In recent years, the number of applications requiring accurate position information has grown steadily; from navigation for autonomous vehicles  to crowd-sensing , location-based advertising  and virtual reality , to name a few. On a two-dimensional surface, a target can be localized if its distance (also known as range) to at least three fixed reference points, called anchors, is known (see Fig. 1
). For wireless systems, ranges can be estimated from the time-of-arrival (ToA) of aranging signal111This requires the targets and anchors to be synchronized. and for wideband systems in particular, ToA-based localization is especially attractive, since the finer time resolution due to the large bandwidth improves the accuracy of the range estimates , thereby resulting in accurate location estimates.
While the principle behind ToA-based localization is fairly straightforward, a variety of operating conditions and propagation phenomena, such as noise, interference, multipath, blocking, target mobility etc. render the task of designing a localization network challenging. In order to provide a reliable quality of service in terms of accuracy (e.g., an error of at most at least of the time, as mandated by the the E911 standard 
), it is important to characterize the probability distribution of the positioning error over an ensemble of operating conditions, especially for safety critical applications like autonomous vehicles or E911 emergency services. A commonly used metric for this purpose is the localization mean-squared error (MSE), which is a function of the anchor locations, the transmit powers, the propagation environment, as well as the choice of ranging and localization algorithms. In this work, we consider the impact of the anchor locations, relative to a target, on the MSE. Specifically, we consider a lower bound for the MSE, known as the squared position error bound (SPEB)[7, 8], which is satisfied by all positioning algorithms that return unbiased222An estimate of a target location is said to be unbiased if . estimates of the target location. The SPEB is a function of the anchor geometry and importantly, does not depend on a specific localization algorithm. As a result, it is well-suited as a metric to analyze the impact of the anchor geometry on the positioning error. If the SPEB exceeds a pre-defined threshold, , then the target is said to be in outage
. Over an ensemble of target and/or anchor locations, the SPEB (and the MSE, as well) is a random variable and characterizing its complementary cumulative distribution function (ccdf) in closed-form (i.e.,, as a function of ) is important from a design perspective, as it can be used to determine a deployment of anchors that can guarantee an outage probability of at most 333For a given error threshold, , an outage probability of at most can be guaranteed if and only if the condition is satisfied, which poses a constraint on the shape of the SPEB ccdf..
Given anchors in a region, a natural model for capturing the randomness in the anchor locations is the well-known binomial point process (BPP) [9, Chap. 2], in which the anchors are distributed independently and uniformly over the region. In this paper, we attempt to derive a closed-form expression for the SPEB ccdf for such an anchor model444Typically, the number of anchors, , is also a random variable, often modeled as having a Poisson distribution. Together with the randomness in the anchor locations, this corresponds to the well-known homogeneous Poisson point process (PPP), which has been used to analyze the localization performance of a variety of wireless networks
, is also a random variable, often modeled as having a Poisson distribution. Together with the randomness in the anchor locations, this corresponds to the well-known homogeneous Poisson point process (PPP), which has been used to analyze the localization performance of a variety of wireless networks[10, 11, 12, 13, 14]. Hence, the results presented in this paper for the BPP anchor model can be readily extended for the PPP case by averaging over the distribution of .. Our approach is summarized below.
For a given target, we assume that the anchors that are within its communication range are distributed according to a BPP over an annular region centered at the target. For this setup, we model the SNR heterogeneity across different anchor-target links using a pathloss model. As a result, the SPEB metric is a function of the anchor distances and angular positions, relative to the target.
Given anchors, we rearrange the SPEB expression and reduce it in terms of the product of two random variables, and . While depends only on the anchor distances, depends on both the distances and angular positions of the anchors. In particular, and are statistically dependent.
We then proceed to demonstrate that the conditional distribution of , given , is difficult to characterize in closed-form. Hence, through constrained moment matching, we derive an approximation for , denoted by , which depends only on the angular positions of the anchors and has the same mean as . In particular, and are statistically independent.
Through simulations, we verify that the derived SPEB ccdf accurately estimates the true ccdf. Thus, from a design perspective, our contribution is useful in determining the number of anchors required in order to satisfy , for any . We also show that the accuracy of our approach is superior to that of other approaches that ignore SNR heterogeneity, which serves to highlight the impact of SNR heterogeneity on the SPEB (and consequently, the MSE) distribution.
I-B Related Work
There have been a number of recent works that have focused on the impact of anchor geometry on the localization error performance; specifically, for the SPEB metric, the impact of the target being situated within the convex hull of the anchors was investigated in , while scaling laws, with respect to the number of anchors within communication range, were derived in . A related, but simpler metric, known as the geometric dilution of precision (GDOP) has been studied extensively for the BPP anchor model. The GDOP corresponds to a special case of the SPEB when all the anchor-target links have the same SNR. The asymptotic distribution of the GDOP, as the number of anchors approaches infinity, was derived using U-statistics in . For the more realistic case of a finite number of anchors, the max-angle metric was proposed and analyzed in  and shown to be correlated to the GDOP. An approximate GDOP distribution was presented in , using the order statistics of the inter-node angles, while the exact GDOP distribution was characterized in . To the best of our knowledge, ours is the first work to consider the more realistic scenario where the anchor-target links may have different SNRs (due to the anchors being situated at different distances from the target), which increases the difficulty of the problem considerably, as highlighted in Section II.
Throughout this work, bold lower case letters are used for deterministic vectors. In particular,denotes the all-one vector. Uppercase letters in serif font are used for scalar random variables (e.g., ), while random vectors are underlined and similarly represented (e.g., ). For square matrices, the trace and inverse operators are respectively denoted by and . represents the real numbers, the complex numbers, the imaginary unit, and the imaginary part of . For random variables and , , and , respectively, while denotes the conditional ccdf of , given . denotes the probability measure, while denotes the expectation operator over the distribution of . A real, parametrized function , with argument and parameters given by a vector, , is denoted by . For , the sine and cosine integrals, denoted by and , respectively, are defined as follows:
denotes the indicator function and finally, the function is defined as follows:
This paper is divided into five sections. The system model is described in Section II, where the anchors are modeled by a BPP over an annular region surrounding the target, and a distance-dependent pathloss model is assumed for the SNRs of the anchor-target links. Under these conditions, we illustrate the difficulty of characterizing the SPEB distribution in Section III, which motivates the derivation of a tractable approximation for the SPEB ccdf later on in the same section. In Section IV, we compare the accuracy of our approach with other bounds and approximations that do not consider SNR heterogeneity. Finally, Section V concludes the paper.
Ii System Model
Consider a target situated in that needs to be localized. Since we are interested in the anchor geometry relative to the target, we can assume, without loss of generality, that the target is situated at the origin, . Centered at the target, consider anchors deployed according to a BPP over an annular region from to 555 can be interpreted as the distance beyond which is too weak to be detected by the target., denoted by , and let denote the location of the -th anchor in polar coordinates . Let
, having Fourier transform, denote the ranging signal transmitted by the anchors666We assume that the anchors coordinate their transmissions to avoid interference at the target. As a result, ToA/range estimation is noise-limited. and let denote the signal received from the -th anchor, which can be modeled as a superposition of a number of multipath components (MPCs) in the following manner:
where the location-dependent quantities , and respectively denote the number of observed MPCs, the complex amplitude of the -th MPC and its ToA. is the measurement noise, which is modeled as a zero-mean complex Gaussian random process, having a power spectral density of . We assume that line-of-sight exists from the target to all the anchors. Hence, the first arriving MPC from each anchor corresponds to the direct path (DP) and depends on the anchor position as follows:
denotes the speed of light in free space. The other MPCs are known as indirect paths (IPs) and we assume no prior knowledge of their statistics. Under these conditions, the MSE of an unbiased estimate of the target location can be bounded using the Cramer-Rao lower bound (CRLB)[7, 8], as follows:
is commonly known as the squared-position error bound (SPEB) [7, 8] in localization terminology. The term is referred to as the ranging information intensity (RII) from the -th anchor and is a measure of the ranging accuracy associated with the -th anchor777 is the reciprocal of the CRLB for an unbiased estimate of .. It is a function of the DP SNR, , the effective bandwith, , and the path overlap factor, , which determines the extent of overlap between the DP and subsequent MPCs, due to finite bandwidth888The expression for can be found in .. For simplicity, we assume for all , which corresponds to the case where the DP does not overlap with any other MPC, thereby resulting in the most accurate estimate of . Furthermore, is a function of the DP attenuation, , for which the following pathloss model is assumed:
For anchors having line-of-sight to the target, the inverse-square law pathloss model in (14) is a reasonable assumption for the DP component if there is zero path overlap, which, in turn, can be assumed when , where denotes the breakpoint distance associated with the ground reflection , since zero overlap between the DP and the ground-reflected path can be rarely achieved.
Apart from the RIIs, which depend primarily on the ranges, also depends on the angular geometry of the anchors, which is captured in (6) by the outer product , where is the unit vector in the direction of the -th anchor. In summary, the -th term in the summation in (6) represents the contribution of the -th anchor to . From (6)-(14), can be expressed as follows:
Since does not depend on any particular positioning algorithm, it is well-suited as a metric to analyze the impact of anchor geometry on the MSE. Moreover, many positioning algorithms have been proposed in recent years that have been shown to satisfy (6) with equality [22, 23, 24, 25]. Hence, for the remainder of this paper, we assume that the MSE is identical to .
For the special case when all the anchors are at the same distance from the target (i.e., all links have the same SNR), reduces to another well-known metric called the Geometric Dilution of Precision (GDOP), which is denoted by and has the following expression999Technically, the GDOP is defined as the square root of  and thus, has the units of distance. However, in order to have a fair comparison with (which has units of distance-squared), we slightly abuse the notation and refer to as the GDOP in this work.:
Compared to , is more tractable for a statistical characterization, since it can be decomposed into a product of two independent random variables, and , as shown in (17). However, since the terms are weighted differently in the denominator of (15), it is, in general, not possible to express as , for some and , in much the same way as it is generally not possible to represent an expression like as , for some scalar-valued real functions, and and arbitrary real values of and . Hence, for the sake of tractability, we formulate an approximation that allows a decomposition of , along the lines of (17), in the following section.
Iii Characterizing SPEB distribution
Although cannot, in general, be decomposed as a product of independent random variables, a partial decomposition can be obtained as shown in the lemma below:
The expression for in (15) can re-written as follows:
See Appendix -A. ∎
While depends only on , is a function of both and . Moreover, since is a function of , the two random variables are statistically dependent. Let denote the ccdf of , which can be expressed as follows:
Before proceeding to derive an expression for , we consider the GDOP special case, for which the evaluation of (III) is relatively simpler.
From (26), the ccdf of , denoted by , can be obtained as follows:
where and denote the zeroth and first order Bessel functions, respectively, while the pdf of is given by
Given , the support (i.e., the feasible set of values) of and is , where the minimum value, , is attained when the anchors are located at the vertices of a regular -sided polygon inscribed within a circle of radius . Due to the common support, can be interpreted as a GDOP-based approximation of , where the averaging over in (III) partially takes into account the SNR heterogeneity, while retaining the GDOP structure. In , the authors considered an alternate GDOP-based approximation for , given below, using the average link SNR:
where is proportional to the average link SNR due to the pathloss model assumed in (14). While also partially accounts for SNR heterogeneity, its minimum value is , since . To illustrate the impact of this support mismatch, consider the ratio between the minimum values of and , provided below:
Since (32) is increasing in , we can reasonably conclude that the approximation given by (31) is unlikely to be accurate when the difference between and is large. Note that (32) holds for the inverse-square law pathloss model only.
We now focus our attention back to the general case of deriving a closed-form expression for from (III), by characterizing the marginal distribution of and the conditional distribution , given .
The characteristic function of is given by:
and is given by (3).
See Appendix -B. ∎
From , the ccdf of can be evaluated as follows :
We have chosen to characterize by its ccdf instead of its pdf, since the ccdf can be obtained from the characteristic function by evaluating a single integral, which is computationally less intensive than the double integral required to obtain the pdf. Since is non-negative, the expected value of , for a differentiable real function , can be expressed in terms of , by considering the following relation:
where denotes the derivative of . Thus, by applying the expectation operator on both sides of (3), we obtain
While the marginal distribution of is fairly tractable, as it is the sum of independent and identically distributed (iid) random variables, the same cannot be said of . To illustrate this, consider the expression for , given :
, the conditional joint distribution of, given , is required, which is not easy to express in closed-form. From (23), it is clear that the dependence between and is induced by the collection of random variables, . For the sake of tractability, we remove this dependence by assuming , for some ; furthermore, to obtain a random variable whose second-order statistics match that of , we approximate as follows:
with the values of and being obtained by moment matching with .
For the special case when , the approximation in (39) reduces to an equality (i.e., ), with and .
), the mean and variance ofare given by:
where (40) follows as a result of being an iid uniform random vector over . Equating and with the corresponding quantities for , i.e., and , we obtain the following expressions for and :
However, since is non-negative, a similar requirement on imposes the following constraint on :
The term in square parentheses in (44) can be interpreted as the squared-distance of an -step two-dimensional random walk with unit step size; thus, it has a maximum value of , obtained when all the steps are in the same direction (i.e., , for all ). Therefore,
However, from Fig. 2, it can be seen that (46) is not satisfied for any ; in fact, the expression on the left-hand side of (46) becomes increasingly negative as increases. Thus, it follows that (42), (43) and (45) are not satisfied simultaneously. In particular, the expression for in (42) is greater than . As a result, we optimize for the values of and in the following manner:
where the above optimization problem can be viewed as constrained moment matching, due to the non-negativity constraint on imposed by (45). From (48) and (40), the objective function in (47) can be represented in terms of a single parameter, , as follows:
For , the expression in (49) is initially a monotonically decreasing function of and attains a minimum value of zero, for given by (42). However, as observed previously, this value of does not lie in the feasible region, . Consequently, the minimum value of (49) over the interval is attained at . Thus, the optimal solutions for and , denoted by and , respectively, are given by:
where the expression for is given by the following lemma:
The mean of is given by
See Appendix -C. ∎
Incidentally, note that . To see this, can be expressed as follows:
Since is a vector of identically distributed random variables, given , we have
While, in retrospect, approximating by its mean may seem like an obvious choice, the optimality of this approach from a constrained moment matching perspective is not self-evident.