Bayesian multi-target localization using blocking ... - IEEE Xplore

IEEE ICC 2015 - Workshop on Advances in Network Localization and Navigation

Bayesian Multi-Target Localization using Blocking Statistics in Multipath Environments (Invited Paper) Sundar Aditya∗ , Student Member, IEEE, Andreas F. Molisch∗ , Fellow, IEEE and Hatim Behairy† ∗

Ming Hsieh Dept. of Electrical Engineering, University of Southern California † King Abdulaziz City for Science and Technology (KACST), Saudi Arabia Email: {sundarad, molisch}@usc.edu, [email protected]

Abstract—Passive localization based on distributed MIMO radar has a wide variety of applications ranging from security/military to consumer support. This paper considers the problem of localizing a (passive) target through time-of-arrival measurements from a number of (distributed) single-antenna transmitters to distributed receivers in the presence of objects that can block the line of sight between transmitter/receiver and target. The key effect investigated in this paper is the correlation of the blocking between an object and different transmitters/receivers. We formulate the problem in a Bayesian framework and suggest a sub-optimum but efficient algorithm to solve this problem; simulations show considerably better performance than can be achieved by ignoring the blocking statistics. Index Terms—Multi-target localization, Correlated blocking, distributed MIMO radar

I. I NTRODUCTION Over the last few years, there has been a steady rise in demand for accurate indoor localization solutions in order to support a multitude of applications, ranging from providing location-based advertising to users in a shopping mall to better solutions for tracking inventory in a warehouse. Many of these applications involve the deployment of multiple transmitters and receivers and hence, can be modeled using the distributed MIMO (multiple-input multiple-output) radar framework. Usually, there are multiple targets present which all have nearly identical radar signatures and hence, cannot be localized on that basis alone. In addition, for reasons of cost and energy efficiency, each transmitter (TX) and receiver (RX) may be equipped with a single antenna only. Hence, the radar cannot exploit the information contained in the angles of arrival/departure from the target-reflected signal. This motivates the study of multi-target localization without angular information in an indoor distributed MIMO radar setting. Apart from the ones mentioned above, other challenges in indoor localization are (i) targets can be blocked by non-target scatterers such as walls, furniture etc., (ii) these, and other, scatterers also give rise to multipath (indirect paths) which needs to be distinguished from direct paths that propagate directly from TX to target to RX. For passive (reflective) targets, each direct path (DP) gives rise to an ellipse passing through the target location, with the TX and RX at the foci and three or much such curves unambiguously determine the

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Fig. 1: Correlated blocking: An example

target location, (iii) in the presence of multiple targets, yet another challenge is to match the DPs to the right targets. An incorrect matching would result in ghost targets [1]. The fundamental limits of localization in MIMO radar networks were studied in [2]. A number of works have dealt with multi-target localization (MTL) using co-located antenna arrays at the transmitter and receiver [3]–[10]. The singletarget localization problem using widely-spaced antenna arrays was investigated in [11], [12] and the multi-target case in [13]. None of these works address the issues of blocking and multipath, common in an indoor environment. The localization and tracking of a single target in the presence of multipath and obstructed line-of-sight (OLOS) using distributed singleantenna TXs and RXs is investigated in [14]–[16]. The work closest to ours is [1] where the problem of multi-target localization in a distributed MIMO radar setting was addressed. While [1] considers the effect of multipath, it relies on the assumption of a constant and independent blocking probability for all DPs. In reality however, the DP blocking events in any environment are not mutually independent. As shown in Fig. 1, the location of the two TXs is such that if one of them has LoS to the target, it is highly likely that the other does as well. Similarly, if one of them is blocked w.r.t the target, it is highly likely that the other is as well. In other words, the DP blocking probabilities are in general correlated and the extent of correlation is a function of the network geometry. In this work, we investigate how the dependent nature of blocking probabilities can be exploited to obtain better location estimates for the targets. Our basic approach is as follows: when three or more ellipses intersect at a point, we first assume that they are DPs. We then compute the joint probability that LoS exists to the TXs and RXs in question at the point of intersection. Only when this probability is high enough do we conclude that a

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target is present. The main contributions of this work are as follows: • The MTL problem with single antenna TXs and RXs is modeled as a Bayesian estimation problem where the joint DP blocking distribution plays the role of a prior. • A sub-optimal polynomial time algorithm to solve this problem is proposed. This paper consists of six sections. In Section II, we model the existence of LoS as a Bernoulli random variable. The MTL problem is modeled as a Bayesian estimation problem in Section III, where the joint DP blocking distribution plays the role of a prior. Section IV proposes an algorithm with polynomial complexity to solve the MTL problem. Simulation results are presented in Section V and finally, Section VI concludes the paper. II. S YSTEM M ODEL Consider a distributed MIMO radar system with MX TXs and Mr RXs, each equipped with a single omni-directional antenna and deployed in an unknown environment. An unknown number of stationary point targets are present and the objective is to localize all of them. We assume that the environment has non-target scatterers too, which can either block some target(s) w.r.t some TX(s) and/or RX(s), and/or give rise to multipath (indirect paths). The location of these non-target scatterers is assumed to be unknown: we assume only the knowledge of TX and RX locations. The number of transmit-receive pairs (TRPs), denoted by n, equals MX Mr . Throughout this work, the convention is that the i-th TRP (i = 1, · · · , n) consists of the iT -th TX and iR -th RX, where iT = (i − 1) modulo MX + 1 and iR = b(i − 1)/MX c + 1. For simplicity, we restrict our attention to the twodimensional (2-D) case, where all the TXs, RXs and targets are assumed to lie on a plane. The extension to the 3-D case is quite straightforward. Let the TX and the RX for the i-th TRP be located at (ci , di ) and (ai , bi ), respectively. We assume that the TXs use orthogonal signals so that the RXs can distinguish between the TXs. Multipath components (MPCs) are extracted whenever their signal strength exceeds a threshold. All MPCs that do not involve a reflection off a target are assumed to be removed by a background cancellation technique. An MPC involving more than two reflections is assumed to be too weak to be detected. Furthermore, DPs can also be blocked by scatterers in the environment and finally, two or more MPCs could have their times of arrivals (ToAs) so close to one another that they can be unresolvable due to finite bandwidth. Under this model, each extracted MPC is either (i) a DP to one or more targets, (ii) an indirect path (TX-scatterer-target-RX or TX-target-scatterer-RX) or (iii) a noise peak. Due to finite resolution, it is possible for an MPC to be a combination of (i), (ii) and (iii) as well. Each MPC gives rise to a ToA estimate which, in turn, corresponds to a range estimate. If only additive white Gaussian noise (AWGN) is present at the RX, then each ToA estimate is perturbed by Gaussian errors having variance σ ˆ 2 . Thus, for a DP, the true range of a target w.r.t its TRP is corrupted by

AWGN of variance σ 2 = c2 σ ˆ 2 , where c is the speed of light in the environment. Suppose the i-th TRP has Ni MPCs extracted from its signal. Let ri = [ri1 ri2 · · · riNi ] ∈ R1×Ni denote the vector of range estimates at the RX of the i-th TRP and let r = [r1 r2 · · · rn ] ∈ R1×N1 N2 ...Nn denote the stacked vector of range estimates from all TRPs. If rij is a DP corresponding to a target at (xt , yt ), then the conditional pdf of rij given (xt , yt ), denoted by fDP (rij |xt , yt ), is normally distributed with σ 2 , where ri (xt , yt ) = p p mean ri (xt , yt ) and variance 2 2 (xt − ai ) + (yt − bi ) + (xt − ci )2 + (yt − di )2 is the true range of (xt , yt ) w.r.t the i-th TRP. In order to model the blocking in the environment, we define the following variables: ( 0, if (xt , yt ) is blocked w.r.t the i-th TRP ki (xt , yt ) = 1, otherwise (1) ki (xt , yt ) can be interpreted as a Bernoulli random variable when considering an ensemble of settings in which the scatterers are placed at random. ki (xt , yt ) = 0 if either the TX or the RX of the ith TRP does not have LoS to (xt , yt ). Hence, ki (xt , yt ) can be decomposed into a product of two terms as follows: ki (xt , yt ) = viT (xt , yt )wiR (xt , yt ), (2) ( 1, if the iT -th TX has LoS to (xt , yt ) viT (xt , yt ) = 0, else ( 1, if the iR -th RX has LoS to (xt , yt ) wiR (xt , yt ) = 0, else For brevity, we shall henceforth refer to ki (xt , yt ) as kit . For the t-th target at (xt , yt ), we define the vectors kt = [k1t · · · knt ], vt = [v1t · · · vMX Nt ] and wt = [w N1t · · · wMr t ]. It can be seen that kt = wt vt , where denotes the Kronecker product. kt is a vector of dependent Bernoulli random variables (Fig. 1) and represents the ground truth as far as the blocking characteristics at (xt , yt ) is concerned. kt can be estimated depending on which MPCs in r are classified as DPs to (xt , yt ). In order to do this, we define the following decision variables, ( 1, if rij is a DP to the t-th target k˜ijt = (3) 0, else ˜ t = [k˜11t · · · k˜1N t · · · k˜ijt · · · k˜nN t ] denote the Let k 1 n vector of decision variables corresponding to the t-th target. It is easy to see that the estimate of the ground truth kit , denoted Ni X ˆ t can be defined by kˆit , is nothing but k˜ijt . The vector k j=1

in a manner similar to kt . III. BAYESIAN L OCALIZATION PROBLEM Assuming the presence of T targets, let x = [x1 · · · xT ], ˆ = y = [y1 · · · yT ] denote the target locations. Also, let k

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ˆ1 · · · k ˆ T ], k = [k1 · · · kT ] and k ˜ = [k ˜1 · · · k ˜ T ]. The [k estimation of all target locations can be formulated as a Bayesian estimation problem in the following manner, ˜ k)P(k|x, ˆ y, k) maximizeT,x,y,k˜ fDP (r|x, y, k, P(k|x, y)f (x, y) s.t k˜ijt , kˆit ∈ {0, 1}, X k˜ijt = kˆit

∀i, j, t

(4) (5) (6)

j

where f (.) and P(.) denote a continuous probability density function (pdf) and a discrete probability mass function (pmf), respectively. We now proceed to simplify the objective function in (4). Firstly, P(k|x, y) is the same as P(k1 ; · · · ; kT ). It captures the blocking statistics in its entirety. As mentioned earlier, any kt is a vector of dependent Bernoulli random variables. In addition, any two blocking vectors kt and ku are also dependent, in general. This is intuitive as we would expect two nearby targets to experience similar blocking characteristics. A consequence of this dependence is that target-by-target localization is not optimal. However, for ease of computation, we resort to target-by-target detection in this paper, thereby implicitly assuming independentQblocking vectors at distinct locations, i.e., P(k1 ; · · · ; kT ) ≈ t P(kt ). The generalization to joint-target detection will be described in a future work. Among the 2n possible values for a n-length binary vector, kt can only take on (2MX − 1)(2Mr − 1) + 1 physically realizableNvalues. Each of these has a decomposition of the form wt vt and shall henceforth be referred to as consistent blocking vectors. All other binary vectors that cannot be decomposed as a Kronecker product are termed inconsistent. Moreover, if kt is inconsistent, P(kt ) = 0. ˆ y, k) Secondly, P(k|x, is the same as ˆ ˆ P(k1 ; · · · ; kT |k1 ; · · · ; kT ). We assume that an estimate kˆit is conditionally independent of other estimates, given its ground truth kit . Thus, ˆ1; · · · ; k ˆ T |k1 ; · · · ; kT ) = P(k

T n Y Y

P(kˆit |kit )

(7)

locations. In the absence of IP statistics, we make the simplifying assumption that P(kˆit = 1|kit = 0) is also a constant, ρ10 . The availability of measurement-based statistics for these quantities would obviously improve estimation performance. ˆ t can in principle take While an estimated blocking vector k n ˆ t is a short on all 2 values, a false alarm is less likely if k Hamming distance away from a consistent blocking vector that has high probability. Let K denote the set of all consistent blocking vectors. To simplify matters, we discard any target location (xt , yt ) if there is no vector in K which is at most a ˆ t . Given k ˆ t , let Kt ⊆ K unit Hamming distance away from k denote the set of consistent blocking vectors which are at most ˆ t . Thus, a unit Hamming distance away from k  if Kt is empty !   0, X Y ˆt) ≈ P(k (8) P(kˆit |kit ) P(kt ), else   kt ∈Kt

˜ the distribution fDP (r|x, y, k, ˜ k) does Thirdly, given k, not depend on k as it is not a decision variable. Therefore, ˜ k) = fDP (r|x, y, k). ˜ The distribution of r fDP (r|x, y, k, ˆ x and y decomposes into product form as the noise given k, on each rij is independent. Thus, ˜ fDP (r|x, y, k)  k˜ijt Y 1 (rij − ri (xt , yt ))2 √ exp − = σ2 2πσ i,j,t

(9)

Finally, the target distribution f (x, y) is assumed to be uniform. Hence, after taking logarithms, and assuming independent blocking vectors at distinct locations, (4) can be simplified as follows:   1 X ˜ minimize kijt (rij − ri (xt , yt ))2  − σ 2 i,j,t   X X √ ˆt) −  log P(k k˜ijt  log 2πσ (10) t

i=1 t=1

Ideally kˆit must equal kit . However, since kˆit is a random estimate of the ground truth kit (also a random variable, albeit one fixed by nature), there can be estimation errors. We distinguish between two kinds of errors: a) We decide kˆit = 0, when in fact kit = 1. This happens when we fail to detect the DP corresponding to the tth target at the i-th TRP because noise pushes the range estimate far away from the true value. If the noise is i.i.d for all TRPs, we may assume that P(kˆit = 0|kit = 1) = ρ01 for all i, t. b) We decide kˆit = 1 when actually, kit = 0. This happens when the DP for the t-th target at the i-th TRP is blocked but we mistake a noise peak/IP for a DP because it has the right range. P(kˆit = 1|kit = 0) depends on the IP distribution and varies according to TRP and target

i

i,j,t

s.t k˜ijt , kˆit ∈ {0, 1}, X k˜ijt = kˆit

∀i, j, t, m

(11) (12)

j

In order to solve the optimization problem in (10)-(12), a mechanism for detecting DPs corresponding to the same target is required. Let qt (r) denote the set of DPs corresponding to the t-th target. In terms of the solution to (10)-(12), qt (r) = {rij ∈ r|k˜ijt = 1}. Henceforth, we shall refer to qt (r) as a matching. In general, a matching q(r) is a set containing at most one MPC per TRP since no point target can give rise to two or more resolvable DPs to a given TRP i. A target can be localized if a matching containing at least three DPs corresponding to it can be identified. For a matching q(r) and a point (x, y), the first term in (10) determines if the ellipses corresponding to MPCs in q(r) pass through (x, y) or not. The

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second term in (10) plays the role of a prior by determining the probability of the blocking vector obtained from q(r), at (x, y). The objective in (10) is minimized only when both these quantities are small. In the next section, we describe our method of obtaining the correct matchings while taking into account the blocking statistics. IV. A LGORITHM The number of matchings possible for
T targets, n

TRPs and N MPCs per TRP is ( n3 N 3 + n4 N 4 · · · + nn N n )T . Hence, the computational complexity of a brute-force search over all possible matchings is O(N nT ), which is intractable for a large number of TRPs and/or targets. In order to obtain the correct matchings in a tractable manner, we build on them iteratively. Consider, without loss of generality, a matching (l−1) qt (r) for the t-th target consisting of MPCs from the first l − 1 TRPs (3 ≤ l ≤ n). Let these MPCs be denoted (l−1) by {r1j1 , ..., rl−1,jl−1 }. Given an initial matching qt (r), (l−1) (l−1) let (ˆ xt , yˆt ) denote its corresponding target location estimate [17]. For an MPC rljl of the l-th TRP, we define the following function, (l−1)

L(qt

(l−1)

(r); rljl ) = rljl − rl (ˆ xt

(l−1)

, yˆt

)

(13)

(l−1) qt (r)

If ∪ {rljl } consists entirely of DPs from a single (l−1) target at (xt , yt ), L(qt (r); rljl ) should be very small in magnitude. Using a Taylor’s series approximation, it can be (l−1) shown that L(qt (r); rljl ) is normally distributed with zero (l) (l) mean [1]. Let (ˆ xt , yˆt ) denote the target location estimate (l−1) ˆ (l) = [kˆ1t · · · kˆlt ] obtained from qt (r) ∪ {rljl } and let k t (l) (l) denote the estimated l-length blocking vector at (ˆ xt , yˆt ). (l−1) If qt (r) ∪ {rljl } consists entirely of DPs from the t(l−1) ˆ (l) ) th target, then both L((qt (r); rljl )) and − log P(k t must be very small. Therefore, we define two thresholds, δ and µ, and a composite blocking-aware likelihood function, (l−1) LB (qt (r); rljl ), as follows: # " L(q (l−1) (r); r ) ljl (l−1) (l) t ˆ ) LB (qt (r); rljl ) = , − log P(k t σ(q (l−1) (r); rlj ) t l (14) where of

(l−1)

σ(qt

(r); rljl )

(l−1) L(qt (r); rljl ). (l)

If

the standard deviation is (l−1) L(qt (r);rlj ) l (l−1) < δ and σ(qt (r);rljl ) (l)

(l−1)

ˆ ) < µ, then q (r) = q − log P(k (r) ∪ {rljl }. t t t (l) (l−1) Otherwise, qt (r) = qt (r). This motivates an algorithmic approach that is divided into stages, indexed by l. In general, let (z1 , z2 , ...zn ), a permutation of (1, 2, · · · , n), be the order in which TRPs are processed. For details on choosing a good TRP order, refer to [1]. At the beginning of the l-th stage (3 ≤ l ≤ n), each (l−1) matching qt (r) has at most l − 1 MPCs. During the l-th stage, we identify all the DPs among the MPCs of the zl -th (l) TRP to obtain qt (r) for all t. It is possible for there to exist multiple matchings for the same target. This occurs whenever (l) a matching is inconsistent. A matching qt (r) is said to be

ˆ (l) , consistent (inconsistent) if the estimated blocking vector, k t is consistent (inconsistent). A finite value of µ ensures that an ˆ (l) is a unit Hamming distance away from inconsistent vector k t (l) being consistent (8). Let Kt denote all the consistent l-length blocking vectors a unit Hamming distance from an inconsistent ˆ (l) . For each k(l) ∈ K(l) , there exists a matching that can be k t t t (l) derived from qt (r) which reflects the ground truth implied (l) by kt . All such matchings are stored in a structure M (t) and (n) at the end of the n-th stage, the matching qt (r) ∈ M (t) with the lowest value of the objective function in (10) is declared as the true matching for the t-th target. The location of the t-th target can then be estimated from this matching. Finally, we describe how to obtain an initial set of matchings that can be used for the third stage. Let P (i, l, ji , jl ) denote the set of points at which the ellipses corresponding to the ji -th MPC of the i-th TRP and the jl -th MPC of the lth TRP intersect. There can be at most four points in any (2) P (i, l, ji , jl ). For a matching qt (r) = {riji , rljl }, the target (2) (2) location estimate (ˆ xt , yˆt ) need not be unique. In fact, all the points in P (i, l, ji , jl ) are ML estimates of the target location (2) for qt (r). For our initial set of matchings, we compute P (z1 , z2 , jz1 , jz2 ) for all 1 ≤ jz1 ≤ Nz1 , 1 ≤ jz2 ≤ Nz2 . Each point (x, y) ∈ P (z1 , z2 , jz1 , jz2 ) gives rise to a prospective target location (ˆ x(2) , yˆ(2) ) = (x, y) and matching q (2) (r) = {rz1 ,jz1 , rz2 ,jz2 }. Essentially, any intersection of two ellipses is a prospective target location to begin with. The pseudocode for the algorithm is presented in Algorithm 1. We use two data structures, P and M to keep track of target locations. P is a collection of all P (., ., ., .)’s defined previously, which stores the points of interesection of pairs of ellipses. M is a collection of all M (t) ’s and contains matchings of size three or more and the corresponding target locations. In each stage, points may move from P to M. Care should be taken not to have the same point in both P and M as this could lead to counting the same target twice. The final target locations will be determined from M. For further details on flow of the algorithm and the structures P and M, refer to [18]. It is easy to see that the concentration of ellipse intersections would be denser around target locations. As a result, it is helpful to visualize Algorithm 1 as a method of clustering ‘closely located’ ellipse intersections. For n TRPs and N MPCs per TRP, the total number of ellipse intersections is O(n2 N 2 ), which is polynomial in the number of TRPs and targets. Hence, it is intuitive that the clustering performed in Algorithm 1 should also have polynomial complexity in the number of TRPs and targets. A more rigorous treatment of complexity is provided in [18]. It is important to note that no assumptions have been made on the blocking distribution and as a result, Algorithm 1 can be applied to empirical blocking distributions (obtained by either measurements or simulations) as well. In the next section, we present our preliminary simulation results.

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MX Mr No. of targets λ L σ Grid size δ µ ρ10 ρ01

Algorithm 1 Algorithm for solving the Bayesian multi-target localization problem 1: 2: 3: 4: 5: 6: 7:

8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40:

Initialize M = N U LL, P = N U LL, t = 0 Obtain the TRP processing order (z1 , z2 , · · · , zn ) [1] for each jz1 and jz2 do Compute all entries of P (z1 , z2 , jz1 , jz2 ) end for for l = 3 to n do for each (x, y) ∈ P (zu , zv , jzu , jzv ), (u, v ∈ {1, ..., l − 1}), ju ∈ {1, ..., Nzu }, jv ∈ {1, ..., Nzv })) and each rzl ,jzl do q (l−1) (r) = {rzu ,jzu , rzv ,jzv } if LB (q (l−1) (r); rzl ,jzl ) < (δ, µ) then t=t+1 (l) qt (r) = q (l−1) (r) ∪ {rzl ,jzl } (l) (l) Compute (ˆ xt , yˆt ) and remove (x, y) from P (zu , zv , jzu , jzv ) (l) Add qt (r) to M (t) ˆ (l) from q (l) (r) Derive k t t ˆ (l) is inconsistent then if k t (l) (l) for each kt ∈ Kt do (l) Obtain the corresponding qt (r) and (l) (l) (ˆ xt , yˆt ) and append to M (t) end for end if end if end for if l > 3 then for each t do (l−1) for each qt (r) ∈ M (t) do (l−1) if LB (qt (r); rzl ,jzl ) < (δ, µ) then (l) (l−1) qt (r) = qt (r ∪ {rzl ,jzl } (l) (l) ˆ Derive kt from qt (r) (l) ˆ is inconsistent then if k t (l) (l) for each kt ∈ Kt do Obtain the corresponding (l) (l) (l) (t) qt (r) and (ˆ xt , yˆt ) and append to M end for end if end if end for end for end if end for for each t do (n) Select the qt (r) in M (t) that minimizes (10) and (n) (n) return the corresponding (ˆ xt , yˆt ) end for

[− log(10−2 )

3 3 3 0.002m−2 0.5m 0.01m 20m × 20m [1 2 3] − log(10−3 ) − log(10−4 ) 2Q(δ) 2Q(δ)

∞]

TABLE I: Simulation Parameters

V. S IMULATION R ESULTS The simulation setting is as follows: the TX, RX and target locations are uniformly and independently distributed across a 20m × 20m grid. To account for the non-zero dimensions of scatterers, we model them as balls of diameter L with their centres drawn from a Poisson point process (PPP) of intensity λ. This model yields a joint blocking distribution that is relatively easy to compute. The validity of this model and the details of the derivation for the blocking distribution are provided in [18]. We have chosen λ = 0.002m−2 and L = 0.5m for our simulations. The standard deviation of ranging error, σ, is assumed to be 0.01m. With these parameters, the performance of Algorithm 1 was tested for three values of the threshold δ (1, 2 and 3) and four values of the threshold µ (− log 10−2 , − log 10−3 , − log 10−4 and ∞). µ = ∞ corresponds to the case where the blocking statistics are not taken into account. Two or more MPCs which are within a distance of 2σ apart are considered unresolvable. In such cases, the earliest arriving MPC is retained and the other MPCs are discarded. Finally, the probabilities ρ01 and ρ10 are both assumed to equal 2Q(δ), where Q(.) is the Q-function. A list of all the parameter values are provided in Table I. A target is considered to be missed if there is no estimate of its location within a distance of 3σ from the actual coordinates. Similarly, a false-alarm is declared whenever there is no target within a radius of 3σ from an estimated target location. The false-alarm and missed-detection probabilities for 3 TXs, 3 RXs and 3 targets are presented in Figs. (2) and (3), respectively. Unsurprisingly, the false-alarm rates decrease as both δ and µ decrease, but this comes at the expense of an increase in miss probabilities. The striking feature in Fig. (2) is the unit false-alarm probability when the blocking statistics are not considered. The reason for this as follows: in the early stages of Algorithm 1, it is possible for two or more sufficiently close MPCs from the same TRP to be mistaken as DPs for the same target, even if they are resolvable. This results in the formation of ghost targets around the true target location. Some ghost targets can be more than 3σ away from the actual target location and as a result, they will have few or no additional ellipse passing through them during the later stages of Algorithm 1. This results in a blocking vector that has low probability whenever the actual target location, which is not very far away, has a lot more ellipses passing through it. Thus, the threshold µ helps eliminate ghost targets whenever

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be obtained via measurements or simulations. Our simulations showed that neglecting the blocking statistics gave rise to unreasonably high false-alarm rates. In conclusion, we believe that exploiting blocking statistics is vital for improved localization accuracy, especially in indoor radar networks which have to deal with blocking and multipath. R EFERENCES

Fig. 2: Ignoring blocking statistics has a detrimental effect when it comes to false-alarm rates

Fig. 3: Missed-detection probabilities

the probability of the blocking vector at a prospective target location is sufficiently low. The fact that false-alarm events occur so often when blocking statistics are neglected emphasizes the importance of their use in localization. VI. S UMMARY AND C ONCLUSIONS In this paper, the MTL problem with single antenna TXs and RXs was formulated as a Bayesian estimation problem, with the joint DP blocking distribution acting as a prior. A sub-optimal polynomial-time algorithm was then proposed to solve this problem. An important feature of the algorithm is that it works even with empirical blocking statistics that may

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