2.13.2.1 Joint location and velocity estimation

Let the signals observed by a MIMO radar system be

(13.11)

which is a realization of the random vector in (13.8), we hope to jointly estimate the target location and velocity in the maximum likelihood (ML) sense. As discussed in [15], the ML estimates of the unknown parameters can be found by examining the corresponding log-likelihood ratio. Stack the parameters of interest into a vector as follows:

(13.12)

and define a bigger vector

(13.13)

to include all the unknown parameters involved. Using the signal model in (13.8), it can be derived that the log-likelihood ratio with respect to is given by [16]

(13.14)

where is a constant independent of the parameters to be estimated,

(13.15)

and³

(13.16)

The vector can be regarded as the output of a matched filter which considers the correlations allowed in the given analysis. It can be shown that, for any value of , the ML estimate of (i.e., and ) is [16]

(13.17)

Then, substituting in (13.14), it yields

(13.18)

Note that in (13.18) we changed the notation in the parenthesis to emphasize that, after we have the ML estimate for , the parameters which need to be estimated are the elements of . Thus, the ML estimate of the unknown parameter vector can be expressed as

(13.19)

It is worth noting that the estimator (13.19) combines signals from different antennas in a coherent way, but we skip the discussion for now and postpone the explanation after (13.22) until some simplifying assumptions are introduced.

In the previous discussions, the clutter-plus-noise was assumed to be temporally white but possibly spatially colored. In order to simplify the analysis, in the following, we introduce two additional assumptions: orthogonal transmitted signals and spatially white clutter-plus-noise. Leveraging these assumptions, we are able to simplify the problem to be tackled and provide a handful of analytical results, which can not only characterize some typical behaviors of the system when these assumptions are satisfied, but can as well render insight into the system behaviors for cases where these assumptions do not hold. Since these analytical results are relatively easy to obtain and explain, we are also able to shed light on the relationship between the system performance and a few important system parameters. Due to the convenience afforded by these assumptions, we will use them repeatedly in the rest of this chapter.

Under Assumption 1, it can be obtained that . Assumption 2 leads to , which can be regarded as a result of perfect whitening if the covariance matrix of , called previously, can be accurately estimated. Then, we have and .

Applying these results into (13.14), we get a simplified log-likelihood ratio

(13.20)

where represents the observed signal corresponding to the received signal model for the th path

(13.21)

Likewise, applying the simplification assumption to (13.19) gives the simplified ML estimate

(13.22)

In the estimator in (13.22), phase shifts imposed on various paths have a measurable impact on the estimation output via the term . Intuitively, we want the terms inside the to add together in phase to maximize (13.22), so as to achieve high estimation performance. This is the motivation for the coherent processing (see Appendix A.2 for a distinction between coherent and noncoherent processing). To make the best use of the phase information, phase synchronization is required, so that all transmitters and receivers employ a common phase reference. Some approaches for achieving phase synchronization are described in Section 2.13.6. Note that under Assumptions 1 and 2, the coherent processing is optimal for the coherent target reflection model in Figure 13.1.

The mean square error (MSE), the average squared difference between an estimate and the true value of the parameter being estimated, is a useful metric for forecasting the performance of an estimator. The MSE of any unbiased estimator is lower bounded by the Cramer-Rao bound (CRB). Since attaining the MSE is often computationally expensive, the CRB that indicates the best MSE an estimator can provide, serves as an important tool for evaluating the estimation performance and system configuration. In (13.22), we have derived the ML estimator for the coherent MIMO radar joint location and velocity estimation problem. It is known that the ML estimator is asymptotically unbiased, and the MSE of the ML estimate asymptotically approaches the CRB. Thus, we can derive the CRBs for the parameters of interest to provide an approximate MSE of the corresponding ML estimates in the asymptotic region. The Fisher information matrix (FIM) with respect to can be derived from the log-likelihood function in (13.20) as follows [15,17]

Under Assumptions 1 and 2, according to the derivations provided in Appendix A.3, we obtain the FIM as shown below:

(13.23)

where

with the terms , , determined by the target position and velocity, and the antenna positions. The terms , , and in are dependent on the characteristics of the received waveforms

(13.24)

(13.25)

(13.26)

(13.27)

and

(13.28)

where represents the Fourier transform of . The CRBs for the estimates of the unknown target locations and velocities are the first four diagonal elements of the inverse of the FIM

(13.29)

The CRBs can be used to optimize system configurations with respect to certain parameters. Collect the parameters of interest () in a vector as . These parameters could be, for example, the number of antennas, the antenna placement, the waveform parameters, and so forth. Suppose we require the optimum system configuration design to minimize a metric formed by the weighted sum of the CRBs of the estimates of , and . Then, the value of that optimizes this metric can be expressed as

(13.30)

where are weighting factors and represents the feasible set of . Numerical methods may be needed to solve this problem. A concrete example of finding the optimum antenna placement for the MIMO radar velocity estimation is provided in [18], where only and are involved and equal weighting is considered, and it is shown analytically that under certain conditions only symmetrical placement can be optimum. Interested readers are referred to [18] for more details.

2.13.2.2 Coherent ambiguity function

Besides the CRB, the ambiguity function (AF) is another useful tool for evaluating the estimation performance for radar systems. The performance of the ML estimate can be reflected by the shape of the AF. Actually, the CRB describes the shape of the AF around its maximum and this information influences the MSE in the high SCNR region or when is large. Other aspects of the AF, including the existence of sidelobes which can not be captured by the CRB, also influence the MSE performance in the low SCNR region when is not large.

Under Assumptions 1 and 2, we develop the AF for coherent MIMO radar system. 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 clutter-plus-noise free received signal can be expressed as

(13.31)

where can be obtained by substituting and into (13.2). Consider the log-likelihood ratio in (13.20), which assumes the expected signals have arbitrary delay and Doppler shift for the th path, corresponding to a target at location with velocity . Substituting the just described clutter-plus-noise free received signals for in (13.20), we obtain

(13.32)

where denotes the time difference for the th path. The quantity in (13.32) measures the likelihood that a ML processor believes the target corresponds to path delays and Doppler shifts for and , when the actual path delays are and the actual Doppler shifts are zero. Of course, the parts of (13.32) that do not depend on the delays and Doppler shifts are not important in this consideration. Here we lump them into the constant . Further, one can define any suitable normalized version of (13.32) to be the AF. In this way, it is clear that the AF is closely related to the performance of a processor that computes ML estimates.⁴ Thus, let us define the coherent AF as

(13.33)

where we have replaced the with to define the coherent MIMO radar AF following the lead of Woodword [19], is introduced as a normalization factor, and the and in (13.33) are functions of and , respectively. An ideal AF has a single peak at and is zero elsewhere, which is however, impossible to realize. In the real world, the AF always comes with a non-zero width mainlobe and several sidelobes, and researchers endeavor to design a better waveform/system with a narrow mainlobe and lower sidelobe peaks.

A coherent MIMO radar AF is studied in [20] and excellent properties are found through numerical investigations.⁵ For some cases with a small number of antennas, advantages are discussed over noncoherent MIMO radar for target location estimation. The coherent AF provided in [20] can be considered as a special case of (13.33) by letting and , such that the resulting coherent AF is reduced to a function of . We refer interested readers to [20] for more details.

2.13.2.3 Localization with phase errors

The previous section talks about location and velocity estimation for MIMO radar with ideal coherent processing, where the target reflections follow the coherent model shown in Figure 13.1 and the phase is assumed to be perfectly aligned across sensors. However, the difficulty in realizing perfect phase synchronization may bring problems for coherent MIMO radar. In this section, we investigate the impact of phase errors (e.g., due to imperfect phase synchronization) on the estimation performance. The focus will be on localization, so here we do not discuss the velocity estimation for simplicity. Assuming frequency synchronization, possibly through reception of a beacon, and white clutter-plus-noise, possibly due to estimating the covariance matrix and whitening the observations, we study the impact of phase errors on the target localization performance for coherent MIMO radar with widely dispersed antennas. We consider cases with sufficiently high SCNR such that the CRB provides accurate performance estimates. The CRBs with phase errors are computed in a few example cases and compared with the CRBs without phase errors. For these examples, using numerical results, we will show that at sufficiently high SCNR, phase errors degrade performance by only a relatively small amount.

Let and denote the phase errors induced by the th transmitter and the th receiver, respectively. Assume the phase errors are static (during the entire CPI) i.i.d. random variables with uniform distribution , where . Using (13.4) and further taking account of the phase errors, the received signal model at the th receiver can be expressed as

(13.34)

where denotes the deterministic, unknown complex reflection coefficient, is the transmitted signal at the th transmitter, is the total transmitted energy, and denotes the time delay between transmitter and receiver as per (13.2). The clutter-plus-noise in (13.34) is assumed to be white complex Gaussian with power spectral density (PSD) , which is assumed to be independent for different .

Collect parameters of interest in a vector as follows

(13.35)

which is assumed to be deterministic and unknown. The random phase errors, regarded as nuisance parameters, are collected in an vector

(13.36)

Based on the results in [23], it is easy to show that the likelihood ratio conditioned on the phase errors is

(13.37)

with defined as

(13.38)

where represents the signal observed over the observation interval , is a constant not dependent on , and . If we remove the conditioning on the phase errors by averaging them out, the likelihood ratio can be written as

(13.39)

which requires an dimensional multiple integration.

Assume an estimator is employed to produce an unbiased estimate, , of the parameters of interest from the observed signal vector . Then the variances of these estimates are bounded from below by the CRBs which are the diagonal elements of the inverse of the FIM. Following the derivation given in [23] for a similar single antenna system without phase errors, the FIM is given by

(13.40)

Since in (13.39) is explicitly a function of the time delays , we introduce an alternative parameter vector

(13.41)

which is a column vector with elements containing the time delays for every transmit-receive path and the real and imaginary parts of the target complex reflectivity. Using the chain rule, we have

(13.42)

The computations of and are provided in Appendix A.4. Substituting and into (13.42) yields the expression for . Thus, for a MIMO radar with transmit and receive antennas, assuming white complex Gaussian clutter-plus-noise and that the phase errors at the transmitters and receivers are static (during the entire CPI) i.i.d. random variables with uniform distribution in , the CRB for the estimates of the target location are the first two diagonal elements of the inverse of the FIM

2.13.3.1 Joint location and velocity estimation

Similar to the discussion in Section 2.13.2.1, suppose the observed signal vector , a realization of the in (13.45), is available. Our goal is to obtain the ML estimates of the target location and velocity based on the knowledge of . Using the signal model in (13.45), the likelihood ratio with respect to conditioned on the reflection coefficient can be derived

(13.47)

where is a constant not dependent on . In (13.47), and , where the integral over time operates on each element of the corresponding matrix. Employing the probability density function (pdf) of the reflection coefficient

(13.48)

to average out, we find the likelihood ratio as a function of the unknown parameter

(13.49)

where is a constant not dependent on implies the multidimensional integral over all possible values of , and

(13.50)

In the calculations, we have assumed that the matrices , , and have full rank, and are hence invertible. Note that the invertibility of is reasonable for MIMO radar with widely spread antennas and a target composed of a large number of scatters [5]. Further, due to thermal noise, is also likely a full-rank matrix. From (13.49), the log-likelihood ratio can be expressed as

(13.51)

Thus, the ML estimate of the unknown parameter vector is

(13.52)

Next we employ some simplifying assumptions,⁷ which are quite reasonable for cases with widely spaced antennas, which simplify matters.

Consider the case where Assumptions 1–3 hold true. Thus, , and the log-likelihood ratio in (13.51) is reduced to

(13.54)

where

(13.55)

In (13.55), is a constant not dependent on and represents the observed signal corresponding to the th path, which is modeled as

(13.56)

Accordingly, the ML estimate can be reduced to

(13.57)

Unlike (13.22), in the estimator from (13.57), the squared magnitude is taken before the summation. Thus, these terms will always add in phase and so we do not need all transmitters and receivers to be synchronized in phase. Therefore, this is a noncoherent processing (across sensors). Note that under Assumptions 1–3, the noncoherent processing is optimal for the noncoherent target reflection model in Figure 13.3.

Having obtained the ML estimate for the joint location and velocity estimation, let us introduce a theorem that shows how the number of antennas affects the estimation performance.

This theorem indicates that increasing the number of antennas properly can improve the estimation performance of noncoherent MIMO radar. Interested readers are referred to [24] for a proof of this theorem.

To evaluate the estimation performance of noncoherent MIMO radar, in the sequel, we develop the CRBs for the joint location and velocity estimation assuming that Assumptions 1–3 hold true. The first step in obtaining the Cramer-Rao bound is to compute the FIM, which is a four-dimensional matrix related to the second order derivatives of the log-likelihood ratio in (13.54)[15]

After lengthy algebraic manipulations, following the steps similar to those in Section 2.13.2, the expression of the FIM can be obtained as below [24]:

(13.59)

where ,

(13.60)

(13.61)

and

(13.62)

with representing the Fourier transform of . Thus, the Cramer-Rao bounds for the estimates of the unknown parameters can be determined by the diagonal elements of the inverse of the FIM as follows:

Note that for any non-singular FIM, a closed-form expression for the Cramer-Rao bound can be easily obtained, since the analytical form of can be derived from (13.71) using Cramer’s rule.

2.13.3.2 Noncoherent ambiguity function

Having studied the CRB, we now consider the ambiguity function (AF) for noncoherent MIMO radar. Under Assumptions 1–3, the noncoherent AF has been developed in [24] in two different ways yielding

(13.64)

where , with determined by the antenna positions which can be obtained by substituting into (13.2). Note that since the and in (13.64) are functions of and , respectively, the AF is essentially a four-dimensional function with respect to variables . Interested readers are referred to [24] for more detailed derivations.

2.13.4.1 Normalized mean square error difference

Let and denote the actual and estimated value of the parameter vector to be estimated . Define the total MSE of the joint estimation as

(13.69)

where defines a constant weighting vector with the same length as and represents the MSE corresponding to the th component of (i.e., ). Note here we consider a weighted MSE that allows one to give extra priority to some components over others. Of course, taking the weight as a constant vector is also possible, which gives equal weight to all the components.

For a detailed proof of the theorem, please refer to [16].

Based on the total MSEs of the joint estimation for the coherent and noncoherent MIMO radars, assuming a known constant , we define the normalized difference of the root mean square errors (NDRMSE) to evaluate the overall difference in the MSE of the coherent and noncoherent MIMO radars as below

(13.70)

where it is assumed that . If , then we would use directly. The smaller the , the closer the MSE performance between the two estimators. For a predetermined value of , which is chosen to satisfy particular requirements, the MSE performance of the noncoherent MIMO radar is considered to be good enough if , and in such cases the noncoherent MIMO radar is preferred for its less stringent synchronization requirements.

Particularly, in the joint location and velocity estimation problem discussed previously, , so we have

(13.71)

(13.72)

and

(13.73)

where it is assumed that . Note that one can easily adjust for units by proper selection of . Later in the numerical examples, we employ for all .

2.13.4.2 Numerical examples

2.13.5.1 Signal model

Consider a radar system that has transmit and receive antennas. The positions of the transmit and receive antennas of the radar system are and , respectively. Denote the waveform transmitted by the th antenna by , where and can be non-orthogonal for , in the sense of [4]. Assume each waveform is normalized to give transmitted power . The radar will break up the monitored space into cells and sequentially probe each cell for a target. Thus, our goal is to find out whether a target is at a particular position or not. Assume a target, if present, is composed of point scatterers, located at . The reflection coefficient, for the th scatterer, is assumed to be constant over the observation interval.

The clutter-plus-noise-free received signal at receive antenna due to the transmission from transmit antenna and the reflection from the th scatterer is modeled as [4]

(13.74)

where denotes the time delay from transmit antenna to receive antenna due to the reflection from the th scatterer, , and is the speed of light. Assume the transmitted waveforms are relatively narrow band, where the bandwidth of the waveforms is given such that they are not capable of resolving individual scatterers [5]. Therefore,

(13.75)

where denotes the time delay from a reflection off a scatterer located at the gravity center of the scatterers . Then (13.74) can be written as

(13.76)

Thus, the received signal at receive antenna due to the propagation from all transmit antennas and all scatterers is given by

(13.77)

where denotes the clutter-plus-noise observed at the th receiver.

2.13.5.2 Effect of signal space dimension on diversity gain

Suppose we attempt to implement the optimum receiver for the Gaussian reflection coefficients and Gaussian clutter-plus-noise case following (13.77), by first projecting the received continuous-time observations onto a basis that spans the space spanned by the first term, the signal term, in (13.77). Let the set of orthonormal basis functions span the dimensional space spanned by the first term, the signal term, of (13.77). The expansion, in terms of this basis, of any delayed signal term appearing in (13.77) can be expressed as

(13.78)

with the coefficients defined by

(13.79)

which are obtained by using (orthonormal basis)

(13.80)

after taking the inner product with (right hand side of (13.79)) on both sides of (13.78). We note that the largest possible value of is and this occurs when each of the delayed signal terms in (13.77), the terms expanded in (13.78), are linearly independent. Now we can project the received signal in (13.77) onto this basis to obtain⁸

(13.81)

where , and is a vector of reflection coefficients for those scatterers. Collecting the outputs and letting , we have

(13.82)

where

(13.83)

and . Further, stacking the outputs across all the receivers into a single vector gives

(13.84)

where

(13.85)

denotes a block diagonal matrix with submatrices , …, on its diagonal, and the size of each submatrix

(13.86)

is is an matrix. Let

(13.87)

then the dimension of matrix is given by . The in (13.84) is an clutter-plus-noise vector with covariance matrix . Based on (13.84), we have the following results which elaborate the diversity gain offered by MIMO radar systems employing non-orthogonal waveforms.

For the special case of Gaussian reflection coefficients (Gaussian signals), it is known from [25] that , so the diversity gain is given by . When the Gaussian optimum detector is employed, it can be shown that , which leads to a diversity gain . Further, suppose the dimension of the space spanned by the target-reflected noise-free received waveforms at different receivers are the same for all , such that . Then the largest possible diversity gain described in Theorem 2 becomes .

Assuming the number of scatterers is very large such that , Theorem 2 implies that to achieve the largest possible diversity gain, , the noise-free received waveforms, due to target reflection, at the th receiver must span the entire dimensional space. If the clutter-plus-noise free received waveforms span a lower dimension, say , the diversity gain will be equal to or smaller than , which is less than in this case. In other words, MIMO radars that employ either orthogonal waveforms or non-orthogonal waveforms can provide the maximum achievable diversity gain, as long as the noise-free received waveforms, due to target reflection, at each receiver span an dimensional space.

2.13.6.1 Master-slave closed-loop phase synchronization

The master-slave closed-loop approach is a very simple scheme. It employs a master-slave architecture as illustrated in Figure 13.12, where sensor is denoted as the master node, and , as slave nodes. All the slave sensors will then synchronize their local phase with that of the master node. We assume no a priori establishment of time synchronization across all the radar elements, at least not one with sufficient accuracy to enable phase synchronization. Each radar element maintains its own time using its own independent local oscillator, and is able to track the elapsed time using its local clock with relative precision. As a result of the absence of a common time reference, the instantaneous phase of each radar element can no longer be precisely determined by the local time at that element. For simplicity, we assume all the radar elements have a similar notion of the length of a specified time interval, e.g., for a time slot. In practice, this can be achieved by using identical stable clocks across all the radar elements. Moreover, we also assume no a priori frequency synchronization among these nodes. But we assume all the radar elements are equipped with identical frequency and phase estimators.

For this master-slave architecture, there exist two basic variations, depending on how the master element interacts with each of these slave elements. The first one is the open-loop approach, in which the master node broadcasts an unmodulated sinusoidal reference signal to all the slave nodes, and each of these slave nodes will then use such a reference signal to estimate and correct its phase offset. In this way, interaction between the master node and slave nodes is minimal. However, the absence of a common time reference, as we pointed out earlier, makes it difficult for each sensor to generate absolute phase with respect to the true time. As a result, phase shift induced by the channel between the master node and each slave node can not be disambiguated at each slave node, and hence cannot be compensated either. So, this open-loop method may lead to inferior accuracy in phase synchronization. On the other hand, suppose the channel phase information can be measured in a certain way and then fed back explicitly to each slave node for appropriate phase compensation, the accuracy of phase synchronization can be much improved. This is the basic idea behind the closed-loop approach, another type of master-slave synchronization. However, note that a fundamental premise that lies underneath this approach is the channel reciprocity, which is assumed in this work, meaning that the channel between a pair of sensors remains the same for both forward and reverse directions.

In the following, we describe a specific time-slotted closed-loop phase synchronization algorithm. In particular, we assume each slave element is assigned a specific time slot, during which they can exchange information with the master element . We further assume each slave element knows the number (or order) of the time slot assigned to it. This time-slotted closed-loop approach is then summarized as follows:

As is evident from the above procedures, it is essential to estimate the pre-compensation phase offset for each slave node when performing this closed-loop phase synchronization scheme. But since the channel between the master node and each slave node may vary from time to time, though at a slow rate, the value of the pre-compensation offset has to be estimated and updated in a periodic manner. A detailed timing analysis and phase synchronization error analysis of this algorithm can be found in [45].

2.13.6.2 Round-trip phase synchronization

The round-trip phase synchronization algorithm, as detailed in [45], was inspired by the all-node-based method proposed in [41] to globally synchronize the clocks in a sensor network, and the time-slotted round-trip approach that was developed in [42] and further improved or modified in [43,44] to attain carrier synchronization for distributed beamforming in multiuser wireless communication systems. The above mentioned approaches, nevertheless, do not appear to be applicable to this MIMO radar phase synchronization problem directly, even though a common feature shared by these approaches, that when in operation, a reference signal or message traverses in loops through all the sensors, remains essential.

The round-trip phase synchronization scheme for coherent MIMO radar [45] employs an unmodulated beacon signal to travel through all these radar elements in a round-trip manner. Each node will record the phase information twice in order to compute the final phase. The equivalence of the cumulative phase shift at each sensor, a quantity that is directly related to the total propagation delay for the round-trip circuit, is the fundamental idea behind this technique. This technique employs pretty much the same assumptions as those in the master-slave closed-loop method. For example, this scheme also assumes no a priori establishment of both time and frequency synchronization across all the radar elements, and assumes the channel reciprocity. However, specific to this round-trip method, we further assume that when one sensor transmits, the signal is only properly received by its intended receiver, i.e., its downstream neighbor. This can be attained in practice, for example, by using directional transmissions or appropriate power control schemes at the transmitter (assuming each radar element loosely knows its and its immediate neighbors’ locations). In what follows, we summarize this round-trip phase synchronization algorithm. Its mathematical characterization, timing analysis, and error analysis can be found in [45]:

2.13.6.3 Broadcast consensus based phase synchronization

It can be easily seen that the aforedescribed master-slave closed-loop and round-trip phase synchronization algorithms are not distributed in nature. Thus, both methods are not very robust against node failures. As far as robustness and scalability is concerned, distributed algorithms have enormous advantages, and may be more desirable for phase synchronization in coherent MIMO radar systems. The essence of these algorithms is that one or more sensors can communicate information with its immediate neighbors in a round, and the computation is distributed over all the sensors involved. In particular, when information averaging is concerned, these distributed algorithms are usually referred to as distributed consensus algorithms or gossip algorithms, e.g., [46–54]. Distributed consensus algorithms have been applied to a variety of network-related scenarios. For example, they have been considered in [53,54] for clock or time synchronization in wireless sensor networks.

Among the various distributed consensus algorithms, of particular note is the broadcast-based consensus algorithm, which has been studied in [50,51] without considering the stochastic disturbances, and more recently in [52] by taking non-zero-mean stochastic perturbations into account. Here we describe a broadcast consensus based phase synchronization algorithm for coherent MIMO radar systems, which was developed in [45], and is a direct extension of the work given in [52]. Similar to the aforedescribed two phase synchronization approaches, this algorithm assumes no a priori time synchronization; thus, there is no common time scale established among all these sensors. Each radar element is assumed to operate according to its own local clock which ticks independently from others. Also, for the ease of exposition, we assume the carrier frequency synchronization is already established, while noting that this assumption can be also relaxed. Thus, when a sensor receives beacon signals from its neighbors, it only needs to estimate the received phase.

In this algorithm, each element broadcasts a reference signal containing its local phase information to all its immediate neighbors. Its neighbors then average their local phase with the received phase. Thus, no global operations or information exchanges are required in the process. As an inherent characteristic, this algorithm exploits the broadcast nature of the wireless communication environment, and obviates the need for sophisticated underlying media access control mechanisms. To this end, we briefly outline this phase synchronization algorithm. Its mathematical characterization and analysis is furnished in [45], which examines in detail how the propagation delays among neighboring sensors and the phase errors induced by inaccurate local estimation affect the phase synchronization accuracy:

Note that the absence of a common time reference implies that none of these sensors know the true time, and as a result, they do not know the “true” initial phase of their local oscillator. Thus, when a sensor receives a broadcast reference signal for the first time, we choose to update its phase using only the estimate of the received phase.

2.13.6.4 Discussion

As can be clearly seen, the master-slave closed-loop method is the simplest one among all these three approaches. It, however, has the potential to lead to the best phase synchronization accuracy. This is because there are only two (the least number) nodes which participate in each round of the phase synchronization process, i.e., the master node and one slave node. Thus, the propagating accumulation of each individual’s phase and frequency estimation errors is very limited, and is less of a concern. This, in fact, is corroborated by a rigorous phase synchronization error analysis included in [45]. Moreover, in this method, the phase offset induced by the propagation channel is explicitly compensated via the closed-loop operation, which further improves the phase synchronization accuracy. Therefore, this approach is particularly useful when a very precise phase synchronization is required. It is generally applicable to situations where a relatively powerful centralized entity exists or can be identified, and can be repeatedly used in systems having tree-like topologies or hierarchical structures. To complete the whole phase synchronization process, a total of transmissions are needed, including the broadcast transmission made by the master node.

Unlike the master-slave closed-loop method, both the round-trip and the broadcast consensus based algorithms do not require a powerful master entity. However, the round-trip scheme does need a global initiator to start the phase synchronization process. In theory, the global initiator can be any radar sensor in the system. This round-trip method is able to synchronize all the radar sensors to the same mean carrier phase, regardless of the topology of these sensors as long as a cycle traversing all these sensors can be identified. But since sensors in the system participate in the phase synchronization process all together in a cooperative fashion, this round-trip method may suffer from the propagating accumulation of each individual’s phase and frequency estimation errors, as mathematically demonstrated in [45]. As a result, this method may only be suitable to coherent MIMO radar systems with a limited number of sensors. In other words, it does not scale very well. This round-trip method needs a total of transmissions to complete the phase synchronization process.

The broadcast consensus based method is distributed and localized in nature. It needs neither a centralized entity nor a particular global initiator. It is scalable and applicable to coherent MIMO radar systems composed of a large number of densely scattered sensors. It is also robust to sensor failures. But unfortunately, as shown in [45], the functionality of this approach is highly dependent on, and varies from topology to topology of the radar sensors. For arbitrary topologies, this approach may be only useful when a coarse phase synchronization is required. Since this algorithm runs in an iterative manner, it generally requires more transmissions than the other two approaches to complete the whole synchronization process. Thus, it has to be implemented at a relaxed time constraint as opposed to a stringent one.

Of further note is that in each of the presented phase synchronization algorithms, the propagation delay for the channel between any pair of MIMO radar elements that are involved in the reference signal exchange, can be critical in determining the resulting phase synchronization accuracy. A small uncertainty in the channel propagation delay, which may result from, for example, the ambiguity in the location of certain radar elements or the mobile scatters and other propagation anomalies, can lead to intolerable phase variations across some or all the radar elements. This is particularly true when high frequency carriers are chosen as the reference signals. One way to alleviate this unfavorable effect is to use relatively low carrier frequency. If the carrier frequency is low enough, the location uncertainty of the MIMO radar elements will not play an important role. Another way is to recalibrate these sensors periodically to track the channel changes. This may induce some extra overhead, but the accompanying benefits are in general well worth it. Besides, the variations of the propagation channels among all these MIMO radar elements, though dependent on the specific operating environment, are usually very slow; it takes a relatively long time to reach the point where the physical changes in the medium are so pronounced that the resulting phase mismatch seriously deteriorates the performance of the coherent processing, especially for applications such as target localization [11]. Therefore, very frequent phase resynchronization may not be needed.

Another practical issue that merits attention is the phase dynamics or phase instability [55] in the coherent MIMO radar system. Note that the oscillator phase noise [56] is always present, and is an irreducible error which can affect the phase synchronization performance. To counter the effect of phase noise in oscillators, periodic phase synchronization remains indispensable. How often the phase synchronization procedure needs to be repeated, however, is dependent on, among other factors, the aggregate impact of the phase noise across all the local oscillators on the phase synchronization accuracy, as well as the phase accuracy or stability requirements of the coherent MIMO radar system and the specific application. It is indicated in [34, p. 261] that the phase accuracy or stability requirements for coherent processing by a monostatic or bistatic radar can range from less than one degree to many tens of degrees of RF phase over a coherent processing interval, depending on the type and duration of coherent processing. We conjecture that similar phase accuracy or stability requirements may hold as well for coherent processing by a MIMO radar system.

2.13.7.1 Signal model and design criteria

Figure 13.14 illustrates a bistatic MIMO radar scenario, where the radar is equipped with transmit elements and receive elements. This model, appropriate for both bistatic and monostatic scenarios, is a slight generalization of that in [2] for the case of extended targets. For simplicity we consider this model in discrete-time and in baseband. For an extended target, the reflection from the signal sent from the th transmit element and captured at the th receive element can be modeled using a finite impulse response (FIR) linear system with order , whose impulse response is . Then the received waveform at the th receive element and discrete-time is given by

(13.97)

where is the waveform transmitted from the th transmit element and is the additive complex Gaussian noise measured at the th receive element.

Let denote the length of the observed signal vector starting at an arbitrary discrete time , and assume that (typically ). Denote and , then using the matrix-vector notation (13.97) can be rewritten into

(13.98)

where is an Toeplitz matrix which contains the waveforms transmitted from the th transmit element, i.e.,

(13.99)

We further define and , so that (13.98) could be reformulated into

(13.100)

By stacking the received waveforms across all the receive elements, we create . Defining , we finally obtain

(13.101)

where and .

To facilitate the ensuing analysis on the waveform design of MIMO radar for extended targets, the model in (13.101) is assumed to have the following properties [9]:

Note that can be diagonalized through its eigenvalue decomposition, i.e.,

(13.102)

where is a unitary matrix whose columns are eigenvectors and is a diagonal matrix with each diagonal entry given by a real and nonnegative eigenvalue.

Now let us introduce the design criteria. The first one is the conditional mutual information (MI) between and given the knowledge of , which is hereafter referred to as MI. We have [76] ( denotes differential entropy)

(13.103)

Conditioned on (equivalent to conditioning on ), we can easily find that is Gaussian distributed with zero mean and covariance . Using (13.103), the MI is [76]

(13.104)

where (13.104) follows from

(13.105)

The design objective is to find those (transmitted waveform) that maximize the MI () between the random target impulse response and the received (reflected) radar waveform under the constraint which effectively limits the total transmit power. Therefore, the problem of waveform design based on MI can be cast as:

Next, let us consider the problem of radar waveform design from the viewpoint of estimation. It is easy to verify that, conditioned on , , and are jointly Gaussian distributed as

The conditional distribution of given and is also Gaussian, with conditional mean given by (using Eqs. (IV.B.53) and (IV.B.55) on p. 156 of [17])

(13.106)

and conditional covariance matrix given by (using Eq. (IV.B.54) on pp. 156 of [17])

(13.107)

where the matrix inversion lemma¹¹ was employed in obtaining both (13.106) and (13.107).

Let denote the Bayes estimate of . When the cost function is defined as , the Bayes estimate will be a linear MMSE estimator [17] which is given by the conditional mean of given (for a given value of ), i.e.,

(13.108)

In this case, the Bayes risk for a given will be

(13.109)

The goal is then to find those that minimize the value of MMSE under a constraint on the total transmit power of . Or equivalently, the problem of waveform design based on MMSE estimation can be expressed as:

2.13.7.2 Optimal waveform design for MIMO radar

As mathematically demonstrated in [9], the maximum value of will be achieved when the matrix is diagonal, and the minimum value of will be attained when is diagonal, also. The following theorem summarizes the result of optimum . A detailed proof of this result can be found in [9].

A review of Theorem 3 indicates that knowledge of the full matrix is needed to perform an eigenvalue decomposition. An asymptotic formulation has been developed in [9] which lessens the required information to just a few samples of the PSD and would be much more suitable in practice. The asymptotic approach employs one important assumption as shown below:

Clearly this assumption is not very restrictive for single transmit and receive antenna cases, implying that the target impulse response samples form a wide-sense stationary random process asymptotically. For multiple transmit and receive antenna cases, this assumption implies that the statistics over space are indistinguishable from the statistics over time. For the extreme cases of complete independence or dependence over both space and time, the assumption is very reasonable, for example. However, it is also straightforward to relax this assumption for multiple-antenna cases by employing a similar asymptotic approximation as that described below. For a further discussion on this assumption, readers are referred to [59].

It is shown in [77] that Toeplitz matrices can be approximated by their associated circulant matrices, and asymptotically (in the dimension of these matrices), they are equivalent. The asymptotic equality of two matrices implies that their eigenvalues and inverses (and certain products) behave similarly [77]. Now, let us denote the th entry of the covariance matrix as , then we have and , where the sequence is absolutely summable, and its -periodic truncated Fourier spectrum

is real valued, finite, and nonnegative, and represents the PSD of the random process when approaches infinity. Then a circulant matrix that is asymptotically equivalent to is [78]

(13.111)

where is the unitary discrete Fourier transform (DFT) matrix with its th entry given by

and the diagonal matrix contains samples of the PSD of along its diagonal, or equivalently

where .

Given the above asymptotic approximation, the waveform design problems posed earlier can be resolved using standard optimization theory. For brevity, we summarize the results in the following theorem. The proof can be found in [9].

In fact, Theorem 4 is simply a modification of Theorem 3 for the asymptotic case. The essential change is that replaces and replaces . Under such asymptotic approximation, the solution for can be explicitly written as

(13.115)

It is obvious that to produce in (13.115), the full covariance matrix is not needed, only samples of the PSD for are necessary. This is a more reasonable assumption in practice.

2.13.7.3 Discussion

Based on Theorem 3, it can be concluded that any meeting the given power constraint which maximizes MI will also minimize MMSE, and this is unique for a given . In fact, can be deemed as the covariance matrix of the transmitted waveforms. Meanwhile, it should be noted that the solution expressed by (13.110) is a closed-form and general solution which does not require any level of approximation. Its optimality is manifested in two ways. On the one hand, only satisfying (13.110) can be optimum in the sense of either maximizing the MI or minimizing the MMSE. On the other hand, any from (13.110) is optimum for both MI and MMSE provided such satisfies . In other words, the condition stipulated by (13.110) is only a necessary condition for to attain the optimality, but not a sufficient one. This is because there may exist some cases where a solution satisfies (13.110) but does not have the desired Kronecker structure. Note for these cases, it can be possible that the equivalence between the MI and the MMSE does not hold any more. However, in a recently published paper [60], an iterative algorithm has been developed, which can identify waveform solutions that not only optimize the performance criteria of interest (i.e., MI and MMSE), but have a specific Kronecker structure. It is also found in [60] that waveform solutions generated through this algorithm can lead to performance which is very close to, and almost indistinguishable from that predicted by Theorem 3. Detailed information about this algorithm can be found in [60].

A close review of Theorem 4 reveals that for the asymptotic simplification approach, exact knowledge of the target PSD samples is required in the waveform design. This requirement appears rather demanding, particulary in practical applications. In order to bring the waveform design even closer to practice and deliver more applicable waveform solutions for MIMO radar, it is necessary to accommodate some issues that may arise in reality, such as the uncertainty in the target’s statistics. In such a circumstance, robust procedures, which can overcome those problems by incorporating modeling uncertainty into the design from the outset [79], seem quite attractive. As a result, a minimax robust waveform design problem for MIMO radar has been formulated and solved in [59], where a band model was adopted by assuming the PSD lies in an uncertainty class of spectra bounded by known upper and lower bounds. This band model might arise in practice, for example, if a confidence band for the spectrum could be determined via spectrum estimation, or the upper and lower bounds can be obtained simply by field measurement and modeling. It is shown in [59] that the resulting minimax robust waveforms can achieve good performance for any PSD in the uncertainty class, and in particular, they can bound the worst-case performance at an acceptable limit. One very interesting finding from the results in [59] is that the minimax robust waveforms for the MMSE criterion are generally different from those for the MI criterion, which is in stark contrast to the results for the completely known PSD case. We direct interested readers to [59] for more details.

A.1 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.

Consider a set of pulsed sinusoidal signals¹²

(A.1)

where , the term 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 Figure A.1. Note that throughout this appendix it is always assumed that . According to Parseval’s theorem

(A.2)

Observing the spectrum, we see that when , the main lobes of and are sufficiently separated so that and (A.2) approximates , which makes and approximately orthogonal. Here we are ignoring contributions from small sidelobes which is similar to truncating the signals in the frequency domain. As , if the frequency increment satisfies

(A.3)

then any two signals from the signal set () are approximately orthogonal.

Assume that and are time delayed by and . The non-zero part of the delayed signal occupies , and the non-zero part of occupies . Hence, the time duration of the overlap between them is , where denotes the time delay difference. Note that when is greater than or approximates , the overlap part of these two signals vanishes, which implies (A.2) is close to and the signals can be considered as orthogonal. Therefore, it is possible for the two delayed signals to be non-orthogonal only if , which makes the following discussion necessary.

We denote the overlap interval as . Thus

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 Figure A.1. In a similar way, for the time delayed signals, the orthogonality is approximately maintained if

(A.4)

Since , the inequality is maintained by requiring . As , the condition becomes , which is the same as in (A.3). Hence, the requirement for (A.3) guarantees (A.4). That is, if (A.3) is met, the time delayed signals are orthogonal.

Next, we further discuss the Doppler shift restriction for orthogonality. The effective time delayed and Doppler shifted signals are given by and . The frequency spectrum of the effective time delayed signals and , i.e., and , are replotted in Figure A.2 using solid curves, the center frequency of which are and . After Doppler frequency shift, the spectrum of the effective time delayed and Doppler shifted signals are and . They are shown in Figure A.2 using dashed curves. Thus, the center frequency of the effective signal is , while the center frequency of becomes . Clearly, provided , that is

(A.5)

the overlapping part between and () can be ignored. Thus the orthogonality is well approximated. Note that since (A.4) is satisfied, we have . Hence the term in (A.5) can be ignored, resulting in or .

It has been shown in [24] that, in the example case of target velocity estimation, the CRBs obtained using untruncated signals under certain conditions that are of practical interest are extremely close to the CRBs calculate by assuming ideal orthogonal signals. Interested readers are referred to [24] for detail.

A.2 Distinguishing coherent and noncoherent processing

In the MIMO radar context, the term coherent processing implies that the relative phase information embedded in the signals traveling over the paths between different transmit and receive antennas will be exploited (e.g., see (13.22)). Ultimately, this requires the oscillators in the up and down converters at different transmit and receive antenna nodes to be locked in phase. This is not the case for noncoherent processing where the relative phase information is not exploited (e.g., see (13.57)).

To distinguish between the coherent and noncoherent processing, one can examine the output of the processor (e.g., (13.22) or (13.57)), say denoted by , as a function of a set of observations coming from different spatial paths, say denoted by , where represents the number of available observations coming from different spatial paths. That is . If an arbitrary constant¹³ phase set is imposed on the observations coming from different spatial paths, but the processing is insensitive to the phase shifts over the different paths, i.e., the output of the processor is not affected, such that

(A.6)

then the processor is regarded as being noncoherent. Otherwise, if the output of the processor is affected by the arbitrary phase set, such that

(A.7)

then the processor is regarded as being coherent.

The estimators in (13.22) and in (13.57), which satisfy (A.7) and (A.6), are examples of noncoherent and coherent processors respectively, where

and

A.3 FIM for coherent MIMO radar joint estimation

Considering that the log-likelihood ratio in (13.20) is explicitly a function of and , but implicitly as a function of , we defined a new parameter vector

(A.8)

Then, the FIM can be derived using the chain rule as

(A.9)

We first compute and obtain

Partitioning into a block matrix, we have

where are submatrices. The in (A.9) can be derived from (13.20) as

which is a matrix. Write in the form of a block matrix

where are matrices. Carrying out the computation, we get

(A.10)

(A.11)

(A.12)

(A.13)

(A.14)

and

(A.15)

where the terms , , dependent on the characteristics of the received waveforms are provided in (A.9)–(A.13). Thus, the FIM can be computed through

(A.16)

After lengthy algebraic manipulations, the expression of the FIM can be obtained as shown in (13.23).

A.4 Calculating FIM for localization with phase errors

First, let us compute the th () element of the matrix. Referring to the vector in (13.41), let refer to the index of the element of corresponding to , then let refer to the index of the element of corresponding to

(A.17)

where

(A.18)

The term in (A.18) is given in (13.39). Further

(A.19)

where is given in (13.38). Assuming an interchange of derivative and integral is valid,¹⁴ then

(A.20)

where

(A.21)

Finally, the remaining term in (A.18) is

where for , and for

where for , and for

Now all terms in (A.18) have been obtained. Further taking the expectation with respect to gives (A.17). Next, we compute the elements related to the target complex reflectivity

(A.22)

for , where

(A.23)

The new terms introduced by (A.23) are

(A.24)

where is given in (13.38) and

(A.25)

and

(A.26)

where is given in (13.38), in (A.20), in (A.25), and

Plugging (13.39), (A.19), (A.24), (A.26) in (A.23) and taking expectation gives (A.22). Then, we compute

(A.27)

for , where

(A.28)

The new terms introduced by (A.28) are

(A.29)

where is given in (13.38) and¹⁵

(A.30)

and

(A.31)

where is given in (13.38), in (A.20), in (A.30), and

Plugging (13.39), (A.19), (A.29), (A.31) in (A.28) and taking expectation gives (A.27). Next, we compute

(A.32)

where

(A.33)

The new term introduced by (A.33) is

(A.34)

where is given in (13.38), in (A.25), and

Plugging (13.39), (A.24), (A.34) in (A.33) and taking expectation, we obtain (A.32). Similarly, we compute

(A.35)

where

(A.36)

The new term introduced by (A.36) is

(A.37)

where is given in (13.38), in (A.30) and

Plugging (13.39), (A.29), (A.37) into (A.36) and taking expectation gives (A.35). Finally, we compute

(A.38)

where

(A.39)

The new term introduced by (A.39) is

(A.40)

where is given in (13.38), in (A.25), and in (A.30). Plugging (13.39), (A.24), (A.29), (A.40) into (A.39) and taking expectation we obtain (A.38).

Now we compute the quantity

using and respectively given in (13.2) and (13.41)

(A.41)

where and .

Relevant Theory: Signal Processing Theory Statistical Signal Processing, and Array Signal Processing