I Introduction
Ia Motivation and Summary of Results
One of the key advantages of multipleinput multipleoutput (MIMO) radar systems is that they enable adaptive transmit beamforming. This technique adaptively steers radar beams to better illuminate a target of interest while reducing the received signal contribution from other targets, jammers, or clutter [2]. There has been significant recent interest in designing these transmit beam patterns, summarized in Section IB below.
The performance of a transmit beam pattern depends on the corresponding receiver architecture. A standard radar receiver architecture for tracking targets consists of two main building blocks: detection of potential targets and track filtering (see, e.g., [3, Chapter 7.3.4], [4, 5]). The two are linked by an association step, which uses the target priors to assign each detection to one of the tracks. The association step ensures that all the detections assigned to the same track are in fact caused by a single target across time. Correct association is critical, since errors in this step can lead to tracking failure.
The simplest transmit beamforming methods focus on optimizing coverage within the field of view. More advanced methods utilize prior information from the tracker, such as estimates of the targets’ azimuth angles. However, to the best of our knowledge, prior work effectively ignores the association step during the beamforming design (see Section
IB).In this paper, we initiate the joint design of the beam pattern and the data association scheme. We show that the coupling of the two is captured by what we term an ambiguity graph. The vertices of this graph are the targets. Two vertices connected by an edge correspond to targets that are “difficult” to disambiguate using the prior distributions during data association. They therefore should not be simultaneously illuminated by the beam pattern. Conversely, two vertices not connected by an edge correspond to targets that are “easy” to disambiguate during data association. They therefore do not place any constraints on the beam pattern. The choice of precisely which targets are “difficult” or “easy” to disambiguate allows to trade the performance of the beamformer and the association scheme. We refer to this as the detectionassociation tradeoff.
The joint beamforming and association design problem can then be broken up into three subproblems: First, designing an optimal beam pattern respecting the constraints of a given ambiguity graph. Second, finding an appropriate association scheme for a beam pattern with the given ambiguity graph. Third, choosing a Paretooptimal ambiguity graph operating on the boundary of the detectionassociation tradeoff.
We term the first subproblem the ambiguityaware beamforming problem. We show that the ambiguityaware beamforming problem can be formulated as a semidefinite optimization problem, which is efficiently solvable using interiorpoint methods. Through analysis and several simulations, we demonstrate two types of performance improvements due to the beamforming being ambiguity aware: an improvement in transmit power gain and an improvement in target identifiability. The transmit power gain increases detection performance for a fixed number of targets. The improvement in target identifiability increases the number of targets that can be tracked for a fixed number of radar antennas. Through analysis and numerical simulations, we demonstrate that both of these gains can be quite substantial.
To solve the second subproblem, we propose an ambiguityaware nearestneighbor association scheme. This association scheme is a generalization of the wellknown nearestneighbor association scheme. Unlike that scheme, it explicitly takes the structure of the beam pattern (captured by the ambiguity graph) into account. The proposed association scheme has a cubic worstcase computational complexity in the number of targets being tracked.
Finally, we consider the third subproblem of choosing a Paretooptimal ambiguity graph operating on the boundary of the detectionassociation tradeoff. This problem turns out to be a computationally difficult combinatorial optimization problem. We therefore propose a design heuristic and show through simulations that it can yield nearoptimal performance.
IB Related Work
Multiantenna radar systems can broadly be classified into two categories: phased arrays
[3, Chapter 3], in which phaseshifted versions of the same signal are emitted from the antennas, and MIMO radar [2], in which general signals can be emitted from the antennas. The increased adaptability of MIMO radar offers significant advantages over phasedarray radar, including increased target identifiability and flexibility in transmit beam pattern design [2].A number of prior works have studied the problem of designing transmit beam patterns for MIMO radar systems. These methods vary in the amount of prior target information, generally obtained from the track filters at the receiver, they utilize. Some methods do not use any prior target information and focus on creating beam patterns with wide coverage [6].
A large body of literature creates beam patterns adapted to the tracked targets’ azimuth angles. One such approach designs the beam pattern using a twostep process, which first selects a crosscorrelation matrix to achieve a desired beam pattern and then chooses signals having that correlation [7]. Another approach decouples the spatial and temporal dimensions of the beamforming design [8, 9]. Yet another class of methods adopts a beamspace approach to beamformer design [10, 11, 12, 13, 14]
, which aims to combine the benefits of phasedarray and MIMO radar. Beamformer designs that are robust to mismatches between the presumed and actual beamforming vectors are studied in
[15]. In the automotive context, beamforming has also been studied from the perspective of interference avoidance [16].A few recent works [17, 18] have addressed the question of adapting beam patterns for improved detection. Some works focus on adapting waveforms to account for change in clutter in the radar scene [19, 20, 21], while other works focus on adaptive beamforming to improve tracking [22, 17, 23].
The different methods described above design beamforming patterns to maximize detection accuracy and are not directly concerned with the resulting association problem. In contrast, in this paper we explicitly take the data association problem into account while designing the beam pattern, and we show that there is in fact a tradeoff between the detection and the association performance.
IC Organization
The remainder of this paper is organized as follows. Section II introduces the problem setting. Section III shows how to solve the ambiguityaware beamforming problem. Ambiguityaware association schemes and how to choose a Paretooptimal ambiguity graph are discussed in Section IV. Section V contains discussions and concluding remarks.
Ii Problem Setting
We consider a monostatic radar system with antennas. Denote by and the transmit and receive gains of antenna at azimuth angle . Assume we emit the complex, basebandequivalent, vectorvalued signal with from the antenna array. The corresponding (noiseless, basebandequivalent) received signal after reflection from a point target with unit radar crosssection at range , Doppler shift , and azimuth angle is
Note that here and in the following we use lowercase bold font for vectors and uppercase bold font for matrices. The vector has components and similar for the vector . The symbol denotes the conjugate transpose. Finally, denotes the speed of light. For general radar crosssection , the reflected signal is . Observe that the radar crosssection includes the path loss.
For targets with parameters and radar crosssection for , the received signal is then
where is Gaussian receiver noise.
We aim to track the parameters of these targets from the received signal . The standard receiver architecture uses a matched filter bank, with each filter tuned to one parameter triple. The output of the matched filter tuned to is
(1) 
Following [8], we make throughout the remainder of this paper the assumption that the transmitted signal has the form
for some matrix satisfying and for some with support and with approximate bandwidth . Here, is assumed to have good crosscorrelation properties, meaning that it has matrixvalued ambiguity function satisfying
(2) 
where
The temporal support and the bandwidth of the waveforms place limits of order on the resolution of and of order on the resolution of . The identity (II) holds up to those resolution limits. See Appendix A for a formal definition of these limits. In the same appendix, we also describe and analyze a construction for waveforms having property (II) for large enough values of timebandwidth product . For simplicity, we make in the remainder of the paper the assumption that (II) holds with equality, i.e., that the ambiguity function is ideal.
Assuming an ideal ambiguity function, the matched filter output (II) can be simplified after some algebra to
(3) 
where
(4) 
is the beamforming matrix and is the filtered receiver noise. Observe that satisfies .
A common procedure for tracking the parameters for the targets consists of two steps, a detection step followed by an association step. In the first step, we detect the targets. This is done by thresholding the matched filter outputs . I.e., we find all such that is strictly above some threshold. In the second step, we associate each such detection with a target track using an association rule.
Assume that we have prior information about the parameters . This prior is typically derived from the output of the prediction stage of a filter tracking these targets. It can be used to define a gate, i.e., a region of parameter space in which we expect the true parameters for a fixed target to fall. Similar to other works in the literature [8], we make for now the simplifying assumption that the prior information on the azimuth angle is precise, so that is in effect known a priori. Section V discusses how this assumption can be relaxed. With this assumption, the gate for target is a subset of in which we expect the parameter tuple to fall.
We now formalize a somewhat stylized association rule that we will use throughout the paper: For each target track , find all detections of the form with , and (which is known a priori by assumption). If there is a single such detection, then we associate it with the track for target . If there are zero or more than one such detections, then we declare an association error. In this association rule, the priors are used to gate the detections. Further, the detectiontotrack association is performed only if this gating removes all association ambiguity. While this association rule is quite simple, it includes several standard rules as special cases.
Example 1 (name=Rectangular Gates, label=eg:gates_rect).
From the prior, we can derive lower bounds and upper bounds for the parameter tuple . For example, could be chosen as the prior mean of
plus three standard deviations (and similarly for the other bounds). We can then construct the standard rectangular gating set
.Example 2 (name=NearestNeighbor Gates, label=eg:gates_greedy).
Let be the prior density on the parameters of target . Set
See Fig. 1 for an illustration. The resulting gating sets partition the space . For Gaussian priors with equal variance, the resulting partition is a Voronoi partition. The corresponding association rule assigns each detection to the track it most likely came from. This is similar to the standard nearestneighbor or greedy association rule.
For future reference, we define
(5) 
as the targets that are apriori ambiguous with respect to target .
To understand the interaction between data association and beamforming, consider initially a scenario with targets with known azimuth angles and . Assume the matched filter outputs , , , are all above the detection threshold. Without further knowledge we cannot unambiguously solve the association problem. Indeed, could be correctly associated with target one and correctly with target two. However, alternatively, could be incorrectly associated with target two and incorrectly with target one. Without additional information, both associations are equally valid from the radar receiver’s point of view.
This ambiguity can be resolved in two ways. First, through additional prior information about and . If the gates and do not intersect (or, equivalently, if ), then one of the two data associations is invalidated by the prior.
Second, by designing the transmitted signals to ensure that two of the matched filter outputs are less than the detection threshold, say . I.e.,
In this case, one of the two data associations is invalidated by the observations.
Consider next the general case with
targets. Assume for the moment that the received signal is noiseless, and set the detection threshold to zero. Under this assumption and using (
3), the matched filter output is nonzero, and hence a potential target detected, if and only if for some satisfyingConsider the resulting association problem. Our association rule (defined just above Example LABEL:eg:gates_rect) will assign all detections to tracks without declaring an error if, for every , there is exactly one tuple inside the gating set for target such that is nonzero.
From the above discussion, the association problem has a unique solution if the following two sufficient conditions hold. First, for every ,
so that the correct target is detected. Second, for every and every ,
so that hard to disambiguate targets are not simultaneously illuminated. This second condition ensures that the association is unambiguous. Crucially, these conditions depend solely on the prior knowledge of the target parameters (i.e., the gates and the azimuths ).
To focus on the essential aspect of the problem, we enforce in the following the slightly stronger condition that
(6) 
for every and that
(7) 
for every and .
To guard against receiver noise, we further want the transmit power gaing (6
) to be as large as possible for all targets. In fact, for fixed detection threshold, we can upper bound the probability of detection error as a decreasing function of the lefthand side of (
6) for the worst target (see, e.g., [3, Chapter 6.3]). The optimal ambiguityaware beamforming problem is therefore^{1}^{1}1To minimize the aforementioned probability of detection error, the term should be added to the objective function of (1). We prefer to drop this term here to simplify the notation, since it does not change the nature of the problem as discussed above. Further, the hard zeroforcing constraint (7) could be replaced by a weaker constraint that the interfering signal contribution is below the noise floor, as is discussed in Section V. R∈C^N×N min_k∈{1,2,…,K} a^†(θ_k) R a(θ_k) a^†(θ_k) R a(θ_k’)= 0, and tr(R)= 1 R ⪰0 Here denotes that the matrix is positive semidefinite (which implies that it is Hermitian).The structure of the ambiguityaware beamforming problem can be conveniently described by a graph with vertices (one for each target) and an edge between vertex and if . Since if and only if by construction, this graph is undirected. We refer to this graph as the ambiguity graph in the following. The next example introduces a special case of this graph that will be used to illustrate the results throughout the paper.
Example 3 (name=Ambiguity Graph, label=eg:graph).
Consider a vehicular scenario with a radar at a crossroad on which several cars are driving in the same direction. This scenario is depicted in Fig. 2(a). The corresponding expected values of the target parameters and are shown in Fig. 2(b) by black dots together with rectangular gates as defined in Example LABEL:eg:gates_rect indicated in gray. Observe from Fig. 2(b) that only the gating sets of neighboring vehicles intersect. The resulting ambiguity graph is depicted in Fig. 2(c).
Remark 1:
If we have no prior information about the parameters and , then all targets are apriori ambiguous. The resulting ambiguity graph is therefore complete (i.e., has and edge between every pair of distinct targets). In this case, the ambiguityaware beamforming problem essentially reduces to the setting in [8]. However, for partial ambiguity graphs, like the one introduced in Example LABEL:eg:graph, the problem and its solution are quite different. This connection is discussed further in Example LABEL:eg:friedlander in Section III.
Iii Optimal AmbiguityAware Beamforming
The ambiguityaware beamforming problem defined in (1) can be solved efficiently as follows. Note that the function is linear in and hence, in particular, concave. Since the minimum of concave functions is again concave, this implies that the objective function of (1) is concave in . The conditions and are linear in . Further, we have the positive semidefiniteness constraint , which describes a convex set. Thus, we are dealing with a convex problem, which can actually be rewritten as a semidefinite program by introducing a slack variable to handle the minimization over . These programs can be solved efficiently using interiorpoint methods [24].
We illustrate this approach with several examples.
Example 4 (label=eg:3x3).
Consider a scenario with uniformly spaced antennas at halfwavelength separation and with targets at azimuth angles , , and . The corresponding transmit antenna gains are
(8) 
Assume first that the ambiguity graph is complete (i.e., for each so that the ambiguity graph has an edge between every pair of targets). The corresponding optimal transmit beam pattern with the solution to (1) as a function of azimuth angle for fixed is shown in red/gray in Fig. 3.
Assume next that the ambiguity graph contains only edges between neighboring targets (as introduced in Example LABEL:eg:graph). The corresponding optimal transmit beam pattern is shown in black in Fig. 3.
Comparing the two curves in Fig. 3, two key differences are apparent. First, the beam pattern for the complete ambiguity graph has an additional null in Figs. 3(a) and 3(c). Second, by not having to enforce these nulls, the beam pattern for the partial ambiguity graph is able to increase the transmit power gain at the desired target by (from to ).
As the previous example shows, taking the reduced ambiguity between the targets (captured by the ambiguity graph being incomplete) into account can improve the transmit power gain. This improvement is explored further in the next example.
Example 5 (name=Transmit Power Gain, label=eg:snr).
Consider uniformly spaced antennas at halfwavelength separation and with targets. Target has azimuth angle . In words, the targets are uniformly spaced in azimuth. For , this reduces to the setting in Example LABEL:eg:3x3.
We compare the optimal beamforming matrices designed according to (1) for a complete ambiguity graph and for a partially connected graph with edges only between neighboring targets and for all (see again Example LABEL:eg:graph). The transmit power gain ratio, i.e., the ratio of the quantity between the two scenarios is shown in Fig. 4 as a function of the number of transmit antennas . As is clear from the figure, explicitly taking the target ambiguity into account can yield a significant improvement in transmit power gain that increases with the problem size.
The last example indicates that ambiguityaware beamforming can substantially improve the transmit power gain. In other words, for the same number of targets, we can expect better detection performance. As we see next, ambiguityaware beamforming can also improve target identifiability, i.e., for the same number of antennas, we can track more targets.
Example 6 (name=Target Identifiability, label=eg:ident).
Consider again the scenario of Example LABEL:eg:snr with uniformly spaced antennas and uniformly spaced targets. However, this time we allow and to differ. We are interested in the maximum number of identifiable targets, which we denote by and define formally as the largest for which the optimization problem (1) has a solution.
We compare again the complete ambiguity graph and the partially connected ambiguity graph with edges only between neighboring targets as introduced in Example LABEL:eg:graph. For the complete graph we show analytically in Appendix B that (see the solid red/gray curve in Fig. 5). In contrast, the solid black curve in Fig. 5 shows the numerically computed value of for the partial graph. The figure suggests that the number of identifiable targets is , i.e., almost doubled. Thus, explicitly taking the target ambiguity into account can also yield a significant improvement in target identifiability that again increases with the problem size.
We next explore the structure of the solution to the ambiguityaware beamforming problem. We start by introducing some additional notation.
Denote by a matrix whose columns are an orthonormal basis for the orthogonal complement of the subspace spanned by . In other words, , and for all . Further, denote by a matrix whose columns are an orthonormal basis for the orthogonal complement of the subspace spanned by . In other words, , and . The choice of and is not unique; any such matrices will work for our purposes.
Consider then the matrix
(9) 
where and are arbitrary positive semidefinite matrices. Note that is positive semidefinite. For any and ,
and for any
Further,
From this discussion, we see that one way to construct a beamforming matrix is to choose , for such that
and so that the constraint is satisfied. Formally, a (potentially suboptimal) solution to the ambiguityaware beamforming problem (1) can be found by solving: F_k, G_k ∀k min_k∈{1,2,…,K} a^†(θ_k)U_kF_kU_k^†a(θ_k) R(F_k, G_k)= R(F_1, G_1), for all tr(F_1) + tr(G_1)= 1 F_k⪰0, for all G_k⪰0, for all .
This reformulation allows to obtain analytical insight into the structure of the solution to the ambiguityaware beamforming problem as the next two examples illustrate.
Example 7 (name=Comparison with [8], label=eg:friedlander).
Let and consider a scenario with a complete ambiguity graph. Then the matrix is in fact simply a vector , and the matrix is simply a positive scalar . Thus,
We need to ensure that
Fix a target and consider another target . Since the ambiguity graph is complete, is orthogonal to . Since the columns of span the orthogonal complement of the subspace spanned by , this implies that can be written as for some . Set
Then
which is identical for all as required. The values of can now be chosen to satisfy the trace constraint and to maximize the transmit power gain. This corresponds to the beamforming solution proposed in [8].^{2}^{2}2To be precise, [8] sets for all with chosen to satisfy the trace constraint. Thus, for the special case of complete ambiguity graphs, the solution to (III) reduces to the one in [8].
Example 8 (name=Identifiability Gain, continues=eg:ident).
We continue Example LABEL:eg:ident, and assume for the moment that is even. As we had seen in Example LABEL:eg:friedlander, for the case of complete ambiguity graphs, the beamforming problem and its solution reduce to the one in [8]. Further, as argued in Appendix B, the largest number of identifiable targets is equal to in this case.
Consider then the case of a partial ambiguity graph with edges only between neighboring targets as defined in Example LABEL:eg:graph. Construct the positive semidefinite matrix
where the columns of are an orthonormal basis for the orthogonal complement of the space spanned by with odd, and where the columns of are an orthonormal basis for the orthogonal complement of the space spanned by with even. Observe that this is of a form similar to (9). This construction ensures that , satisfying the zeroforcing constraint between neighboring targets. Further, it is easily seen that as required.
Iv AmbiguityAware Association and ParetoOptimal Ambiguity Graphs
So far, we have assumed that the ambiguity graph is given (defined through (5) by the gates ). In Section III, we have seen how to solve the ambiguityaware beamforming problem (1), which maximizes the transmit power gain for the worstcase target as a function of the ambiguity graph . Denote by the value of this objective function for the optimal beamforming design.
We now turn to the problem of designing the ambiguity graph itself, or, equivalently, the gating sets , for the given prior distributions on the target parameters. Recall that our association rule is to find all detections with received filter tuned to azimuth angle falling into the gate . If there is a single such detection, the rule assigns it to track . Otherwise, the rule declares an association error. Correct association occurs if the association rule runs to completion (i.e., does not declare an association error) and assigns each detection to the correct target.
Assume for the moment that the received signal is noiseless and a detection threshold of zero is used. The probability of detection error is then equal to zero. Further, making use of the zeroforcing constraint (7), we can lower bound the probability of correct association by the quantity
(10) 
where the second equality follows from the definition of in (5). The randomness in this expression is due to and
being random variables with distributions given by the prior.
Now consider noisy received signals. In this case, the probability of error can be upper bounded by the sum of two terms: The first term is an upper bound on the probability of detection error and is a decreasing function of the power gain as mentioned in Section III. The second term is , capturing the performance of the association rule.
There is a tradeoff between these two terms: Choosing to increase causes to decrease . At one extreme consider the complete graph containing all edges. For this graph, the received filter tuned to a particular azimuth angle nulls out the interference of all other targets . Therefore, the association problem becomes simple, and the probability of correct association is large. However, since we need to create beams with nulls, the optimization problem (1) is maximally constrained and hence the optimal transmit power gain is minimized. At the other extreme, consider the empty graph containing no edges. For this graph, the beams can be chosen without any nulling constraints. Therefore, the transmit power gain is maximized. However, the received filter tuned to a particular azimuth angle contains interference from all other targets . Therefore, the association problem is difficult, and the probability of correct association is small.
Formally, the detectionassociation tradeoff is given by the set of all Paretooptimal^{3}^{3}3A pair is Pareto optimal if no other pair dominates it, meaning that there exists no other graph such that simultaneously and , with at least one of the two inequalities being strict. pairs of the form parameterized by graphs with vertices . We call an ambiguity graph optimal, if the corresponding pair is Pareto optimal. Since there are possible graphs with vertices , finding optimal ambiguity graphs represents a computationally nontrivial combinatorial optimization problem. We next describe a design heuristic for constructing close to optimal ambiguity graphs.
For the expression (IV) to be well defined, we need to describe how to choose the gating sets corresponding to the ambiguity graph . In order for the two to be consistent, we need for all and (see (5)). We introduce here an extension of the nearestneighbor gating sets seen in Example LABEL:eg:gates_greedy. Define
(11) 
where is the probability density for the prior on . Further, define
as the set of pairs that are more likely to occur from target than from any of the nonneighboring targets in the graph . See Fig. 6 for an illustration. Clearly, for all and , as required.
With this choice of gates , our generic association rule recalled at the beginning of this section can be rewritten (after some algebra) more compactly as in Algorithm 1. Since the maximizing in the innermost loop can be computed in time, the entire algorithm has run time . We refer to this association rule as ambiguityaware nearestneighbor association. We use this association rule for the evaluation of .
Continuing from (IV), we can now lower bound using the union bound as
(12) 
This last lower bound is a function of only the most difficult to disambiguate pair of targets not connected by an edge in the ambiguity graph .
Our design heuristic for choosing the graph is as follows. Instead of optimizing directly, we use the lower bound (IV) on as a substitute. With this substitution, the optimizing can be readily found. Observe that does not decrease (and usually increases) whenever we remove an edge from . At the same time, (IV) stays constant whenever we remove an edge from that is not achieving the maximum probability of pairwise error. This implies that all optimal (for the modified criterion) ambiguity graphs have edge sets of the form
(13) 
for some threshold parameter and with as defined in (11). In words, all target pairs that are pairwise more difficult to disambiguate than the threshold parameter form an edge in the ambiguity graph. Let be the corresponding graph. By sweeping from to , we create a sequence of up to optimal graphs , which in turn trace out the detectionassociation tradeoff . Fig. 7 shows two examples of this approach. The figure indicates that the proposed design heuristic can be quite close to optimal.
To summarize this section, our proposed approach is as follows. Fix a value of the threshold parameter to some value between and depending on the application requirements. Construct the corresponding ambiguity graph with edges given by (13). Solve the corresponding optimal ambiguityaware beamforming problem (1) and steer the radar beams accordingly. From the radar returns, find all the detections and associate them using the nearestneighbor association rule in Algorithm 1 parameterized by the graph .
V Discussion and Conclusion
In this paper, we studied the problem of jointly designing transmit beam patterns and association rules for tracking multiple targets using MIMO radar. We introduced the concept of the ambiguity graph used to describe which targets are hard to disambiguate based on prior information. We first formulated a semidefinite program to solve the problem of designing beam patterns that avoid simultaneous illumination of hard to disambiguate targets. We then designed the association rule assigning detections to the different targets being tracked. Finally, we solved the problem of designing ambiguity graphs by using a heuristic to approximate the optimal detectionassociation tradeoff.
In our analysis we made a number of simplifying assumptions. Some of these can be readily relaxed. For instance, the ambiguityaware beamforming problem (1) completely nulls interfering target signal contributions, i.e., for all and all . This constraint may be too restrictive, and it may be sufficient to ensure that the interfering target signals are below the noise floor, i.e., for all and all with . The resulting optimization problem remains convex and can be solved efficiently.
Similarly, we have assumed that the target azimuth angles are known a priori. To relax this, assume that we have prior upper and lower bounds and on each target azimuth . The ambiguityaware beamforming problem (1) can be modified to accommodate this uncertainty by replacing the objective with and the constraints with for all and all . Here is a small but strictly positive number that controls the amount of interference as discussed in the last paragraph. This optimization problem is again convex.
There are several directions for future work. Our analysis assumed that the ambiguity function is ideal, and we have argued how to construct corresponding waveforms in the largebandwidth regime. Joint beamforming and association design under nonideal ambiguity function is an interesting open problem. Further, while we have jointly solved the beamforming and association problems, we have not directly considered the tracking problem. Solving all three problems jointly is an interesting direction for future work involving several new aspects such as how to handle association failures and how to initiate tracks.
Appendix A Waveforms with Approximately Ideal Ambiguity Function
We construct waveforms of support and (approximate) bandwidth that have close to ideal ambiguity function as defined in (II). We formally state the result as the following theorem.
Theorem 1.
Let with , and let , with , satisfying
(14) 
There exists waveforms of support and approximate (i.e., Rayleigh or equivalently half the nulltonull) bandwidth with ambiguity function having the following properties:

for all . 
for all and for all with either or . 
for all with and for all .
While Theorem 1 only proves existence of such a set of waveforms, the proof shows that if is slightly larger than the minimum necessary to satisfy (14), then the vast majority of waveforms generated by the randomized construction described below will have the above properties. The ambiguity function for one such set of waveforms generated at random is shown in Fig. 8.
Example 9.
For a numerical example, consider temporal support , bandwidth , and antennas. Then the condition (14) is satisfied for any sidelobe level of . Choosing , and assuming a carrier frequency of , the limits on and imposed by Property 2 in Theorem 1 translate into a range resolution of and a Doppler velocity resolution of with an upper bound of .
Proof:
As alluded to above, we use a randomized construction. Let with and
be independent random variables uniformly distributed over the complex unit circle. Set
(15) 
where denotes the indicator function for the set , i.e.,
In words, is piecewise constant over intervals of length , and the value on each such interval is uniformly distributed over the complex circle with magnitude . These waveforms therefore fall into the class of polyphase codes [3, Section 4.10.2]. Clearly, has support . Further, since each term in has approximate bandwidth , so does itself.
We next analyze the entries of the ambiguity function. By symmetry of the ambiguity function, we can assume without loss of generality that . The analysis proceeds in three main steps. In the first step, we argue that all waveforms of the above type have the desired ambiguity properties for and and for all large enough and . In the second step, we argue that a set of waveforms randomly generated as above has the desired ambiguity properties with positive probability at all sample points of the form and for integer . Hence, there exists at least one such set of waveforms. In the third step, we argue that if a set of waveforms has the desired ambiguity properties at all the sample points, then it also has them at all points in between.
Before proceeding to the main arguments we present the following two technical lemmas that we will use throughout the proof.
Lemma 2.
Let , , , and . Then can be expanded as
(16) 
where
and with the convention that
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