical investigations, given later, indicate similar behavior for some reasonable cases violating the simplifying assumptions. In these initial investigations, we study unconstrained systems, in terms of complexity and energy, where each added transmit antenna employs a fixed energy so that the total transmitted energy is allowed to increase as we increase the number of transmit antennas. Following this, we also look at constrained systems, where the total system energy and complexity are fixed. To approximate systems of fixed complexity in an abstract way, we restrict the total number of antennas employed to be fixed. Here, we show numerical examples which indicate a preference for receive antennas, similar to MIMO communications, but where systems with multiple transmit antennas yield the smallest possible mean-square error (MSE). The joint Cramér–Rao bound (CRB) is calculated and the MSE of the ML estimate is analyzed. It is shown for some specific numerical examples that the signal-to-clutter-plus-noise ratio (SCNR) threshold, indicating the SCNRs above which the MSE of the ML estimate is reasonably close to the CRB, can be lowered by increasing . The noncoherent MIMO radar ambiguity function (AF) is developed in two different ways and illustrated by examples. It is shown for some specific examples that the size of the product controls the levels of the sidelobes of the AF. Manuscript received June 03, 2009; accepted February 10, 2010. Date of publication March 01, 2010; date of current version June 16, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ljubisa Stankovic. Q. He and R. S. Blum were supported by the Air Force Research Laboratory under Agreement FA9550-09-1-0576, the National Science Foundation under Grant CCF-0829958, the U.S. Army Research Office under Grant W911NF-08-1-0449, and by the China Scholarship Council. A. M. Haimovich was supported in part by the U.S. Air Force Office of Scientific Research under Agreement FA9550-09-1-0303. Q. He is with the Electrical Engineering Department, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China, and also with the Lehigh University, Bethlehem, PA 18015 USA (e-mail: [email protected]/[email protected]). R. S. Blum is with the Electrical and Computer Engineeing Department, Lehigh University, Bethlehem, PA 18015 USA (e-mail: [email protected]. edu). A. M. Haimovich is with the Electrical and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: [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/TSP.2010.2044613

Index Terms—Ambiguity function (AF), Cramér–Rao bound (CRB), joint estimation, mean-square error (MSE), noncoherent MIMO radar.

I. INTRODUCTION

T

HE study of multiple-input multiple-output (MIMO) radar has received much attention from researchers in recent years. Target estimation performance for MIMO radar has been investigated for several applications based on different antenna deployments and target models. For MIMO radar with widely separated antennas, both coherent processing and noncoherent processing have been considered. In the noncoherent case, time synchronization is required, but phase synchronization is not required. This simplifies implementation. A fluctuating extended target model is proposed in [1], and the spatial diversity gain is analyzed for noncoherent processing. The Cramér–Rao bound (CRB) for target localization in a noncoherent MIMO radar is discussed in [2] and [3]. The direction finding performance is investigated for four types of Swerling scattering models in [4]. Optimal joint target detection and location estimation is studied in [5]. In the coherent case, both time synchronization and phase synchronization are required. In [6] and [7], the CRB for localization estimation performance is studied for a stationary point target for coherent MIMO radar, and the CRB for velocity estimation for a moving point target is presented in [8]. The average CRB for bearing estimation in MIMO radar is analyzed in [9]. In [10], the outage CRB is introduced to evaluate MIMO radar direction finding performance. For MIMO radar with co-located antennas, target estimation techniques are investigated in [11] and [12]. In this paper, we discuss the joint estimation of target location and velocity for noncoherent MIMO radar for the first time. A MIMO radar with transmit and receive antennas is considered. In our formulation, we assume the target has already been detected using a radar with a fairly coarse set of range cells. This is often the case in realistic radar settings. Then an estimation procedure, which we study in this paper, will attempt to produce a more accurate target location. It is assumed that using the detection information allows us to estimate the propagation losses with reasonable accuracy. Thus, one can develop specific relative values for these propagation losses based on this. The signals traveling over all paths from transmit to receive antennas could be modeled as complex Gaussian random variables with different variances whose relative values are known, due to the target reflection. We derive the joint CRB and show that the joint CRB decreases as the number of antennas increases in a few examples.

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The CRB study1 is important. However CRB has limitations that a mean-square error (MSE) analysis for the particular approach we intend to employ, the maximum-likelihood (ML) estimator, does not. For our problem, as we will show, the CRB indicates the approximate MSE for the ML estimator in the asymptotic region, where the product of the number of transmit and receive antennas is large. This will be shown in our paper, along with a particular consistency result which shows that both the CRB and the MSE of the ML estimate improve with increasingly large , implying perfectly accurate estimates as . However, in the nonasymptotic region, the MSE can deviate far away from the CRB. Such things have been observed in the past for similar problems, e.g., [15]–[19]. Another important point that is nicely illustrated in our results is that the relationship between the CRB and the MSE of the ML estimate also depends on the SCNR as might be expected. Thus, for a fixed and sufficiently large SCNR, we find the CRB and the MSE of the ML estimate are quite close, although they do not approach one another with increasingly large SCNR in the problem studied here. The fact that these two important performance measures do not approach each other as SCNR has been observed before in cases with in, for example, [16]. On the other hand, for sufficiently small SCNR the two performance metrics (CRB and MSE) are generally much different. We will show plots that illustrate these two ranges of SCNRs and define an SCNR threshold which divides these two regions. Given the previous discussions, it is necessary to be informed of when a system operates with performance close to the CRB. To this end, the analysis of the MSE is required. In [16], the method of interval errors (MIE) is proposed to predict the MSE of the ML estimate. The MIE is also presented in [17], and is used in [18] and [19] to analyze the MSE for direction of arrival estimation. Although the MIE has been shown to be useful for approximating MSE, it still demands complicated numerical calculations, nearly as complicated as numerically computing MSE. In this paper, we numerically analyze the MSE of the ML estimator, which is the most essential and useful criterion to guide system design. We study the SCNR threshold phenomenon and show the effect of and on the MSE performance. Numerical results for particular examples show that increasing lowers the MSE for SCNRs above the SCNR threshold and also lowers the SCNR thresholds. While our analytical results on consistency help explain these numerical results, the gains seen for relatively small , for sufficient large SCNR, are particularly enlightening. The signal model initially employed assumes each transmit antenna uses a fixed energy so that total energy increases when the number of transmit antennas is increased. However, we later consider systems with energy and complexity constraints. In particular we consider cases with fixed total system energy such that total system energy is held fixed as additional transmit antennas are added. To approximate systems of fixed complexity in an abstract way, we restrict the total number of antennas employed to be fixed. Here we show numerical examples which indicate a preference for receive antennas, similar to MIMO communications, but where systems with multiple transmit antennas yield the smallest possible MSE.


The ambiguity function (AF) is an important tool for radar analysis that reveals the inherent trade-off between the ability to estimate the range (time delay) and range rate (Doppler) of a moving target. The AF for MIMO radar is discussed in [20] and [21] for coherent processing. The noncoherent AF for MIMO radar is defined in [22]. Here, we develop the noncoherent MIMO radar AF in two ways, by relating it to the ML estimate and the squared norm difference between two target returns. For some specific examples, we will show that using more antennas gives a more favorable AF, which implies better performance. The remainder of the paper is organized as follows. The initial signal model for jointly estimating the location and velocity of a complex Gaussian extended target is introduced in Section II, considering the simplifying assumptions of temporally and spatially white noise-plus-clutter, independent reflections, and orthogonal waveforms. Under these assumptions, the ML estimate is analyzed, and the asymptotic behavior of the ML estimate is presented in a theorem in Section III. Section IV derives the joint velocity-position CRB. The MSE of the ML estimate is studied in Section V, where numerical results are provided. Then the noncoherent AF is developed in Section VI. Section VII extends the discussion to more general cases, where the simplifying assumptions are removed. Finally, Section VIII concludes the work. Throughout this paper, bold symbols are used to denote matrices or vectors. II. SIGNAL MODEL transmitters and reConsider a MIMO radar that has ceivers. The th, transmitter and the th, receiver are placed at known positions and respectively, in a two-dimensional Cartesian coordinate system. The lowpass equivalent of the signal transmitted , where denotes the transfrom the th transmitter is mitted energy for each transmit antenna, and the waveform is normalized . The reflection coefficient corresponding to the th path is modeled as a zero-mean complex Gaussian random variable as per [1], which is constant over the observation interval. This is often called the Swerling 1 model. The relative values of the for various , are known, assuming the target has already been detected using a radar with a fairly coarse set of range cells. Assume the target location and velocity are and , which are assumed to be deterministic unknowns. Further, the following assumptions are made to simplify analysis. Assumption 1: Assume that the transmitted signals are approximately orthogonal2 if if

and maintain approximate orthogonality for time delays , and Doppler shifts , of interest (see Appendix for some justification for this approximation)

if if

1Alternate

derivations seem possible, possibly using approaches like those given in [13] and [14] for different applications.

(1)

2The

superscript 3 denotes the conjugate operator.

(2)

HE et al.: NONCOHERENT MIMO RADAR FOR LOCATION AND VELOCITY ESTIMATION

such that the signals contributed from different transmitters can be separated at each receiver. Assumption 2: The noise-plus-clutter corresponding to the th path is a temporally white, zero-mean complex , Gaussian random process where is a constant, and is a unit impulse function. The noise-plus-clutter components are spatially white, such that if or . Since scaling the observations changes nothing, we set without loss of generality. Assumption 3: The antennas are sufficiently separated [9] so that each path provides an independent observation of the target, such that the reflection coefficients are independent for different and/or . Assumption 2 might be accomplished by prefiltering. For example, assuming zero-mean Gaussian noise-plus-clutter with known or accurately estimated covariance matrix, the noiseplus-clutter can be temporally and spatially whitened without any loss of optimality, by a space-time whitening filter applied before the ML processing. Such processing is common in the popular radar space-time processing research and development. We note that in such cases a proper predistortion procedure should be considered for waveform design to preserve the desired signal properties at the output of the whitening filter. In Section VII, we discuss cases which remove Assumptions 1–3. Under these assumptions, the received signal model at receiver due to the signal transmitted from transmitter is (3) and represent the time delay and Doppler shift, where respectively, corresponding to the th path. The time delay is a function of the unknown target location

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III. MAXIMUM-LIKELIHOOD ESTIMATION As discussed in [26], the ML estimate of the unknown parameter vector can be found by examining the likelihood ratio for corresponding to the target presthe hypothesis pair, with ence hypothesis modeled in (3) and corresponding to the noise only hypothesis, given by

(7) denotes the observed signal at receiver due to the where signal transmitted from transmitter . The log-likelihood ratio obtained from (7) is

(8)

(9) where

is not dependent on .

A. ML Estimate of the Joint Time-Delay and Doppler Estimation Under Assumptions 1–3, and assume the noise-plus-clutter and the reflection coefficients are mutually independent, the joint likelihood ratio can be expressed as the product of the individual likelihood ratios

(10) (4) where denotes the speed of light, the distance between the target and the th transmitter, and the distance between the target and the th receiver. The Doppler shift is a function of the unknown target location and velocity

where (11) collects the observed signals from the entire set of the received antennas. The joint log-likelihood ratio is thereby

(5) where denotes the wavelength of the carrier. Define an unknown parameter vector that collects the parameters to be estimated3 (6)

(12)

, and is a four-dimensional space consisting where of the possible values of . Next we consider ML estimation of . 3The

superscript y denotes the transpose operator.

(13)

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where is not dependent on . Thus, the ML estimate of the unknown parameter vector is

(14)

It follows that the ML estimate is asymptotically unbiased and its variance must asymptotically approach the CRB as . Proof of Theorem 1: Firstly we show that the term to be maximized on the right-hand side of (16) converges almost surely to its mean value. Since the , as per [24] for and are independent random variables with bounded mean and bounded variance, by the Kolmogorov Strong Law of Large Numbers [34, pp. 67]

(15) No further simplification seems possible for the ML estimator from (15), and a four dimensional search over is required for obtaining the ML estimate of in (6) numerically. It is worth noting that increasing the number of transmit or receive antennas, or , imposes more system complexity and computational load. There may be some suboptimum approaches that may reduce complexity. This is a topic for future paper.

(18) that is

(19)

B. Asymptotic Analysis In this section we show that MIMO radar with a sufficiently large (the product of the number of transmit and receive antennas) achieves perfect performance in the sense that the ML asymptotically converges to the true parameter estimate value , under Assumptions 1–3. This seems to indicate very good performance with a large number of antennas. We will give some discussion on the benefits of using more antennas for nonasymptotic cases in Section V by analyzing the MSE. Considering that multiplying the right-hand side of (14) by a , does not change , one can write the constant, say ML estimate as (16) Theorem 1: For a MIMO radar with transmit and receive antennas, under Assumptions 1–3 and assume the observations follow the assumed model in (3), the ML estimate as described in (15) converges almost surely to the true parameter value when is sufficiently large (17)

Thus, it can be inferred from (19) that when the number of antennas is sufficiently large

(20) Next we show that the right-hand side of (20) is maximized at the true value of the target parameter . Considering that the observations follow the assumed model in (3) for a target inducing the delays , and Doppler frequencies , we can write (9) as (21), shown at the bottom of the page, where denotes the clutter-plus-noise at the output of the matched filter for the th path, is a Gaussian random variable with zero mean and variance , and (22)

(21)


is a function of . Since the reflection coefficients and the noise components are mutually independent, the quantity we are attempting to maximize in (20) becomes

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IV. JOINT CRAMÉR–RAO BOUND In this section, we provide the CRB for jointly estimating the and velocity under Assumptions target location 1–3. The first step in obtaining the CRB is to compute the Fisher information matrix (FIM), which is a 4 4 matrix related to the second-order derivatives of the joint log-likelihood [26]

(23) which is a deterministic function of . For signals that satisfy Assumption 1, applying the Cauchy–Schwarz inequality to (22) we obtain

Considering the likelihood (7) is explicitly a function of and , but implicitly as a function of , we define a new parameter vector (28) According to the chain rule, the Fisher information matrix (FIM) can be derived by

(24) (29)

(25) only at , i.e., where the equality holds for all and . Then, due to the fact that is independent of and due to the positivity of the other terms in (23), we have

(26) and equality occurs only at . Thus, the left-hand side of (26) must have a unique maximum at . To say this another way, the right-hand side of (23) is maximized only when each positive term in the sum is individually maximized, since this is possible, and this occurs at . Therefore,

(27) Substituting (27) into (20), we get the correct estimate in the limit

Since the estimate converges almost surely to a deterministic unknown when is sufficiently large, then so does its mean. Thus, the estimate is asymptotically unbiased. Further, since the estimate approaches the correct value, then the estimate implied in the CRB calculation, since it is optimal, must also. Thus, the variance of our estimate must converge almost surely to the CRB for large enough . In the following parts, we derive the CRB of the estimate analytically, and show the asymptotic approach of the MSE to the CRB by providing numerical examples.

We first compute

. Recalling (28) and (6), we obtain

the elements of which define the following symbols based on (4) and (5)

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Note that , , , , , are determined by the target location and velocity, as well as the position of the radar aninto a block matrix tennas. For later use, we partition

(35) and4

where , , are submatrices, and is a zero matrix. The in (29) can be derived from (13) using

which is a a block matrix

matrix. We write

(36)

in the form of

where , , , and are matrices. Note that contains the second-order derivatives with respect to for all possible and , and contain the second-order derivatives with respect to both and for all possible and , and contains the second-order derivatives with respect to for all possible and . Carrying out the computation, we get (30) where is an identity matrix, denotes the Hardmard product, denotes the Kronecker product, and

represents the Fourier transform of . where After lengthy algebraic manipulations, the expression of the FIM can be obtained as given in (37), shown at the bottom of the page. The CRBs for the estimates of the unknown parameters are determined by the diagonal elements of the inverse of the FIM evaluated at the true parameter value

For any nonsingular FIM, a closed-form expression for the CRB can be easily obtained (though we are not going to write down the lengthy formula here), since the analytical form of can be easily derived from (37) using Cramer’s rule. The size and increase as we increase or . This will of increase the computational complexity of the CRB computation. Note that, in our formulation, the size of the FIM, , does not change with or , which is fortunate since we need to invert the FIM. In Section V, we provide numerical examples which show the MSE approaches the CRB for large .

(31) is an

diagonal matrix. Further (32)

and (33) where , , and are dependent on the characteristics of the received waveforms as described by (34)

V. MEAN-SQUARE ERROR INVESTIGATION The best achievable performance, smallest variance estimates of any unbiased estimate, is indicated by the CRB. The MSE of the ML estimate is close to the CRB only if certain conditions are satisfied. Knowing when this occurs is important for system design, which requires the analysis of the MSE. As shown in Fig. 1, a threshold phenomenon exists for the MSE of the nonlinear estimator. The SCNR threshold is defined as the SCNR at which the MSE performance changes slope drastically. Above the threshold, is the high SCNR region, where the estimation errors are small and the MSE is close to the CRB. Below the threshold, the MSE rises quickly with decreasing SCNR and 4The

symbol =f1g denotes the imaginary operator.

(37)


Fig. 1. Schematic illustration of the threshold phenomenon for the MSE of nonlinear estimation.

Fig. 2. MSE (in m for position and (m/s) for velocity) versus SCNR for a MIMO radar with 2 3 antennas uniformly placed in [0; 2 ). T = 0:1.

2

the MSE generally deviates significantly from the CRB. Various numerical examples will be provided in this section. In these examples, we first consider single pulse signals and then multiple pulse signals. A. Single Pulse Signals Assume the lowpass equivalents of the transmitted waveforms are frequency spread single Gaussian pulse signals , where is proportional to the pulsewidth and is the frequency increment between and , here assumed large enough so that Assumption 1 holds. As a specific example, assume a target moving with velocity m/s is present at m. Assume the reflection coefficients have approximately the same variance, such that . For simplicity, let so that the signal-to-clutter-plus-noise ratio is SCNR . Now we can simply vary to vary SCNR.

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Fig. 3. MSE (in m for position and (m/s) for velocity) versus SCNR for a MIMO radar with 5 4 antennas uniformly placed in [0; 2 ). T = 0:1.

2

The performance of three MIMO radar systems with a are compared in different number of antennas Figs. 2–4. The transmit antennas are uniformly distributed in , i.e., the angle of the th transmit antenna is , , and the receive antennas are also uniformly distributed in , i.e., the angle of the th receive antenna is , , where the angles are measured with respect to the horizontal axis. The systems related to Figs. 2–4 have 2 3, 5 4, and 9 9 antennas, respectively. Choose for the transmitted single Gaussian pulse. 5 Set carrier frequency 1 GHz and the frequency increment to 500 KHz. Suppose the distance between each antenna and the origin is 7000 m. The CRB and the MSE curves of the ML estimate versus SCNR are investigated under Assumptions 1–3. We see that increasing the number of antennas decreases the CRB uniformly, and lowers the threshold (the SCNR at which MSE performance changes slope drastically, see arrows in the figure) to a smaller value. We also find that the MSE curves get more favorable (closer to CRB in the asymptotic region) with more antennas. These results show that more antennas means better performance in both asymptotic (large ) and nonasymptotic cases. We note that the performance of the ML estimate can be quite poor, especially if for cases with very small puts us well below the SCNR threshold. This the given can be seen in most of the figures after Fig. 1. In the 5 4 MIMO radar system, suppose the transmit antennas are placed uniformly in , i.e., , , and the receive antennas are also placed uniformly in , , , where i.e., the angles are measured with respect to the horizontal axis. 5In these examples, the CRBs and MSEs for target velocity estimation are smaller than those for target location estimation. This is due to the choice of the parameter T for the single Gaussian pulse. Note that a larger T implies narrower bandwidth and wider pulsewidth which gives good velocity estimation accuracy but is not as good for the location estimation. If one reduces T to a certain value which is more favorable for location estimation, then the curves for location estimation error will be lower than those for velocity estimation.

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Fig. 4. MSE (in m for position and (m/s) for velocity) versus SCNR for the 9 9 MIMO radar with transmit antennas placed uniformly in [0; 2 ) and receive antennas also placed uniformly in [0; 2 ).

2

, and one receiver placed at 0 . The other parameters are the same in Figs. 2–4. Both the CRB and MSE curves look similar for the two figures, which illustrates that systems with similar antenna placements and the same value of the product of the number of transmitters and receivers yields similar CRB and MSE performance. We have also looked at many other cases with the uniform placement , and the results examples, equally spaced antennas over are similar. These results can also be seen to be intuitively reasonable from the form of the FIM given in (37). Further, note that both of the systems in Fig. 6 have a total of 31 antennas, but both have performance inferior to the 9 9 system shown in Fig. 4, which has fewer total antennas but a larger . This again demonstrates the importance of the product of the number of transmit and receive antennas. Later in this section we consider cases where the total system energy is fixed and not increased with the number of transmit antennas. We feel these two comparisons, nonfixed and fixed total system energy, are useful and very informative when taken together. B. Multiple Pulse Signals

Fig. 5. MSE (in m for position and (m/s) for velocity) versus SCNR for the 5 4 MIMO radar with transmit antennas placed uniformly in [=6; 5=6] and receive antennas also placed uniformly in [=6; 5=6].

2

Other parameters are assumed the same as before. The CRB and MSE curves for this case are shown in Fig. 5. It is seen that the performance is degraded compared with Fig. 3, where the same number of antennas are spread fully around . This seems to justify that larger sensor separations, corresponding to larger angular spreads, lead to better performance. For the model, we have initially assumed, where the energy per transmit antenna is fixed, there is a symmetry between transmit and receive antennas which we now discuss. The system considered in Fig. 6(a) has one transmitter placed at direction 0 and 30 receivers uniformly placed in , i.e., , . The system considered in Fig. 6(b) has 30 transmitters uniformly placed in , i.e.,

Assume the lowpass equivalents of the transmitted waveforms are frequency spread Gaussian pulse trains, composed by first repeating several Gaussian pulses with pulse repeto develop a pulse train. Then the entire tition interval pulse train is modulated in amplitude by a broad Gaussian envelope. The resulting Gaussian pulse train can be de, where scribed by is the narrow Gaussian pulse with pulsewidth , is the broad Gaussian envelope with pulsewidth , and is the frequency increment between and , again assumed large enough so that Assumption 1 applies. Fig. 7 shows the MSE and CRB curves for a 5 4 MIMO radar with antennas uniformly spaced in , assuming , and . Compared with Fig. 3, Fig. 7 has lower CRB and smaller SCNR threshold. In the latter case, the performance is improved due to the use of multiple pulses. With the same parameters, the curves for a 9 9 MIMO radar is plotted in Fig. 8. Again, it is observed that using more antennas leads to better performance by comparing with Fig. 7. Further, multiple pulses are also helpful, as we see by comparing Fig. 8 with Fig. 4. It is seen in Figs. 2 and 3 that the CRBs and the MSEs decrease with the number of antennas. Now, for fixed SCNR, we plot the MSE curves versus antenna numbers in Figs. 9 and 10. Consider the MIMO radar has 1 transmit antenna placed at . direction 0 and receive antenna uniformly placed in Note that virtually identical curves, which fall right on top of the curves shown, are obtained if we use one receive antenna and increase instead of . Assume SCNR 3 and 10 dB. Other parameters are the same as those in Fig. 2. We see that in both figures, the MSE decreases with the number of antennas. Further, in Fig. 9 the MSE starts departing from the CRB when , while in Fig. 10 the MSE starts departing from the CRB when . Here, we see that the exact beginning of the asymptotic region moves lower for higher SCNR. This shows the dependence on both SCNR and as we discussed previously.


Fig. 6. MSE (in m for position and (m/s) for velocity) versus SCNR for (a) 1

Fig. 7. MSE (in m for position and (m/s) for velocity) versus SCNR for a MIMO radar with 5 4 antennas uniformly placed in [0; 2 ), using Gaussian pulse train with T = 0:1, T = 0:5, and T = 2.

2

C. Systems With Energy and Complexity Constraints So far, we have not said much about the system resources employed to achieve these performance gains. Here, we consider constraints on both total system energy and total system complexity. These are both very reasonable and realistic constraints. Thus, we assume fixed total system energy, so that as we add transmit antennas the energy per transmit antenna, which is the same for each transmit antenna, must decrease. Bounding complexity in a simple and abstract way, with a complexity definition which will not be different for each different implementation and the exact components chosen in the system design, is more complicated. We employ an approximate method. To approximate systems of fixed complexity in an abstract way, we

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2 30 and (b) 30 2 1 MIMO radars. T = 0:1.

Fig. 8. MSE (in m for position and (m/s) for velocity) versus SCNR for a MIMO radar with 9 9 antennas uniformly placed in [0; 2 ), using Gaussian pulse train with T = 0:1, T = 0:5, and T = 2.

2

restrict the total number of antennas employed, the sum of the transmit and receive antennas, to be fixed. Consider a MIMO radar system that has a fixed total number of antennas, . The position of the transmit antennas is m, , and the position of the receive antennas m, , is . The other parameters are set the same as Fig. 2. Assume the SCNR 20 dB. First consider the unconstrained energy case. As we did in the previous section, consider that the transmitted energy per transmit antenna is fixed, such that increasing the number of transmit antennas increases the total in Fig. 11. transmitted energy. The CRB is plotted versus It is seen that, in this case, the effects of the number of and

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Fig. 9. MSE (in m for position and (m/s) for velocity) versus the product of the number of transmit and receive antennas. SCNR = 3 dB. T = 0:1.

Fig. 10. MSE (in m for position and (m/s) for velocity) versus the product of the number of transmit and receive antennas. SCNR = 10 dB. T = 0:1.

are symmetric. Since and are symmetric, one can get (the important quantity) for the same number of larger total antennas by employing both and larger than one. In fact the maximum of for and fixed is achieved if and then . This follows from since this implies, after algebra, that . Now consider the constrained energy case. Since the total transmitted energy is fixed, increasing decreases the energy per transmit antenna in this case. The CRB is plotted versus in Fig. 12 assuming the other parameters, besides energy per transmit antenna, are unchanged from those adopted in Fig. 11. It is observed that the effects of and are asymmetric. For small , increasing (i.e., decreasing ) improves performance. This occurs until the energy is spread too thinly between the transmit antennas. After this, further increases in tend to degrade the CRB. The results in Fig. 12 indicate that multiple transmit antennas, , achieve the best performance if


Fig. 11. CRB (in m for position and (m/s) for velocity) versus M for a MIMO radar with a total number of M + N = 50 antennas uniformly placed in [0; 2 ). The transmitted energy per transmit antenna is fixed, such that adding M increases the total transmitted energy. T = 0:1. SCNR = 20 dB. Red line indicates the minimum.

Fig. 12. CRB (in m for position and (m/s) for velocity) versus M for a MIMO radar with a total number of M + N = 50 antennas uniformly placed in [0; 2 ). The total transmitted energy is fixed, such that adding M decreases the energy per transmit antenna. T = 0:1. SCNR = 20 dB. Red line indicates the minimum.

is fixed. The minimum CRB in this example occurs at small , indicated by the red lines. We see that the difference in performance between the best case, , and that for in Fig. 12 is quite large, where the former is more than 12 times better than the later for either position or velocity was best estimation. We tried many cases and found that for many reasonable antenna placements. Thus, without considering system resources employed, then performance can always be improved by increasing as per Theorem 1. When we do consider system resources employed, then the story is a bit more complicated, but in fact the conclusions are quite obvious. First, if we fix the total transmitted energy (sum over all the transmitters), then we expect an asymmetry between and . The reason is that as we increase we must split the total energy among more antennas and so the energy for each transmit antenna is smaller. This is not the case


when we increase . If we limit complexity by fixing , larger than one our results indicate that it is better to pick which seems very reasonable. Taken together, these resource constrains, the total transmitted energy and total number of antennas, imply one should generally pick both and larger than 1 but one might want to pick slightly larger than . In fact these suggestions are quite similar to those made in MIMO communications by the group at Bell Labs that proposed the Blast system [25]. Next we present an alternative view of the capability of ML estimation with MIMO radar which employs techniques that are very popular in the radar community.

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in (38) measures the likelihood that an The quantity ML processor believes the target corresponds to path delays and , for and , Doppler shifts when the actual path delays are and the actual Doppler shifts are zero. Of course, the parts of (38) that do not depend on the delays and Doppler shifts are not important in this consideration. Here, we lump them into the constant . For convenience, we change variables to obtain

(39) VI. NONCOHERENT AMBIGUITY FUNCTION Since the pioneering work of Woodward [31], the AF has been used to study the resolution and ambiguity of radar signals, as well as the estimation accuracy and clutter suppression ability determined by the radar signals. In [31], the AF is introduced as a graphical method to evaluate the capabilities of the ML estimate in a very intuitive way. These ideas are also very nicely explained in [26] and the AF is very frequently employed in the radar community. For example, in [32], the AF is defined as the time response of a filter matched to a given finite energy signal when the signal is received with a certain delay and Doppler shift relative to the nominal values expected by the filter. In the recent years, the AF has been further extended to MIMO radar, e.g., [20]–[22]. The performance of the ML estimate can be related to the AF. Actually, the CRB only describes the shape of the AF around its maximum and this information influences the MSE in the high is large. The other parts of the AF, SCNR region or when such as the existence of sidelobes, which can not be captured by the CRB, influence the MSE performance in the low SCNR region when is not large. A. Alternative Definitions In this section, under Assumptions 1–3, we develop the noncoherent AF for MIMO radar in two different ways. Note that energy unconstrained case is considered here, but the energy constrained case can be obtained easily from the results. First, we derive the AF by relating it to the ML estimate. For the noiseplus-clutter free received signals, assume a stationary target is present at the origin of the Cartesian coordinate system, causing time delay and zero Doppler shift to the signal transmitted over the th path, thus, the received signal is represented by , if the noise-plus-clutter term in (3) is omitted. Further assume that the reflection coefficients have approximately the same variance, such that . Now consider the log-likelihood ratio in (13), which assumes the exand Doppler shift for pected signals have arbitrary delay the th path, corresponding to a target at general with velocity . Substitute the just described noise-plus-clutter free received signals for in (13) to obtain

(38)

where

denotes the time difference for the th path. Further, we can define any suitable normalized version of (39) to be the AF. Defined in this way, it is clear that the AF is closely related to the performance of a processor that computes ML estimates. Thus, we define the AF as

where is introduced as a normalization factor. However, such an AF will depend on the exact realization of the for , , which will certainly complicate things. We define a simplified AF by taking the expected value of the previous equation with respect to and considering the special case, where is constant for , . This simplified AF, after scaling, becomes6

(40) Alternatively, we can introduce the AF function in a different way. Consider two targets, one located at the origin inducing path delays for and and zero Doppler shifts, and the other located at with velocity inducing path delays and Doppler shifts . The received signal for unit reflection7 in the first case is

and in the second case is

6There are clearly other ways to handle the reflection coefficients and this is a good topic for future study. The method described here is chosen to promote simplicity. 7Similar to the analysis just described, we aim to eliminate dependence on the reflection coefficients.

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Let

and . The squared norm difference between the two received signal vectors and is described by8

Fig. 13. Further simplified scenario for visualizing MIMO radar AF.

(41)

Fig. 13, where the triangle denotes the th transmitter, the circle denotes the th receiver, and the pentagrams denote possible targets in the monitored area. Plugging (43) and (44) into (42) gives the simplified MIMO AF

denotes the time difference for the th where path between the two target returns. Note that the first two integrals in (41) represent the signal energy, which is a constant. The third integral is a varying function of . The ability to resolve the two signals can be measured by the third term. Folwith lowing the lead of Woodword [31], we replace the to define the noncoherent MIMO radar AF as (45)

(42)

Assume frequency spread single Gaussian pulse signals are used for transmission, whose complex envelope are . Plugging into (45), the simplified MIMO AF for the single Gaussian pulse is

Comparing (42) with (40), we find that the same noncoherent MIMO radar AF can be derived from either the ML estimate or the squared norm difference point of view. B. Simplified MIMO Radar Ambiguity Function To make the AF a simple two-dimensional function that we can plot and easily interpret, we assume , which only allows the target to move along the -axis. Thus, the MIMO AF can be plotted versus and in a three-dimensional figure. The expression for the MIMO AF can be simplified more by further assuming that the monitored area is relatively small and all antennas are located sufficiently far away. In this case, it can be proved that the , in the argument of the MIMO AF in (42) are approximately linearly related to and

(43) (44) where and are the look angles of the th transmitter and the th receiver, respectively. This scenario [28] is depicted in 8The

symbol