Bayesian Identifying One or Two Close Sources by Gaussian Estimates of Planar Location under Double Emission

Having two parameter estimates the problem to identify whether single source emits twice or two sources at some distance between parameters below resolution limit emit once takes place in application of the resolution methods. The resolution limit of the estimator is described by the Statistical Resolution Limit ($SRL$) defined as the minimal separation at which estimates are resolved correctly [1]. If the separation is less than $SRL$, then there will exist ambiguity about one or two sources. The concepts of $SRL$ are primarily formulated in the detection theory [2,3,4,5] and the estimation accuracy approach [1], [6,7,8]. Herein, we rely on the estimation accuracy utilizing the Cramer-Rao Bound ($CRB$) matrix-function of parameter, which under mild conditions represents a narrow lower bound on covariance matrix of any unbiased estimator [9]. The equality to $CRB$ is achieved in the class of asymptotically efficient estimators to which we will address in the paper. While true parameter(s) is unknown, $CRB$ can be well approximated by the same $CRB$ matrix-function but taken at parameter estimate to gain its covariance matrix and thereby sufficient approximation of the $SRL$ in the domain of small estimation errors.

We propose novel Byes formalism which can identify one or two sources below $SRL$ by two planar location estimates and their covariance matrices. A handling with positional estimates on the plane is relevant in mobile communication, astrometry, microscopy and many other fields.

The sources have different spectrum, power, others features of the signal but they may be identical. The existing approaches in identifying closely spaced sources typically invoke the probability of number of sources – either from subjective expert experience or as posterior one obtained from Bayes inference, which operates with physical characteristics of the emissions. We come across the closest example of Bayes signal classification in radar [10], seismology [11] and so on, where, having the probabilities of each class for each emission, it is not difficult to deduce probabilities of all possible number of sources for a given number of emissions using probability theorems. In infrared optics, Bayesian calculation is applied to obtain probability of number of neighboring screen images to overcome the smearing effect [12]. A similar problem arises in localization of blinking objects in microscopy where Bayesian Information Criteria are to achieve probability of number of objects for the analysis below the diffraction limit [13]. We will refer to a solution that is able to support the required probabilities of one and two sources as the prior solution (PS). Prior probabilities in the Bayesian formalism will be the ones extracted from PS.

When resolving, the separation $\left|p_1-p_2\right|$ to the parameters $p_1$ and $p_2$ is not to be less than some quantitative measure of random distance scattering between their estimates: $\left|{p^{\prime }}_1-{p^{\prime }}_2\right|$. For the design of identifier, resolution criteria are reformulated by the means of the notation of the resolving inequality: $\left|{p^{\prime }}_1-{p^{\prime }}_2\right|\le SRL$ (RI) in order to catch the probability that distance between parameter estimates is not bigger than $SRL$. Following this reformulation we propose new $SRL$ concept for planar decoupled estimates when confidence circles of high (near to unity) probability around each of the two parameters are tangent. The concept provides approximately the same probability of the RI ($PRI$) as in scalar resolution criteria in the widest range of covariance matrix spectrum of difference between estimates.

Two mutually exclusive events constitute the Bayesian sample space: confidence circles around each of the sample estimates are either disjoint or intersectional if the distance between estimates is over or below $SRL$. Bayesian technique recalculates a posteriori given prior probabilities with regard to presumed distance between true locations. Doing that, it can change the probabilistic decision induced by PS. To the best of our knowledge, no solution aimed to discriminate single source emitted twice and two close sources emitted once for a given pair of location estimates, which would be parameterized by the distance between hypothesized sources is mentioned previously.

We illustrate the job of the Bayesian formalism by the example of distinguishing two positional estimates between one and two close each other users, obtained in the basic stations (BSs) network by the means of time difference of arrival (TDOA) technique. The signals are there classified in such a way that probabilities of one and two users can be derived. The solution must answer the following question: does one user emit twice or do two users at varying distance between them, carry out emissions consecutively. The identifier implementation is based on the radius of high confidence circle obtained in the paper for the confidence probability of 0.99. This radius will subsequently be denoted as $R99$.

The paper is organized as follows. In Section 2 the resolution criteria are surveyed. The problem is described in Section 3. New concept of $SRL$ is founded in Section 4. Section 5 contains the design of Bayesian identifier. The algorithm for estimation of $R99$ is presented in Section 6. The application of the identifier in the BSs network is quantitatively studied in Section 7. Section 8 briefly draws the summary of the paper.

One of the earliest was Lee resolution criterion [6] for scalar decoupled estimates, in our reformulation $\left|{p^{\prime }}_1-{p^{\prime }}_2\right|\le SR{L}_L$, $SR{L}_L=qmax\left\{{\sigma }_1,{\sigma }_2\right\}$, ${\sigma }_1=\sqrt{CRB\left(p_1\right)}$, ${\sigma }_2=\sqrt{CRB\left(p_2\right)}$, where ${\sigma }_1$, ${\sigma }_2$ are standard deviations of estimates ${p^{\prime }}_1$, ${p^{\prime }}_2$, which provide the resolving with “high probability” due to big factor $q=8$: it is evident that the bigger $q$ is chosen, the higher the probability is that a separation between estimates is not bigger than $SR{L}_L$ (authors referencing to Lee consider $q=2$). Delmas and Abeida [7] expressed $SR{L}_{D\&A}$ via standard deviation of the estimates difference: $SR{L}_{D\&A}={\sigma }_{{p^{\prime }}_1-{p^{\prime }}_2}=\sqrt{{\sigma }_1^2+{\sigma }_2^2}$, hence the probability of the inequality $\left|{p^{\prime }}_1-{p^{\prime }}_2\right|\le SR{L}_{D\&A}$ is smaller than with Lee, because inequality $SR{L}_{D\&A}<SR{L}_L$ being satisfied even at $q=2$. Regarding coupled estimates, Smith [1] offered $SR{L}_S={\sigma }_{{p^{\prime }}_1-{p^{\prime }}_2}=\sqrt{{\sigma }_1^2+{\sigma }_2^2-2cov\left({p^{\prime }}_1,{p^{\prime }}_2\right)}$, $cov\left({p^{\prime }}_1,{p^{\prime }}_2\right)=CRB\left(p_1,p_2\right)$, where $CRB\left(p_1,p_2\right)$ has been interpreted in [8] as an element of extended $CRB$ matrix by using the change of variable formula. Both $SR{L}_{D\&A}$ and $SR{L}_S$ are articulated in standard deviation of a difference between estimates. This fact offers an opportunity to achieve $PRI$ in a scalar case for a known distribution of estimates.

In the work of Korso, Boyer, Renaux and Marcos [8] the general problem for multiple estimates was considered. Therein, $SRL=CRB\left(\delta \right)$, where separation $\delta$ between sets of parameters is the $\kappa$-norm distance (Minkowsky distance). Let us represent two scalar parameters $p^{\left(1\right)}$, $p^{\left(2\right)}$ in a vector form, $p={\left[p^{\left(1\right)},p^{\left(2\right)}\right]}^T$ and denote estimates of vector $p$ obtained from the decoupled emissions 1 and 2 as ${p^{\prime }}_1$, ${p^{\prime }}_2$ respectively. Then, the RI is written for coupled estimates in metric of 1-norm ($\kappa =1$) of a difference vector $\Delta p={p^{\prime }}_1-{p^{\prime }}_2$ as $\delta \left(\Delta p\right)={‖\Delta p‖}_1=\left|{p^{\prime }}_1^{\left(1\right)}-{p^{\prime }}_2^{\left(1\right)}\right|+\left|{p^{\prime }}_1^{\left(2\right)}-{p^{\prime }}_2^{\left(2\right)}\right|\le SR{L}_{K,B,R\&M}$,

$SR{L}_{K,B,R\&M}^2={\left({\sigma }_1^{\left(1\right)}\right)}^2+{\left({\sigma }_2^{\left(1\right)}\right)}^2-2cov\left({p^{\prime }}_1^{\left(1\right)},{p^{\prime }}_2^{\left(1\right)}\right)+{\left({\sigma }_1^{\left(2\right)}\right)}^2+{\left({\sigma }_2^{\left(2\right)}\right)}^2-2cov\left({p^{\prime }}_1^{\left(2\right)},{p^{\prime }}_2^{\left(2\right)}\right)$, where variances and covariance of estimates are defined by the means of underlying $CRB$ functions at true parameters $p^{\left(1\right)}$ and $p^{\left(2\right)}$, which are obtained by applying change of variable formula to the extended measurement model of the estimator. The result distinctively extends scalar $SRL$ on vector case, however $PRI$, i.e. the probability of $\delta \left(\Delta p\right)\le SR{L}_{K,B,R\&M}$, could hardly be determined here in comparison with scalar resolution criteria.

Clark [14], analyzing the resolution of estimator that produces vector decoupled Gaussian estimates, presented the metric of a distance between ellipsoidal confidence regions of probability 0.9 around each of the two parameters. The estimates are resolvable when ellipsoids are disjoint. $SRL$ of the estimator is the metric from which they are tangent.

In the study, we transform Clark’s idea about the ellipsoidal confidence region to develop the concept of $SRL$ based on circular one of high confidence probability which is probabilistically consistent with conventional resolution criteria [1,7]. As with Clark, the estimates will be resolvable when the circles are disjoint, but their tangency will correspond to the $SRL$ equal to the sum of their radii.

During the two consecutive time intervals $t^I\in T^I$ and $t^{II}\in T^{II}$, $T^I\cap T^{II}=\varnothing$, $L$ signals from emission I and $L$ signals from emission II are received. The $k-$th signal depends upon vector parameter $q$, $q\in \left\{q^I,q^{II}\right\}$, which gathers physical characteristics of the emissions, and $m-$ dimensional measurement parameter $p_k\left(\phi \right)$ which is known smooth function of two-dimensional unknown vector of the source location, $\phi \in \left\{{\phi }^I,{\phi }^{II}\right\}$: where $g\left(t;q,p_k\right)$ is a waveform vector-function of time and parameters $q$, $p_k$; $n_k\left(t\right)$ is unbiased noise with $E\left\{n_i\left(t^I\right)n_j^T\left(t^{II}\right)\right\}=0$ for the all $i,j\le L$, $E\left\{\cdot \right\}$ denotes mathematical expectation operator.

Unbiased estimator ${\Im }_p$ establishes the inverse connection between signals (1) and related to emissions I and II measurement vectors $p^{\prime }$ and $p^{″}$, $\tilde{p}={\Im }_p\left\{V\left(t\right)\right\}$, $\tilde{p}\in \left\{p^{\prime },p^{″}\right\}$: where ${\upsilon }^{\prime }$ and ${\upsilon }^{″}$ are the measurement errors corresponding to emissions I and II with $E\left\{{\upsilon }^{\prime }{{\upsilon }^{″}}^T\right\}=0$, i.e. $p^{\prime }$ and $p^{″}$ are decoupled.

Unbiased estimator ${\Im }_{\phi }$ uses measurements (2) to produce estimates ${\phi }^{\prime }$ and ${\phi }^{″}$ over emissions I and II: $\tilde{\phi }={\Im }_{\phi }\left(\tilde{p}\right)$, $\tilde{\phi }\in \left\{{\phi }^{\prime },{\phi }^{″}\right\}$, $\tilde{\phi }=\phi +\varsigma$, $\varsigma \in \left\{\Delta {\phi }^{\prime },\Delta {\phi }^{″}\right\}$, where $\Delta {\phi }^{\prime }$ and $\Delta {\phi }^{″}$ are the corresponding errors, which are also decoupled.

We consider (a) “regular enough” algorithms ${\Im }_p$ and ${\Im }_{\phi }$ such that $\tilde{p}$ and $\tilde{\phi }$ are both normally distributed [15]: $\tilde{p}\sim$N($p$,$\Pi$), $\tilde{\phi }\sim$N($\phi$,$\Phi \left(\phi \right)$) with a given covariance matrix $\Pi$ of estimates $\tilde{p}$ and an unknown one $\Phi \left(\phi \right)$ of estimates $\tilde{\phi }$. The estimators ${\Im }_p$ and ${\Im }_{\phi }$ are assumed (b) to be efficient, hence $\Phi$ is calculated as $CRB$ by use of Fisher Information Matrix $FIM\left(\phi \right)$: $\Phi \left(\phi \right)=CRB\left(\phi \right)=FI{M}^{-1}\left(\phi \right)$. For Gaussian noise, $FIM\left(\phi \right)={\left[\frac{\partial p\left(\phi \right)}{\partial \phi }\right]}^T{\Pi }^{-1}\left[\frac{\partial p\left(\phi \right)}{\partial \phi }\right]$.

Hypotheses $H_{\alpha }$ and $H_{\beta }$ serve to specify the sources of emissions: hypothesis $H_{\alpha }$ means that one source at ${\phi }_1={\phi }^I={\phi }^{II}$ with $q_1=q^I=q^{II}$ emits twice, hypothesis $H_{\beta }$ – that each of the two sources at ${\phi }_1={\phi }^I$ and ${\phi }_2={\phi }^{II}$ with $q_1=q^I$ and $q_2=q^{II}$ emit once. The probabilities $P\left\{H_{\alpha }\right\}$ and $P\left\{H_{\beta }\right\}$ of the hypotheses come from a preceding step of data processing where parameter $q$ can be involved.

We define the circles $S_1=\left\{\phi :‖\phi -{\phi }_1‖\le R_1\right\}$ and $S_2=\left\{\phi :‖\phi -{\phi }_2‖\le R_2\right\}$, $‖·‖$ denotes Euclidian norm, with radii $R_1$ and $R_2$ where estimates ${\phi }^{\prime }$ and ${\phi }^{″}$ fall with high probability $P_S$: $\tilde{\phi }\in S_1$ for $H_{\alpha }$ and ${\phi }^{\prime }\in S_1$, ${\phi }^{″}\in S_2$ for $H_{\beta }$. Due to negligible probability $1-P_S^2$ of outliers beyond circles we will treat estimates ${\phi }^{\prime }$ and ${\phi }^{″}$ so that as if ${\phi }^{\prime }$, ${\phi }^{″}\in S_1$ or ${\phi }^{\prime }\in S_1$, ${\phi }^{″}\in S_2$. Assumption (c) on small estimation errors is quantified inside the confidence circles as $\Phi \left(\tilde{\phi }\right)\approx \Phi \left({\phi }_1\right)$ at $\tilde{\phi }\in S_1$ and $\Phi \left({\phi }^{″}\right)\approx \Phi \left({\phi }_2\right)$ at ${\phi }^{″}\in S_2$.

The identification is performed in the area where positions ${\phi }_1$ and ${\phi }_2$ will be placed at the distance of not more than $R_S+\delta R_S$, where $R_S=R_1+R_2$, $\delta R_S<R_S$. We take the assumption (d) that $\Phi \left({\phi }_2\right)$ is sufficiently approximated by a constant for the all locations ${\phi }_2$ from the annulus $A=\left\{{\phi }_2:R_1<‖{\phi }_2-{\phi }_1‖\le R_S+\delta R_S\right\}$. Thus, it is permissible to use in that area sample estimates $\stackrel{\wedge }{{\phi }^{\prime }}$ and $\stackrel{\wedge }{{\phi }^{″}}$ of ${\phi }^{\prime }$ and ${\phi }^{″}$ for the calculation of sample approximations $\stackrel{\wedge }{{\Phi }^{\prime }}\equiv \Phi \left(\stackrel{\wedge }{{\phi }^{\prime }}\right)$, $\stackrel{\wedge }{{\Phi }^{″}}\equiv \Phi \left(\stackrel{\wedge }{{\phi }^{″}}\right)$ of unknown true covariance matrix or matrices. As shown in the application, the stronger condition $\stackrel{\wedge }{{\Phi }^{\prime }}\approx \stackrel{\wedge }{{\Phi }^{″}}$ will be there fulfilled in the specific domain of closely spaced sources with a good accuracy.

Our purpose is to find the probabilities of that estimates ${\phi }^{\prime }$ and ${\phi }^{″}$ refer to one source in position ${\phi }_1$ and two sources in positions ${\phi }_1$ and ${\phi }_2$ for a given separation $r=‖{\phi }_1-{\phi }_2‖$ on conditions (a), (b), (c) and (d).

In the frame of this Section, random variables ${\phi }^{\prime }$ and ${\phi }^{″}$are considered as just the Gaussian two-dimensional estimates, not necessarily of location coordinate, with $E\left\{{\phi }^{\prime }\right\}={\phi }_1$ and $E\left\{{\phi }^{″}\right\}={\phi }_2$. The difference between estimates $\psi ={\phi }^{″}-{\phi }^{\prime }=\overline{\psi }+\Delta \psi$, $\overline{\psi }={\phi }_2-{\phi }_1$, $\Delta \psi =\Delta {\phi }^{″}-\Delta {\phi }^{\prime }$, is Gaussian with a mean $\overline{\psi }$ and covariance matrix $W_{\psi }=\Phi \left({\phi }_1\right)+\Phi \left({\phi }_2\right)$ with eigenvalues $l_1>l_2>0$. We propose the concept of $SRL$ based on a tangency of two confidence circles $S_1$ and $S_2$ of probability $P_S$ when $‖{\phi }_1-{\phi }_2‖=R_S$, which comes from the following Theorem.

Theorem.

1. The region $\Sigma$ encompassing $\overline{\psi }$ where $\psi$ falls with probability $P_S^2$ in polar coordinates $\left(r^{″},{\vartheta }^{″}\right)$, $\left(r^{\prime },{\vartheta }^{\prime }\right)$ originated correspondently in ${\phi }_2$ and ${\phi }_1$ is $\Sigma =$$=\left\{\overline{\psi }+\left[\begin{array}{c}{r}^{″}\cos {\vartheta }^{″}-r^{\prime }\cos {\vartheta }^{\prime }\\ r^{″}\sin {\vartheta }^{″}-r^{\prime }\sin {\vartheta }^{\prime }\end{array}\right]\right\}$, $r^{″}\in \left(0,R_2\right]$, $r^{\prime }\in \left(0,R_1\right]$ and ${\vartheta }^{″}$, ${\vartheta }^{\prime }\in \left[0,2\pi \right]$. The squared distance from $\overline{\psi }$ to any point of $\Sigma$ is ${\left(r^{″}\right)}^2+{\left(r^{\prime }\right)}^2-2r^{″}{r}^{\prime }\cos \left({\vartheta }^{″}-{\vartheta }^{\prime }\right)$ maximum of which is achieved at $r^{″}=R_2$, $r^{\prime }=R_1$ for ${\vartheta }^{″}-{\vartheta }^{\prime }=\pi$ and is equal to ${\left(R_1+R_2\right)}^2$. Consequently, $\Sigma$ is a circle centered at $\overline{\psi }$ with radius $R_S$ and bounded by the circumference ${\Sigma }_{С}$ including origin of coordinates.

2. The region $\Xi$ where variable $\psi$ falls with desired probability is formed by the intersection of the circles $\Sigma$ and $\Lambda =\left\{\psi :‖\psi ‖\le R_S\right\}$, so long as $\psi \in \Sigma$, see Figure 1a. Straight line $L$ connecting the intersection points of corresponding circumferences ${\Sigma }_{С}$ and ${\Lambda }_{С}$ is collinear in virtue of symmetry to the tangent $T$ to ${\Lambda }_{С}$ at the point $\overline{\psi }$. It divides the line from origin to $\overline{\psi }$ on two identical segments, hence the angle $\theta$ is equal to $\pi /6$, see Figure 1a again. We successively translate $\overline{\psi }$ in origin and rotate coordinates to obtain probability integral for $\Xi$ in terms of spectrum of $W_{\psi }$. Changing to polar coordinates ($s$,$\varphi$) we get where $g\left(\varphi ;l_1,l_2\right)\equiv \left(l_1^{-1}{\cos }^2\varphi +l_2^{-1}{\sin }^2\varphi \right)$. The ordinate divides $\Sigma$ on two semicircles ${\Sigma }_{left}$ and ${\Sigma }_{right}$, but two identical regions ${\Sigma }_u$ and ${\Sigma }_d$ associated with $\pi /6$ complement $\Xi$ to semicircle ${\Sigma }_{left}$, Figure 1b. From the symmetry of probability density in integral (3) with respect to angle $\varphi$ one has $P\left\{\psi \in {\Sigma }_{left}\right\}=P\left\{\psi \in {\Sigma }_{right}\right\}=0.5P_S^2$ and $P\left\{\psi \in {\Sigma }_u\right\}=P\left\{\psi \in {\Sigma }_d\right\}$.

3. We represent probability (3) as where $s(\varphi )=2R_S\sin \left(\varphi -\pi /2\right)=-2R_S\cos \varphi$, see Figure 1b again.

Performing the integral in (4) over variable $s$ leads to

We maximize it over $\varphi$ to evaluate $P\left\{\psi \in \Xi \right\}$ from below:

It can be easily verified that the derivative of the exponential function from (5) is non-positive, thus required maximum is achieved at $\varphi =2\pi /3$ and is equal to $\left(1-{\mathrm{e}}^{-\frac{R_S^2}{2}\left(\frac{1}{4l_1}+\frac{3}{4l_2}\right)}\right)$. We substitute $l_1$ with $l_2$ in exponent and perform integration in (5) [16] to achieve the inequality where $\overline{e}\equiv \sqrt{l_1/l_2}$. The expression in parentheses with exponent is the probability that realizations of $\psi$ will be in the concentration ellipsoid centered at origin with principal semi-minor axis $R_S$ and semi-major axis $\overline{e}{R}_S$. Region $\Sigma$ which contains $\psi$ with probability $P_S^2$ is the circle inscribed in the ellipsoid, so its confidence probability will be closer to unity than $P_S^2$. And if so, we substitute it with unity and end the Proof □.

Proceeding from the Theorem, we establish $SRL=R_S$ in RI $‖\psi ‖\le SRL$ ensuring $PRI<0.5P_S^2$, low bound of which depends on spectral characteristic $\overline{e}$. Function $1/2+$$+{\pi }^{-1}arctg\left[\overline{e}tg\left(2\pi /3\right)\right]$ is monotone decreasing by $\overline{e}$: it approaches $1/6$ at $\overline{e}\to 1$, hence $PRI>0.4905-0.1667=0.3238$ when $P_S=0.99$. In other hand, it approaches zero at $\overline{e}\to \infty$, hence $PRI\to 0.4905$. Since, say $\overline{e}>10$ we get $PRI>0.4721$ and can declare that $PRI$ occurs here in a proximity of $1/2$. When $\overline{e}\to 1$ concentration ellipsoid is degenerating into the circle with radius $l=l_1=l_2$ yet $g\left(\varphi ;l_1,l_2\right)=l^{-1}$ therefore the contribution of $2P\left\{\psi \in {\Sigma }_u\right\}$ in probability $P\left\{\psi \in {\Sigma }_{left}\right\}$ is in fact $2\left(\pi /6\right)\times$$\times 0.5P_S^2/\pi =P_S^2/6$. On the contrary, at $\overline{e}\to \infty$ it elongates along the semi-axis $l_1$, $l_1\gg l_2$, and $2P\left\{\psi \in {\Sigma }_u\right\}$ hereby approaches zero.

The consistency of the proposed $SRL$ with the $SRL$ for scalar estimates in the sense of $PRI$ is obvious for stretched concentration ellipsoid: $SRL$ from [1,7] for scalar $\psi$ is achieved at $\left|\overline{\psi }\right|={\sigma }_{\Delta \psi }$ with $PRI=P\left\{-{\sigma }_{\Delta \psi }\le \psi \le {\sigma }_{\Delta \psi }\right\}=0.477$ which also is not far from $1/2$. In general, $PRI$ lies in the interval (0.3238, 0.4905) at $P_S=0.99$.

Sample analogues $\stackrel{\wedge }{{\Phi }^{\prime }}$ and $\stackrel{\wedge }{{\Phi }^{″}}$ of one or two true covariance matrices are employed to achieve the approximation ${\overline{R}}_S$ of $R_S$ as ${\overline{R}}_S={R^{\prime }}_S+{R^{″}}_S$ where estimates ${R^{\prime }}_S$, ${R^{″}}_S$ approximate $R_1$ twice or $R_1$, $R_2$ once. Bayes sample space $C$ is defined to consist of two events, when the confidence circles encompassing sample estimates $\stackrel{\wedge }{{\phi }^{\prime }}$ and $\stackrel{\wedge }{{\phi }^{″}}$ intersect: $‖\stackrel{\wedge }{{\phi }^{″}}-\stackrel{\wedge }{{\phi }^{\prime }}‖\le {\overline{R}}_S$ at $C=C_1$ or do not intersect: $‖\stackrel{\wedge }{{\phi }^{″}}-\stackrel{\wedge }{{\phi }^{\prime }}‖>{\overline{R}}_S$ at $C=C_2$. Probabilities of the events $C_1$ and $C_2$ are: $P\left\{\left.C_1\right|H_{\alpha }\right\}=P\left\{\left(\left.{‖\psi ‖}^2\right|\overline{\psi }=0\right)\le {\overline{R}}_S^2\right\}$ and $P\left\{\left.C_1\right|H_{\beta }\right\}=P\left\{\left(\left.{‖\psi ‖}^2\right|\overline{\psi }\ne 0\right)\le {\overline{R}}_S^2\right\}$, $P\left\{\left.C_2\right|H_{\alpha }\right\}=P\left\{\left(\left.{‖\psi ‖}^2\right|\overline{\psi }=0\right)>{\overline{R}}_S^2\right\}$ and $P\left\{\left.C_2\right|H_{\beta }\right\}=$$=P\left\{\left(\left.{‖\psi ‖}^2\right|\overline{\psi }\ne 0\right)\right.\left.>{\overline{R}}_S^2\right\}$. Probability $P\left\{\left.C_1\right|H_{\alpha }\right\}$ is very near to unity thus $P\left\{\left.C_2\right|H_{\alpha }\right\}$ is very near to zero. Probability $P\left\{\left.C_1\right|H_{\beta }\right\}$ decreases asymptotically with the rise of $‖\overline{\psi }‖$ while $P\left\{\left.C_2\right|H_{\beta }\right\}$ asymptotically increases. We define all possible $\overline{\psi }$ on the circumference ${ℂ}_r=\left\{\overline{\psi }:‖\overline{\psi }‖=r\right\}$ in order to pass from unknown $P\left\{\left.C\right|H_{\beta }\right\}\left(r\right)\equiv {\left.P\left\{\left.C\right|H_{\beta }\right\}\right|}_{\overline{\psi }\in {ℂ}_r}$ to its minimum $P_C^{-}\left(r\right)=\underset{\overline{\psi }\in {ℂ}_r}{min}P\left\{\left.C\right|H_{\beta }\right\}\left(r\right)$ and maximum $P_C^{+}\left(r\right)=\underset{\overline{\psi }\in {ℂ}_r}{max}P\left\{\left.C\right|H_{\beta }\right\}\left(r\right)$ on ${ℂ}_r$. To develop Bayesian scheme we need the following Lemma.

Lemma. $P_{C_1}^{-}\left(r\right)>P_{C_1}^{-}\left(r^{\prime }\right)$, $P_{C_1}^{+}\left(r\right)>P_{C_1}^{+}\left(r^{\prime }\right)$ at $r<r^{\prime }\le {\overline{R}}_S$, and $P_{C_2}^{-}\left(r^{\prime }\right)>P_{C_2}^{-}\left(r\right)$, $P_{C_2}^{+}\left(r^{\prime }\right)>P_{C_2}^{+}\left(r\right)$ at ${\overline{R}}_S<r<r^{\prime }$.

Proof of Lemma. For the pair of some vectors ${\overline{\psi }}_r=r{\overline{\psi }}_1$ and ${\overline{\psi }}_{r^{\prime }}=r^{\prime }{\overline{\psi }}_1$, where ${\overline{\psi }}_1\in {ℂ}_1$, one has ${\overline{\psi }}_r\in {ℂ}_r$ and ${\overline{\psi }}_{r^{\prime }}\in {ℂ}_{r^{\prime }}$, and ${‖{\psi }_r‖}^2=r^2+r2{\overline{\psi }}_1^T\Delta \psi +{‖\Delta \psi ‖}^2$, ${‖{\psi }_{r^{\prime }}‖}^2={r^{\prime }}^2+r^{\prime }2{\overline{\psi }}_1^T\Delta \psi +$$+{‖\Delta \psi ‖}^2$. Variances of $r^{\prime }2{\overline{\psi }}_1^T\Delta \psi$ and $r2{\overline{\psi }}_1^T\Delta \psi$ are increased by factor ${r^{\prime }}^2$ and $r^2$ correspondently as compared to the variance of $2{\overline{\psi }}_1^T\Delta \psi$, hence due to the properties of Gaussian distribution $P\left\{\left(r^2+r2{\overline{\psi }}_1^T\Delta \psi \right)\le {\overline{R}}_S^2\right\}>P\left\{\left({r^{\prime }}^2+r^{\prime }2{\overline{\psi }}_1^T\Delta \psi \right)\le {\overline{R}}_S^2\right\}$ and $P\left\{\left({r^{\prime }}^2+r^{\prime }2{\overline{\psi }}_1^T\Delta \psi \right)>{\overline{R}}_S^2\right\}>P\left\{\left(r^2+r2{\overline{\psi }}_1^T\Delta \psi \right)>{\overline{R}}_S^2\right\}$. Thus, $P\left\{{‖{\psi }_r‖}^2\le {\overline{R}}_S^2\right\}>$$>P\left\{{‖{\psi }_{r^{\prime }}‖}^2\le {\overline{R}}_S^2\right\}$ and $P\left\{{‖{\psi }_{r^{\prime }}‖}^2>{\overline{R}}_S^2\right\}>P\left\{{‖{\psi }_r‖}^2>{\overline{R}}_S^2\right\}$ that completes the Proof □.

As comes from Lemma, $P\left\{\left.C_1\right|H_{\beta }\right\}\left(r\right)$ is smaller than $P\left\{\left.C_1\right|H_{\alpha }\right\}$ and approaches it from below at $r\to 0$; both $P_{C_1}^{-}\left(r\right)$ and $P_{C_1}^{+}\left(r\right)$ decrease monotonically with the rise of $r$ while $P_{C_2}^{-}\left(r\right)$ and $P_{C_2}^{+}\left(r\right)$ monotonically increase.

We apply binary Bayesian theorem to construct maximum $P_{\max }^{\beta }\left(C;r\right)\equiv$ $\equiv \underset{\overline{\psi }\in {ℂ}_r}{\max }P\left\{\left.H_{\beta }\right|C\right\}$ and minimum $P_{\min }^{\beta }\left(C;r\right)\equiv \underset{\overline{\psi }\in {ℂ}_r}{\min }P\left\{\left.H_{\beta }\right|C\right\}$ of the posterior probability of two sources, and minimum $P_{\min }^{\alpha }\left(C;r\right)$ and maximum $P_{\max }^{\alpha }\left(C;r\right)$ of the posterior probability of one source,

The preference of a hypothesis at small $r$ depends on which probability, $P\left\{H_{\alpha }\right\}$ or $P\left\{H_{\beta }\right\}$ is bigger: if a) $P\left\{H_{\beta }\right\}>P\left\{H_{\alpha }\right\}$ then $P_{\min }^{\beta }\left(C_1;r\right)\ge P_{\max }^{\alpha }\left(C_1;r\right)$, otherwise b) $P_{\max }^{\beta }\left(C_1;r\right)<P_{\min }^{\alpha }\left(C_1;r\right)$. Both $P_{\max }^{\beta }\left(C_1;r\right)$ (6) and $P_{\min }^{\beta }\left(C_1;r\right)$ (7) decrease as $r$ goes up while probabilities $P_{\min }^{\alpha }\left(C_1;r\right)$ (8) and $P_{\max }^{\alpha }\left(C_1;r\right)$ (9) grow. Hence, hypothesis $H_{\alpha }$ becomes more and more probable with $r$ from zero up to ${\overline{R}}_S$, while hypothesis $H_{\beta }$ is less and less probable. Starting with a certain $r$, we get for the case a) that $P_{\min }^{\beta }\left(C_1;r\right)<P_{\max }^{\alpha }\left(C_1;r\right)$ but still $P_{\max }^{\beta }\left(C_1;r\right)>P_{\min }^{\alpha }\left(C_1;r\right)$, at which preference is ambiguous, however with a further increase in $r$ hypothesis $H_{\alpha }$ begins to prevail, $P_{\max }^{\beta }\left(C_1;r\right)<P_{\min }^{\alpha }\left(C_1;r\right)$. For the case b) hypothesis $H_{\alpha }$ prevails at all $r\le R_S$. As $P_{C_2}^{-}\left(r\right)\gg P\left\{\left.C_2\right|H_{\alpha }\right\}$, probability $P_{\min }^{\beta }\left(C_2;r\right)$ is much bigger than $P_{\max }^{\alpha }\left(C_2;r\right)$ even though $P\left\{H_{\alpha }\right\}>P\left\{H_{\beta }\right\}$ for sensible prior probabilities.

We define for $C=C_1$ identification probabilities (IPs) $P_{C_1}^{\alpha }\left(r\right)\equiv P_{\min }^{\alpha }\left(C_1;r\right)$ of one and $P_{C_1}^{\beta }\left(r\right)\equiv P_{\max }^{\beta }\left(C_1;r\right)$ of two sources if $P\left\{H_{\beta }\right\}{P}_{C_1}^{+}\left(r\right)<P\left\{H_{\alpha }\right\}P\left\{\left.C_1\right|H_{\alpha }\right\}$, and $P_{C_1}^{\alpha }\left(r\right)\equiv$$\equiv P_{\max }^{\alpha }\left(C_1;r\right)$, $P_{C_1}^{\beta }\left(r\right)\equiv P_{\min }^{\beta }\left(C_1;r\right)$ if $P\left\{H_{\beta }\right\}{P}_{C_1}^{-}\left(r\right)>P\left\{H_{\alpha }\right\}P\left\{\left.C_1\right|H_{\alpha }\right\}$. IPs are not defined if $P_{C_1}^{-}\left(r\right)\le P\left\{\left.C_1\right|H_{\alpha }\right\}P\left\{H_{\alpha }\right\}/P\left\{H_{\beta }\right\}\le P_{C_1}^{+}\left(r\right)$ when a hypothesis could not be preferred. As for $C=C_2$, $P_{C_2}^{\alpha }\left(r\right)\equiv P_{\max }^{\alpha }\left(C_2;r\right)$ and $P_{C_2}^{\beta }\left(r\right)\equiv P_{\min }^{\beta }\left(C_2;r\right)$. To achieve preference the function ${\delta }_H\left(r;C\right)$ of relative difference between IPs is compiled: which is compared with a threshold $t_H>0$. At $\left|{\delta }_H\left(r;C\right)\right|<t_H$ preference is ambiguous. If ${\delta }_H\left(r;C_1\right)\ge t_H$ or ${\delta }_H\left(r;C_2\right)\ge t_H$ then hypotheses $H_{\alpha }$ or $H_{\beta }$ respectively prevail. When ${\delta }_H\left(r;C_1\right)\le -t_H$ we select $H_{\beta }$.

Probability distribution of squared distance between two Gaussian vectors (in our case it will be ${‖\psi ‖}^2$) is referred to the distribution of a quadratic form ${\psi }^{T}A\psi$, where $\psi$ is non-central Gaussian vector with a mean $\overline{\psi }$ and covariance matrix $W$ (for us $W=\stackrel{\wedge }{{\Phi }^{\prime }}+\stackrel{\wedge }{{\Phi }^{″}}$), $A$ is symmetric and nonnegative definite matrix. We obtain the desired distribution by equating $A$ to the identity matrix. Although its mathematical structure is complicated, modern computer resources allow implement supporting functions precisely [17].

The separation $r$ is to be chosen everywhere in the range of $0<r\le {\overline{R}}_S$ for the event $C_1$ or of $r>{\overline{R}}_S$ for the event $C_2$ in the use of identifier (10). Nevertheless, when it approaches ${\overline{R}}_S$ probability of both events may grow nearer to $1/2$, where they tend to be equally probable. To improve identification reliability we invent the probability threshold $t_C>0.5$ if necessary for determining such distances $r_1\left(t_C\right)<{\overline{R}}_S$ or $r_2\left(t_C\right)>{\overline{R}}_S$ that would afford a more valid event: $\left.P^{+}\left(r_1\right)\right|=t_C$ for $C_1$ or $\left.P^{-}\left(r_2\right)\right|=t_C$ for $C_2$. Beginning from ${\overline{R}}_S$ we decrease the distance $r$ up to $r_1$ or increase it up to $r_2$. Probability $t_C$ is assigned to be the probabilistic measure of event validity, which prescribes that events $C_1$ at $r\le r_1$ or $C_2$ at $r\ge r_2$, are both valid with probability no smaller than $t_C$. The region of $r\in \left(r_1,r_2\right)$ is defined to be of invalid event.

To summarize the results of Section 5 we present the core pseudo-code of the identification procedure:

if $C=C_1$ then

 Step 1. Identify one/two sources at $r\in (0,r_1]$

if$P_{C_1}^{-}\left(r\right)\le P\left\{\left.C_1\right|H_{\alpha }\right\}P\left\{H_{\alpha }\right\}/P\left\{H_{\beta }\right\}\le P_{C_1}^{+}\left(r\right)$

then No preference of a hypothesis else

if ${\delta }_H\left(r;C_1\right)\ge t_H$thenPrefer$H_{\alpha }$

else if${\delta }_H\left(r;C_1\right)\le -t_H$thenPrefer$H_{\beta }$

else No preference of a hypothesis

end if end if end if

else /$C=C_2$/

 Step 2. Identify one/two sources at $r\ge r_2$

if ${\delta }_H\left(r;C_2\right)<t_H$ thenNo preference of a hypothesis

else Prefer $H_{\beta }$

end if end if

The topic of a confidence circle has been addressed in a number of publications regarding circular error probability ($CEP$) integral [18,19,20,21,22], occurring in navigation and surveillance systems. $CEP$ is the radius of a confidence circle which contains a random variable with probability $0.5$.

The probability density for zero mean bivariate Gaussian variable $z={\left[\begin{array}{cc}x& y\end{array}\right]}^T$ with covariance matrix ${\rm K}$ is defined as where ${\sigma }_x$, ${\sigma }_y$ and ${\rho }_{\mathrm{cv}}$ are standard deviations and covariance of $x$ and $y$. Hence, function (11) is where $\rho ={\rho }_{cv}/\left({\sigma }_x{\sigma }_y\right)$ is a correlation coefficient.

Among others, Krempasky $CEP$ estimator [22] is the most attractive both owing to its numerical simplicity (it is closed-form) and high accuracy (the deviation of $CEP$ estimate from the exact value is less than 2 percent). Krempasky first rotates the $x$ and $y$ coordinates by the angle $\Gamma$: $tg\Gamma ={\left({\sigma }_y^2-{\sigma }_x^2\right)}^{-1}\left[-2{\rho }_{cv}+\sqrt{4{\rho }_{cv}^2+{\left({\sigma }_y^2-{\sigma }_x^2\right)}^2}\right]$ such that ${{\sigma }^{\prime }}_x^2={{\sigma }^{\prime }}_y^2={\left({\sigma }^{\prime }\right)}^2={\sigma }_x^{2}co{s}^2\Gamma +{\sigma }_y^{2}si{n}^2\Gamma +{\rho }_{cv}sin2\Gamma$ in transformed coordinates $x^{\prime }$ and $y^{\prime }$. Correlation coefficient is ${\rho }^{\prime }={\left({\sigma }^{\prime }\right)}^{-2}\left[0.5\left({\sigma }_y^2-{\sigma }_x^2\right)sin2\Gamma +{\rho }_{cv}cos2\Gamma \right]$ in the new coordinates. With polar coordinates the originating in (12) integral for $CEP$, expressed through variables $x^{\prime }$ and $y^{\prime }$, is written with angular coordinate $\vartheta$ after the integral over radial coordinate is performed:

The integral (13) is then expanded to the fourth order in the correlation coefficient ${\rho }^{\prime }$, and after much manipulation the final $CEP$ estimate is found to be

We substitute $0.5$ with $0.99$ in (13) to rework Krempasky $CEP$ estimator to estimate $R99$. Repeating the same manipulation as in [22], one can make sure that $R99$ estimate is deduced from (14) by substitution $\ln 2$ with $2\ln 10$. Finally, the expression for $R{99}_K$ is derived: