A Unifying Framework for Adaptive Radar Detection in the Presence of Multiple Alternative Hypotheses

Nowadays, modern radar systems incorporate sophisticated signal processing algorithms which take advantage of the computational power made available by recent advances in technology. This growth in complexity is dictated by the fact these systems have to face with more and more challenging scenarios where conventional algorithms might fail or exhibit poor performance. For instance, in target-rich environments, structured echoes contaminate data used to estimate the spectral properties of the interference (also known as training or secondary data) leading to a dramatic attenuation of the signal of interest components and, hence, to a nonnegligible number of missed detections [1]. In such case, radar system should be endowed with signal processing schemes capable of detecting and suppressing the outliers in order to make the training data set homogeneous [2, 3, 4, 5]. Another important example concerns high resolution radars, which can resolve a target into a number of different scattering centers depending on the radar bandwidth and the range extent of the target [6, 7]. The classic approach to the detection of range-spread targets consists in processing one range bin at a time despite the fact that contiguous cells contain target energy. As a consequence, classic detection algorithms do not collect as much energy as possible to increase the Signal-to-Interference-plus-Noise Ratio (SINR). To overcome this drawback, architectures capable of detecting distributed targets by exploiting a preassigned number of contiguous range bins have been developed [8, 9, 10, 11, 12]. These energy issues also hold for multiple point-like targets. In fact, detection algorithms which can take advantage of the total energy associated with each point-like target are highly desirable.

In the open literature, the existing examples concerning the detection of either multiple point-like or range-spread targets share the assumption that the number of scatterers (or at least an upper bound on it) is known and are based upon the Maximum Likelihood Approach (MLA) [13, 14]. However, in scenarios of practical value, such a priori information is not often available especially when the radar system is operating in search mode. Moreover, the problem of jointly detecting multiple point-like targets is very difficult since target positions and, more importantly, target number are unknown parameters that must be estimated. Thus, conversely to the conventional detection problems that comprise two hypotheses, namely the noise-only (or null) and the signal-plus-noise (or alternative) hypothesis, this lack of a priori information naturally leads to multiple alternative hypotheses with the consequence that the radar engineer has to face with composite multiple hypothesis tests.

Besides target-dense environments, another operating situation leading to multiple hypothesis tests is related to possible electronic attacks by adversary forces (jammers). These attacks comprise active techniques aimed at protecting a platform from being detected and tracked by the radar [6] through two approaches: masking and deception. More precisely, noncoherent jammers or Noise-Like Jammers (NLJs) attempt to mask targets generating nondeceptive interference which blends into the thermal noise of the radar receiver degrading the radar sensitivity due to an increase of the Constant False Alarm Rate (CFAR) threshold [6, 15, 16]. On the other hand, the Coherent Jammers (CJs) illuminate the victim radar by means of low duty-cycle signals with specific parameters that, when estimated by the radar processor, force the latter to allocate resources to handle false targets. In fact, CJs are equipped with electronics apparatuses capable of receiving, modifying, amplifying, and retransmitting the radar’s own signal to create false targets with radar’s range, Doppler, and angle far away from the true position of the platform under protection [16, 6].

A possible way to react to this kind of interference relies on the use of decision schemes devised by modifying the conventional detection problem with additional hypotheses associated with the presence of such threats [17, 18]. In [17], adaptive detection and discrimination between useful signals and CJs in the presence of thermal noise, clutter, and possible NLJs is addressed by considering an additional hypothesis under which data contain the CJs only. In addition, the latter is assumed to lay on the orthogonal complement of the subspace spanned by the nominal steering vector (after whitening by the true covariance matrix of the composite disturbance). The resulting multiple hypothesis test is solved resorting to an approach based upon a generalized Neyman-Pearson criterion [19]. However, from a computational point of view, detection threshold setting might require an onerous load and, more importantly, such solution is effective when the multiple hypotheses are not nested. As a matter of fact, in the presence of nested hypotheses, the MLA and, hence, the Generalized Likelihood Ratio Test (GLRT), may fail because the likelihood function monotonically increases with the hypothesis order (or model order). As a consequence, the MLA experiences a natural inclination to overestimate the hypothesis order. An alternative approach consists in looking for detection schemes that incorporate the expedients of the so-called Model Order Selection (MOS) rules [20, 21, 22, 23], which include the parameter number diversity to moderate the overestimation attitude of the MLA. In [18], the authors follow the last approach to conceive two-stage detection architectures for multiple NLJs whose number is unknown. Specifically, the first stage exploits the MOS rules to provide an estimate of the number of NLJs, whereas the second stage consists of a jammer detector that uses the estimate obtained at the first stage. Finally, it is important to observe that MOS rules can be adapted to accomplish detection tasks by also considering the model order “$0$” which is associated with the null hypothesis [24, 25, 26]. However, in this case, it is not possible to set any threshold in order to guarantee a preassigned Probability of False Alarm ($P_{fa}$) and, more importantly, the CFAR property, which is of primary concern in radar, cannot be a priori stated.

In this paper, we develop an elegant framework relying on the Kullback-Leibler Information Criterion (KLIC) [27] to address multiple hypothesis testing problems where there exist many alternative hypotheses. This framework provides an important interpretation of both the Likelihood Ratio Test (LRT) and the GLRT from an information theoretic standpoint and, remarkably, it lays the theoretical foundation for the design of new one-stage decision schemes for multiple hypothesis tests. This result represents the main technical contribution of this paper and, interestingly, such new detection architectures share a common structure that is given by the sum of a conventional decision statistic and a penalty term as well as the generic structure of a KLIC-based MOS rule consists of the compressed log-likelihood plus a penalty term. The starting point of the developed framework is the case where all the parameters are known showing that under suitable regularity conditions the LRT approximates a test which selects the hypothesis with the associated probability density function (pdf) minimizing the Kullback-Leibler Divergence (KLD) with respect to the true data distribution. In addition, the LRT coincides with such test when the KLD is measured with respect to the Empirical Data Distribution (EDD). Then, we guide the reader towards more difficult scenarios where the distribution parameters are no longer known. Specifically, we resort to the Taylor series approximations of the KLD, that are also used to derive the MOS rules, in order to come up with decision schemes capable of moderating the overfitting inclination of the MLA. From a different perspective, the same results can be obtained by regularizing the pdf under the generic alternative hypothesis through a suitable prior for the unknown model order and applying a procedure that combines the MLA and the Bayesian estimation [28, 21] (see the appendix for further details). The proposed theoretical framework is, then, completed with the investigation of the CFAR behavior of these decision schemes framing the analysis in the more general context of statistical invariance and providing two propositions which allows to state when the newly proposed decision architectures are invariant with respect to a given group of transformations and, possibly, enjoy the CFAR property. Finally, we present numerical examples obtained over simulated data and concerning three different radar detection problems also in comparison with two-stage architectures where the first stage is aimed at estimating the model order while the second stage is a conventional detector. The analyses highlight that the developed MOS-based detectors are capable of providing good detection capabilities in several contexts of practical interest.

The remainder of this paper is organized as follows: the next section contains preliminary definitions and formulates the detection problem at hand in terms of a multiple hypothesis test. Section III describes the proposed framework and introduces the new architectures, whereas in Section IV some discussions about the CFAR properties ensured exploiting the Principle of Invariance are also provided. Then, Section V provides some illustrative examples to assess the performance of the new architectures also in comparison to natural competitors. Concluding remarks and future research tracks are given in Section VI. The derivation of an alternative framework leading to the same architectures is confined to the Appendix.

Let us consider a radar system, equipped with $N$ space and/or time identical channels, which collects a data matrix ${\mbox{\boldmath$Z$}}=[{\mbox{\boldmath$z$}}_{1},\ldots,{\mbox{\boldmath$z$}}_{K}]\in{\mathds{C}}^{N\times K}$ whose¹¹1Observe that in the case of Space-Time Adaptive Processing (STAP), $N$ represents the number of space-time channels, whereas when the system transmits either a single pulse through an array of sensors or exploits a single antenna to transmit a pulse train, $N$ represents the number of elements of either the spatial array or the number of transmitted pulses, respectively [29]. Finally, $K$ may represent the number of range bins or the number of pulses when a slice of the STAP datacube is processed. columns can be modeled as statistically independent random vectors whose distribution belongs to a preassigned family. For simplicity and in order not to burden the notation (a point better explained below), we assume that these vectors share the same distribution parameters (identically distributed) and that their joint unknown pdf is denoted by $\bar{f}({\mbox{\boldmath$Z$}};{\mbox{\boldmath$\theta$}})$ with ${\mbox{\boldmath$\theta$}}\in{\mathds{R}}^{p\times 1}$ the parameter vector taking on value in a specific parameter space, $\Theta\subseteq{\mathds{R}}^{p\times 1}$ say. Now, a conventional binary decision problem partitions the latter into two subsets $\Theta_{0}$ and $\Theta_{1}$ corresponding to the null ($H_{0}$) and the alternative ($H_{1}$) hypothesis, respectively. Thus, denoting by ${\mbox{\boldmath$\theta$}}_{0}$ the elements of $\Theta_{0}$ and by ${\mbox{\boldmath$\theta$}}_{1}$ the elements of $\Theta_{1}$, then the pdf of $Z$ under the $i$th hypothesis can be written as $f_{i}({\mbox{\boldmath$Z$}};{\mbox{\boldmath$\theta$}}_{i})$, $i=0,1$. It is important to stress here that $\bar{f}({\mbox{\boldmath$Z$}};{\mbox{\boldmath$\theta$}})$ is the actual pdf and is unknown, whereas $f_{i}({\mbox{\boldmath$Z$}};{\mbox{\boldmath$\theta$}}_{i})$ is the pdf of $Z$ when²²2In what follows, we assume correctly specified models, namely that data distribution family is known. ${\mbox{\boldmath$\theta$}}={\mbox{\boldmath$\theta$}}_{i}\in\Theta_{i}$.

Generally speaking, in many detection applications, the actual value of the parameter vector is unknown or, at least, partially known. In addition, not all the entries of $\theta$ are useful to take a decision for the specific problem at hand. As a consequence, we can split $\theta$ as ${\mbox{\boldmath$\theta$}}=[{\mbox{\boldmath$\theta$}}_{r}^{T},{\mbox{\boldmath$\theta$}}_{s}^{T}]^{T}$, where ${\mbox{\boldmath$\theta$}}_{r}\in\Theta_{r}\subseteq{\mathds{R}}^{p_{r}\times 1}$ contains the parameters of interest, while the components of ${\mbox{\boldmath$\theta$}}_{s}\in\Theta_{s}\subseteq{\mathds{R}}^{p_{s}\times 1}$ represent the parameters that do not enter into the decision process and are called nuisance parameters [30]. From a more general perspective, it would be possible to associate a parameter vector of interest to each column of $Z$ with the consequence that ${\mbox{\boldmath$z$}}_{k}$s are no longer identically distributed. However, as stated before, we prefer to proceed assuming that ${\mbox{\boldmath$z$}}_{k}$s share the same parameter vector in order to maintain an easy notation and because the extension to the more general case is straightforward as we will show in Section V. Thus, with the above remarks in mind, a conventional binary hypothesis testing problem can be expressed as³³3Note that $p_{r}$ is the number of the parameters of interest and $p_{s}$ is the number of the nuisance parameters. It follows that $p=p_{r}+p_{s}$.

where ${\mbox{\boldmath$\theta$}}_{i}=[{\mbox{\boldmath$\theta$}}_{r,i}^{T},{\mbox{\boldmath$\theta$}}_{s}^{T}]^{T}\in{\mathds{R}}^{p\times 1}$, $i=0,1$, with ${\mbox{\boldmath$\theta$}}_{r,i}\in\Theta_{r}^{i}\subseteq{\mathds{R}}^{p_{r}\times 1}$ the parameter vector of interest under $H_{i}$, $g_{i}(\cdot;\cdot)$, $i=0,1$, is the pdf of ${\mbox{\boldmath$z$}}_{k}$, $k=1,\ldots,K$, under $H_{i}$.

Two remarks are now in order. First, note that $\{\Theta_{r}^{0},\Theta_{r}^{1}\}$ is a partition of $\Theta_{r}$. Second, the above problem assumes that $p_{r}$, $p_{s}$, and, hence, the total number of unknown parameters $p$, are perfectly known. However, in many radar (and, generally speaking, signal processing) applications, it is not seldom for $p$ to be unknown under the alternative hypothesis due to the fact that the size of ${\mbox{\boldmath$\theta$}}_{r,1}$ might depend on the specific operating scenario [21]. For instance, radar systems might face with situations where an unknown number of targets are present in the surveillance area [31, 13] or be under the attack of an unknown number of jammers [18, 32, 33]. Under this assumption, the hypothesis test can be modified as

where $p_{r,0}$ and $\Omega_{r}=\{p_{r,1},\ldots,p_{r,M}\}$ with $p_{r,1}\leq\ldots\leq p_{r,M}$ are known, while $p_{r,m}$ is unknown. It follows that the pdf of $Z$ (and, hence, that of ${\mbox{\boldmath$z$}}_{k}$) under $H_{1}$ depends on $p_{r,m}$ and, more important, the uncertainty on the latter leads to a testing problem formed by multiple (possibly nested) $H_{1}$ hypotheses, i.e.,

where $f_{1,m}({\mbox{\boldmath$Z$}};{\mbox{\boldmath$\theta$}}_{r,1},{\mbox{\boldmath$\theta$}}_{s},p_{r,m})=\prod_{k=1}^{K}g_{1,m}({\mbox{\boldmath$z$}}_{k};{\mbox{\boldmath$\theta$}}_{r,1},{\mbox{\boldmath$\theta$}}_{s},p_{r,m})$ is the pdf of $Z$ under $H_{1,m}$, namely when ${\mbox{\boldmath$\theta$}}_{r,1}\in\Theta_{r}^{m}$, with $g_{1,m}({\mbox{\boldmath$z$}}_{k};{\mbox{\boldmath$\theta$}}_{r,1},{\mbox{\boldmath$\theta$}}_{s},p_{r,m})$ the pdf of ${\mbox{\boldmath$z$}}_{k}$ under $H_{1,m}$.

In the next section, we propose an Information-theoretic based approach to deal with problem (3) exploiting the KLIC [27]. Specifically, this criterion relies on the measurement of a certain distance, the so-called KLD, between a candidate distribution belonging to the family of densities $\mbox{$\mathcal{F}$}=\left\{f_{0},\ f_{1,m},\ m\in\{1,\ldots,M\}\right\}$ and the actual distribution of $Z$, which is assumed to lie in $\mathcal{F}$ and denoted by

where $\bar{g}({\mbox{\boldmath$z$}}_{k};{\mbox{\boldmath$\theta$}})$ is the true pdf of ${\mbox{\boldmath$z$}}_{k}$, $k=1,\ldots,K$. Besides, we suppose that the inequalities required to invoke the Kintchine’s Strong Law of Large Numbers [34] are valid, namely

as well as the following “regularity conditions” [21]

and ${p}/{T}\overset{T\rightarrow\infty}{\longrightarrow}0$, which are required to suitably approximate the KLD. In the above equations, $T$ represents the total number of real-valued observations, that, for the problem at hand, is equal to $2NK$ since we are dealing with complex vectors.

Note that a “minimum information distance” selection criterion has already been successfully applied for model order estimation giving rise to the so-called MOS rules [21, 35]. The resulting selection architectures share the same structure consisting of a fitting term (the compressed log-likelihood function) plus an adjustment which also depends on the number of parameters. As a consequence, the parameter number diversity comes into play to moderate the overfitting inclination of the compressed likelihood in the case of nested hypothesis. In fact, in this context, the KLIC-based rules can provide satisfactory classification performance whereas the Maximum Likelihood (ML) approach may fail because the likelihood function monotonically increases with $p_{r,m}$ and the ML estimate (MLE) of $p_{r}$ will always be $p_{r,M}$ (or, equivalently, the MLE of $p$ will be $p_{r,M}+p_{s}$).

To conclude this preliminary section, for the reader ease, we recall that the KLD [27] (also called relative entropy) between $\bar{f}$ and $f_{1,m}$, namely the pdf of a generic candidate model under $H_{1}$, can be written as [36]

where $d{\mbox{\boldmath$Z$}}=dz^{r}_{1,1}dz^{i}_{1,1}\ldots dz^{r}_{N,K}dz^{i}_{N,K}$ with $z^{r}_{n,k}$ and $z^{i}_{n,k}$ the real and imaginary parts of the $n$th component of ${\mbox{\boldmath$z$}}_{k}$, and can be decomposed into the sum of two terms⁴⁴4We assume that the considered pdfs exist with respect to a Lebesgue measure.

where $h(\bar{f})$ is the differential entropy of $\bar{f}$ and $h(\bar{f},f_{1,m})$ is the cross entropy between $\bar{f}$ and $f_{1,m}$. Note that unlike $h(\bar{f})$, $h(\bar{f},f_{1,m})$ depends on the $m$th model (or hypothesis). Analogously, we can write the KLD with respect to the pdf under $H_{0}$ as

Recall that $D(\cdot,\cdot)$ is not a true distance between distributions since it is not symmetric and does not satisfy the triangle inequality [36]. Nonetheless, it is often useful to think of the KLD as a “distance” between distributions. Finally, the KLD can be interpreted as the information loss when either $f_{1,m}$ or $f_{0}$ is used to approximate $\bar{f}$ [20].

In this section, we exploit the KLIC to devise decision schemes for problem (3). To this end, we proceed by considering the case where all the parameters are known and then evolve to more difficult cases where the parameters become unknown. It is important to observe that we define here an information theoretic framework where well-established decision rules as the Likelihood Ratio Test and the GLRT are suitably encompassed.