1

A Statistical Characterization of Localization
Performance in Wireless Networks
Christopher E. O’Lone, Graduate Student Member, IEEE,

arXiv:1710.07716v1 [cs.NI] 20 Oct 2017

Harpreet S. Dhillon, Member, IEEE, and R. M. Buehrer, Fellow, IEEE

Abstract
Localization performance in wireless networks has been traditionally benchmarked using the CramérRao lower bound (CRLB), given a fixed geometry of anchor nodes and a target. However, by endowing
the target and anchor locations with distributions, this paper recasts this traditional, scalar benchmark
as a random variable. The goal of this work is to derive an analytical expression for the distribution of
this now random CRLB, in the context of Time-of-Arrival-based positioning.
To derive this distribution, this work first analyzes how the CRLB is affected by the order statistics
of the angles between consecutive participating anchors (i.e., internodal angles). This analysis reveals
an intimate connection between the second largest internodal angle and the CRLB, which leads to
an accurate approximation of the CRLB. Using this approximation, a closed-form expression for the
distribution of the CRLB, conditioned on the number of participating anchors, is obtained.
Next, this conditioning is eliminated to derive an analytical expression for the marginal CRLB
distribution. Since this marginal distribution accounts for all target and anchor positions, across all
numbers of participating anchors, it therefore statistically characterizes localization error throughout an
entire wireless network. This paper concludes with a comprehensive analysis of this new network-wideCRLB paradigm.

Index Terms
Cramér-Rao lower bound, localization, order statistics, Poisson point process, stochastic geometry,
Time of Arrival (TOA), mutual information, wireless networks.
The authors are with Wireless@VT, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg,
VA, USA. Email: {olone, hdhillon, buehrer}@vt.edu.
This paper was presented in part at the IEEE ICC 2017 Workshop on Advances in Network Localization and Navigation
(ANLN), Paris, France [1].

2

I. I NTRODUCTION
The Global Positioning System (GPS) has for decades been the standard mechanism for position location anywhere in the world. However, the deployment locations of recent and emerging
wireless networks have begun to put a strain on the effectiveness of GPS as a localization solution.
For example, as populations increase, precipitating the expansion of urban environments, cell
phone use in urban canyons, as well as indoors, is continually increasing. The rise of these GPSconstrained environments highlight the need to fall back on the existing network infrastructure
for localization purposes.
Additionally, with the emergence of Wireless Sensor Networks (WSNs) and their increased
emphasis on energy efficiency and cost-effectiveness, [2], [3], the possibility of equipping each
potential target node with a GPS chip quickly becomes impractical. Furthermore, the deployment of these networks in GPS-constrained environments further necessitates a reliance on the
terrestrial network for a localization solution. Thus, localization within a network, performed by
the network itself in the absence of GPS, has begun to garner attention, [4], [5].
Benchmarking localization performance in wireless networks has traditionally been done using
the Cramér-Rao lower bound, which provides a lower bound on the position error of any unbiased
estimator [6]. Common practice has been to analyze the CRLB in fixed scenarios of anchor nodes
and a target, e.g., [7], [8], [9], [10]. This strategy produces a scalar/fixed value for the CRLB and
is specific to the scenario being analyzed. While this idea does provide insight into fundamental
limits of localization performance, it is rather limited in that it does not take into account all
possible setups of anchor nodes and target positions within a network.
In order to account for all possible setups, it is useful to appeal to the field of stochastic
geometry. Whereas in the past stochastic geometry has been applied towards the study of
“connectivity, capacity, outage probability, and other fundamental limits of wireless networks,”
[11], [12], we now however apply it towards the study localization performance in wireless
networks. Modeling anchor node and target placements with point processes opens the possibility
of characterizing the CRLB over all setups of anchor nodes and target positions. Thus, the CRLB
is no longer a fixed value, but rather a random variable (RV) conditioned on the number of
participating anchors, where the randomness is induced by the inherent randomness of the anchor
positions. Upon marginalizing out the number of participating anchors, the resulting marginal
distribution of the CRLB will characterize localization performance throughout an entire wireless

3

network.
A. Related Work
The quest for a network-wide distribution of localization performance comprises two main
steps. The first step involves finding the distribution of the CRLB conditioned on the number
of participating anchor nodes, and the second step involves finding the probability that a given
number of anchors can participate in a localization procedure.
With regards to the first step, there have been several attempts in the literature to obtain this
conditional distribution. An excellent first attempt can be found in the series of papers, [13],
[14], [15], in which approximations of this conditional distribution were presented for ReceivedSignal-Strength (RSS), Time-of-Arrival (TOA), and Angle-of-Arrival (AOA) based localization,
respectively. These approximations were obtained through asymptotic arguments by driving the
number of participating anchor nodes to infinity. While these approximate distributions are very
accurate for larger numbers of participating anchors, they are less than ideal for lower numbers.
However, it is desirable that this conditional distribution be accurate for lower numbers of
participating anchors, since this is the dominate case in terrestrial networks, e.g., cellular.
This conditional distribution of the CRLB was also explored in [16]. Here, the authors were
able to derive the true expression for this conditional distribution through a clever re-writing of
the CRLB using complex exponentials. This distribution was then used to derive and analyze the
so-called “localization outage probability” in scenarios with a fixed number of randomly placed
anchor nodes. While this expression represents the true conditional distribution, its complexity
puts it at a disadvantage over simpler approximations (discussed further in Section III-D).
The second step, which involves finding the participation probability of a given number of
anchor nodes, was explored in [17]. In this work, the authors modeled a cellular network with a
homogeneous Poisson point process (PPP), which consequently allowed them to derive bounds
on L-localizability, i.e., the probability that a mobile device can hear at least L base stations for
participation in a localization procedure.1 Further, by employing a “dominant interferer analysis,”
they were able to derive an accurate expression for the probability of L-localizability, which
can easily be extended to give the probability of hearing a given number of anchor nodes for
participation in a localization procedure.
1 The

term hearability (or hearable) is used to describe anchor nodes that are able to participate in a localization procedure,

i.e., their received SINR at the target is above some threshold.

4

B. Contributions
This paper proposes a novel statistical characterization of a wireless network’s ability to
perform TOA-based localization. Using stochastic geometry to model target and anchor node
placements throughout a network, and using the CRLB as the localization performance benchmark, this paper presents an analytical derivation of the CRLB’s distribution.2 This distribution
offers many insights into localization performance within wireless networks that previously were
only attainable through lengthy, parameter-specific, network simulations. Thus, this distribution
1) offers a means for comparing networks in terms of their localization performance, by
enabling the calculation of network-wide localization statistics, e.g., avg. localization error,
2) unlocks insight into how changing network parameters, such as SINR thresholds, processing
gain, frequency reuse, etc., affect localization performance throughout a network, and
3) provides network designers with an analytical tool for determining whether a network meets,
for example, the FCC E911 standards [18].
In pursuit of this distribution, this paper makes four key contributions. First, this work presents
an analysis of how the CRLB is affected by the order statistics of the internodal angles. This
analysis reveals an intimate connection between the second largest internodal angle and the
CRLB, which leads to an accurate approximation of the CRLB. Second, this approximation is
then used to obtain the distribution of the CRLB conditioned on the number of hearable anchors.
Although this is a distribution of the CRLB approximation, its simplicity and accuracy offer clear
advantages over the true distribution presented in [16] and the approximate distribution presented
in [14]. These advantages are discussed further in Section III-D.
Third, this work then takes a major step by combining this conditional distribution of the CRLB
with the distribution of the number of participating anchors. This eliminates the conditioning
on a given number of anchors, allowing for the analytical expression of the marginal CRLB
distribution to be obtained. Since this marginal distribution now simultaneously accounts for
all possible target and anchor node positions, across all numbers of participating anchor nodes,
it therefore statistically characterizes localization error throughout an entire wireless network,
and thus signals a departure from the existing literature. Additionally, since the two component
distributions are parameterized by various network parameters, our resulting marginal distribution
2 We

will be using the square root of the CRLB as our performance benchmark, however, we just state the CRLB here as to

not unnecessarily clutter the discussion. This will be described further in Section III-A.

5

of the CRLB will also be parameterized by these network parameters. Consequently, our final
contribution involves a comprehensive analysis of this new network-wide CRLB paradigm, where
we examine how varying network parameters affects the distribution of the CRLB — thereby
revealing how these network parameters affect localization performance throughout the network.
II. P ROBLEM S ETUP
This section details the assumptions we propose in determining the network layout as well
as the localization procedure. Additionally, we describe important notation and definitions used
throughout the paper and conclude with how the assumptions impact the network setup.
A. Network Setup and Localization Assumptions
Assumption 1. We assume a ubiquitous, two-dimensional wireless network with anchor nodes
distributed according to a homogeneous PPP over R2 . Further, we assume potential targets to be
distributed likewise, where the anchor and target point processes are assumed to be independent.
Remark. This assumption for modeling wireless networks is common in the literature, e.g., [19],
[20], [21], [22], [23].
Assumption 2. We assume a 2-Way TOA-based positioning technique is used within the twodimensional network.
Remark. Although Time-Difference-of-Arrival (TDOA) is also commonly implemented, 2-Way
TOA represents a viable approach, and additionally offers a lower bound on TDOA performance.
Furthermore, the 2-Way assumption eliminates the need for clock synchronization between the
target and anchor nodes.
Assumption 3. Range measurements are independent and exhibit zero-mean, normally distributed range error.
Remark. A reader familiar with wireless positioning will recognize this as a classic Line-of-Sight
(LOS) assumption. However, those familiar with localization in terrestrial networks will realize
that Non-Line-of-Sight (NLOS) measurements are more common. Thus, while we move forward
under this LOS assumption in order to make progress under this new paradigm, we will see in
Section IV that we can adapt our model to accommodate NLOS measurements by selecting
ranging errors consistent with NLOS propagation.

6

TABLE I
S UMMARY OF N OTATION
Symbol
fX (·)
h(X)
u(x)
k·k
1[·]
tr(X)
L
S
Θk
∠k
X(k)
α
β
K

Description
Probability distribution function (pdf) of X
Differential entropy of X
Unit step function, 0 when x < 0, 1 when x ≥ 0
`2 -norm
Indicator function; 1[A] = 1 if A true, 0 otherwise
Trace of the matrix X
Number of participating anchors, L ∈ {0, 1, 2, . . . }
The localization performance benchmark
Anchor node angle k, a random variable
Internodal angle k, a random variable
k th order statistic of a sequence {Xi } of RVs
Path loss exponent, α > 2
Post-proc. SINR threshold
Frequency reuse factor

Symbol
FX (·)
I(X; Y )
N (µ, σ 2 )
P[A]
b·c
XT
N
M
θk
ϕk
λ
γ
q
σr2

Description
Cumulative distribution function (cdf) of X
Mutual Information between X and Y
Normal distribution, mean µ, variance σ 2
Probability of event A
Floor function
Transpose of the matrix X
Max anchors tasked to perform localization
Fixed localization error, in meters, M ∈ R+
A realization of Θk
A realization of ∠k
Density of PPP of anchor locations
Processing gain
Average network load (i.e., network traffic)
Common range error variance

Assumption 4. The range error variance, σr2 , is common among measurements from participating
anchor nodes and is considered a known quantity.
Remark. This assumption is often made in the literature for range-based localization, e.g., [10],
[25], and although not realistic in every scenario, it allows us to gain initial insight into the
problem and will be relaxed in future work.
B. Notation
The notation used throughout this paper can be found in Table I.
C. Localizability
We now define the terms localizable and unlocalizable, which were introduced in [17].
Definition 1. We say that a target is localizable if it detects localization signals from a sufficient
number of anchor nodes such that its position can be determined without ambiguity.
Remark. Under Assumption 2, this implies that L ≥ 3. We also define unlocalizable to be the
negation of Definition 1.
For the purposes of this setup and subsequent derivations, we will initially only consider
scenarios in which the target is localizable, to avoid unnecessary complication. Later in Section

7

III-E and those which follow, we will account for scenarios in which the target is unlocalizable,
and will modify our results accordingly.
D. Impact of Assumptions
With these assumptions in place, we now describe how they impact the network setup.
From Assumption 1, since the anchors and potential targets are distributed by independent,
homogeneous PPPs over R2 (i.e., stationary), then without loss of generality, we may perform
our analysis for a typical target placed at the origin of the xy-plane [21]. This is due to the
fact that the independence and stationarity assumptions imply that no matter where the target is
placed in the network, the distribution of anchors relative to the target appears the same.
Next, we assume that the number of hearable anchors is some fixed value, L, and begin by
numbering these anchors in terms of their increasing distance from the origin (target position).
This is depicted in Fig. 1 for a particular realization of a homogeneous PPP in which there
are four hearable anchor nodes. Fig. 1 also depicts how their corresponding angles, measured
counterclockwise from the +x-axis, are labeled accordingly. Assumption 1 further implies that
these angles of the hearable anchors are i.i.d. random variables that come from a uniform
distribution on [0, 2π).
Definition 2. If the target is placed at the origin of an x y-plane, then the term anchor node
angle, Θ k , is defined to be the angle corresponding to hearable anchor node k, measured
i.i.d.

counterclockwise from the +x-axis. Note that Θ k ∼ unif[0, 2π), ∀k ∈ {1, . . . , L}.
In later sections we will see that the distances between the anchor nodes and the target are
important for determining how many anchors are able to participate in a given localization procedure. However, under Assumption 4, once particular anchor nodes are identified as participating
in a localization procedure, then they are endowed with the common range error variance. As
we will see in the following section, this assumption, along with Assumptions 2 and 3 will
lead to the CRLB (with L fixed) as being dependent on only the angles between participating
anchor nodes (i.e., internodal angles). Thus, since the CRLB expression is only dependent on
the internodal angles, the distances between participating anchor nodes and the target need not
be considered. Hence, we may view the participating anchors as being placed on a circle about
the origin. This is depicted in Fig. 2, under the same PPP realization as in Fig 1.

8

y

1

y

θ1
θ3

2

θ2

θ (3)

4
θ4
x

ϕ(2)

θ (2)
ϕ(1) θ (1)
ϕ(3)

ϕ(4)

x

θ (4)

3

Fig. 1. I NITIAL L ABELING S CHEME . The dots represent

Fig. 2. E QUIVALENT S ETUP. This is the realization from

a particular realization of anchors placed according to a

Fig. 1, where only participating anchor nodes and the in-

homogeneous PPP. The origin represents the location of the

ternodal angles they trace out are considered, whereas their

target. The anchors participating in the localization procedure

distances from the target are not. The realizations of the RVs

are labeled in increasing order w.r.t. their distance from the

are given by Θ(k) = θ (k) and ∠(k) = ϕ(k) . Note, the anchor

origin. Their corresponding anchor node angles are labeled

node angle order stats “renumber” the participating anchors

accordingly. Note the realization of the RV Θk is Θk = θ k .

in terms of increasing angle c.c.w. starting from the +x-axis.

Next, we formally define the term internodal angle. Since the anchor node angles are such
i.i.d.

that Θ k ∼ unif[0, 2π), for k ∈ {1, . . . , L}, we may examine their corresponding order statistics,
Θ(1), Θ(2), . . . , Θ(L) , where 0 ≤ Θ(1) ≤ Θ(2) ≤ · · · ≤ Θ(L) ≤ 2π by definition. Thus, the order
statistics of the participating anchor node angles effectively “renumber” the nodes in terms of
increasing angle, starting counterclockwise from the +x-axis. This is also depicted in Fig. 2.
Definition 3. If participating anchor nodes are considered according to their anchor node angle
order statistics, then an internodal angle, ∠ k , is defined to be the angle between two consecutive
participating anchor nodes. That is, ∠ k = Θ(k+1) − Θ(k) for 1 ≤ k < L and ∠ L = 2π − (Θ(L) − Θ(1) ).
Remark. Since the internodal angles are functions of RVs, they themselves are RVs. Thus, we
may also consider their order statistics, ∠(1), ∠(2), . . . , ∠(L) . These order statistics of the internodal
angles are depicted for the particular PPP realization in Fig. 2.
In summary, Fig 2 depicts an example of a typical setup realization, given L = 4, once all of
the assumptions are taken into consideration.

9

III. D ERIVATION OF THE N ETWORK -W IDE CRLB D ISTRIBUTION
In this section, we first formally define our localization performance benchmark: the square
root of the CRLB. Using this definition and assuming a random placement of anchors, as well
as a random L, we then describe how this work generalizes localization performance results
currently in the literature. In what follows, we present the steps necessary to derive the marginal
distribution of our localization performance benchmark.
A. The Localization Performance Benchmark
Consider the traditional localization scenario, where the number of participating anchor nodes
(L) and their positions, as well as the target position, are all fixed. We represent the set of
coordinates of these anchors by
n
o

T
ΨL = ψi ∈ R2 ψi = xi, yi , i ∈ {1, 2, 3, ..., L} .

T
The coordinates of the target are denoted by ψt = xt, yt .
Next, under Assumptions 2, 3, and 4, the 2-Way range measurements between the target and
the L participating anchors are given by
ri = di + ni ,
where ri is the measured distance divided by 2 (i.e., the measured 1-Way distance),
q
2
2
xi − xt + yi − yt
di = kψi − ψt k =
i.i.d.

is the true 1-Way distance, and ni ∼ N (0, σr2 ) for all i ∈ {1, 2, . . . , L}.
Remark. Note that under Assumption 4, σr2 is common among the range measurements. For
2
2
2-Way TOA we set this to be σr2 = σ2-Way
/4. If 1-Way TOA is considered, then σr2 = σ1-Way
.

Thus, moving forward we may consider the range measurements to always be 1-Way, regardless
of whether 1-Way or 2-Way TOA is used.
Continuing, Assumption 3 enables the likelihood function to be easily written as a product.
Denoting the vector of range measurements as r = [r1, . . . , rL ]T , the likelihood function is
!
L

 Ö
2
(r
−
d
)
1
i
i
exp −
.
L ψt r, ΨL, σr2 =
√
2
2σ
2πσ
r
r
i=1
From this likelihood function, we obtain the following Fisher Information Matrix (FIM)
ÍL
 ÍL

2

i=1 cos θ i
i=1 cos θ i sin θ i 
1 
J L (φt ) = 2 
,
ÍL

σr Í L cos θ sin θ
2
sin θi 
i
i
i=1
i=1



10

where cos θi =

xi −xt
di

and sin θi =

yi −yt
di .

Remark. Note that if the target is placed at the origin, then the angles here, i.e., the θi ’s, are a
particular realization of the anchor node angles from Definition 2.
Taking the inverse of the FIM above, then the CRLB for any unbiased estimator, ψ̂t = [ x̂t, ŷt ]T ,
of the target position, ψt = [xt, yt ]T , is given by


CRLB = tr J−1
(ψ
)
t ,
L

(1)

as defined in Ch. 2.4.1 of [24].
Definition 4. We define our localization performance benchmark, S, to be the square root of
the CRLB in (1):
r 

S , tr J−1
(ψ
)
t .
L

(2)

Remark. This benchmark is often referred to as the position error bound (PEB) in the literature,
[25], [26].
To conclude, notice that from (2), a closed-form expression for S can be obtained:
√
L
S = σr v
u
!2 ,
t L
L
L
Õ
Õ
Õ
cos2 θi
sin2 θi −
cos θi sin θi
i=1

i=1

(3)

i=1

which is a function of the anchor node angles.
B. Departure From the Traditional Localization Setup
In the previous section, we assumed a traditional setup, in which the number of participating
anchor nodes (L) and their positions (ΨL ), as well as the target position (ψt ), were all fixed. In
this section however, we now assume that both ΨL and L are random and briefly describe how
the random placement of anchors impacts our localization performance benchmark, S, and how
the randomness of L signals a departure from the existing literature.
We begin by describing how the random placement of anchors affects our localization performance benchmark. This is accomplished by now invoking Assumption 1 and examining
the impact that this has on the expression for S given by (3). Recall from Section II-D that
Assumption 1 implies that ψt = [0, 0]T and that the anchor node angles are i.i.d. on [0, 2π).
Thus, the θi realizations in (3) are now replaced with the random variables, Θi , from Definition

11

2. Since S is now a function of random variables, it itself becomes a random variable, and we
may now seek its distribution.
While work in the past has sought this distribution for S, e.g., [14], [16], there always
remained one implicit assumption: a fixed L. Therefore, the results presented thus far in the
literature have only applied to localization setups with a fixed number of anchor nodes, and hence
were not applicable network-wide. To address this issue, we now consider L to be a random
variable, whose distribution statistically quantifies the number of anchor nodes participating in
a localization procedure. This new interpretation of L will consequently allow us to decouple S
from L, thereby enabling the marginal distribution of S to account for all possible positioning
scenarios within a network. In addition to the contributions outlined in Section I-B, taking
advantage of this new interpretation of L, to subsequently obtain the marginal distribution of S,
is the main contribution setting this work apart from the existing literature.
C. Approximation of the CRLB
Before deriving the conditional distribution of S given L, we use this section to acquire an
approximation of our expression for S in (3). This will consequently allow for an accurate,
tractable, closed-form expression for the conditional distribution of S to be obtained.
1) Approximation Preliminaries and Goals: To facilitate the search for an accurate approximation, we recall that (3) is now in terms of the random variables Θi , and thus can be rewritten
using the internodal angles from Definition 3. This given by
√
L
S = σr v
u
! ,
tÕ
j−1
L−1 Õ
L
Õ
∠k
sin2
i=1 j=i+1

(4)

k=i

Definition 5. We define the terms underneath the square-root in the denominator in (4) by the
random variable D. That is,
D=

L−1 Õ
L
Õ
i=1 j=i+1

sin2

j−1
Õ

!
∠k .

(5)

k=i

Thus, we would like to find an approximation for D which comprises two key traits: (1) it
allows for a straightforward transformation of random variables, i.e., the number of sin2 (·) terms
does not change with L (and would ideally only involve a single term), and (2) it simultaneously
does not sacrifice accuracy, i.e., the approximation should preserve as much “information”

12

as possible about the setup of anchors, implying that the approximation should dominate (or
contribute the most to) the total value of D.
2) Initial Approach and Intuition: In trying to find an approximation that satisfies both
traits, we consider the following possibilities. First, consider approximating D with the sine
squared of an arbitrary internodal angle, sin2 (∠ k ), or with a sum of consecutive internodal angles,
sin2 (∠ k +∠ k+1 +...), where the starting angle, ∠ k , is arbitrary. While these possible approximations
may seem like reasonable candidates for satisfying the first trait, they unfortunately fall short
of satisfying the second trait. To see why, it is illustrative to examine Fig. 2 under different
realizations/placements of anchor nodes. By only looking at the same unordered internodal angle
on every realization, little knowledge is gained about the total setup of anchors. For example,
on one realization, the arbitrary internodal angle being examined might be large and would
therefore give a strong indication of how the rest of the anchors are placed, however, on another
realization, this same internodal angle might be small, thus giving little information about the
placement of the remaining anchors. Hence, in general, arbitrary internodal angles do not provide
accurate approximations due to their inconsistency in describing the anchor node setup, which
consequently leads to their (sine squared terms’) inability to capture the total value of D across
all realizations of anchors for a given L.
3) A Quantitative Approach Using Mutual Information: Taking advantage of the intuition
gained above, it should now be clear that we would like an approximation that utilizes angles
which tend to consistently dominate any given setup. Therefore, it makes sense to examine
larger internodal angles, and thus, the use of internodal angle order statistics follows naturally.
Since we ideally desire a single-term approximation, we examine the possibility of using the
sine-squared of the largest, second largest, and third largest internodal angles, i.e.,3
W(L) = sin2 (∠(L) ),

W(L−1) = sin2 (∠(L−1) ),

and

W(L−2) = sin2 (∠(L−2) ).

(6)

Note that these larger internodal angles should intuitively contain more “information” about the
setup of anchors, and consequently D, since they greatly restrict the placement of the remaining
anchors. Since each of the order statistic approximations above might seem viable under this
3 We

will not examine sums of the larger internodal angle order statistics, e.g., sin2 (∠(L) + ∠(L−1) ) or sin2 (∠(L) ) + sin2 (∠(L−1) ),

for the following two reasons: (1) they would lead to a more complex expression for the conditional distribution of S than
desired, and (2) they offer no gain in accuracy over using the sine squared of just a single internodal angle order statistic, as
evidenced through simulation.

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1.6
I(D; W(L) |L)
I(D; W(L−1) |L)
I(D; W(L−2) |L)

Mutual Information (Bits)

1.4
1.2
1
0.8
0.6
0.4
0.2
0
3

4

5

6

7

Number of Participating Anchor Nodes (L)

Fig. 3.

J USTIFYING A PPROXIMATIONS T HROUGH M UTUAL I NFORMATION. The mutual informations were calculated

numerically by computing (8) and (9), where the necessary distributions were generated using a Monte Carlo simulation of
10 million anchor node realizations. The bin width of these distributions was chosen to be 0.01, Matlab’s ‘spline’ option was
used to interpolate the integrands in (8) and (9), and the supports of D and W(i) are given by D = [0, L 2 /4] and W(i) = [0, 1],
where i ∈ {L, L − 1, L − 2}. Furthermore, we adopt the convention: 0 log2 0 = 0 “based on continuity arguments” [29].

qualitative notion of information, we thus turn towards a quantitative notion, in order to justify
the use of one of these approximations.
Towards this end, we utilize the concept of mutual information. The reason behind this choice,
over that of correlation for example, is because mutual information captures both linear and
nonlinear dependencies between two random variables, since it is zero if and only if the two
random variables are independent [28]. Hence, we examine the mutual information between D
and the random variables W(L) , W(L−1) , and W(L−2) , so that we may quantify which approximation
carries the most information about D. Thus, we condition on L equaling some integer `(≥ 3)
and calculate:
I(D; W(i) |L = `) = h(D|L = `) − h(D|W(i), L = `),
where i ∈ {L, L − 1, L − 2}, and the differential entropies are given by
∫
1
h(D|L = `) = fD (d|`) log2
dd,
and
fD (d|`)
D
∫ ∫
1
h(D|W(i), L = `) =
fDW(i) (d, w|`) log2
dd dw,
fD (d|w, `)

(7)

(8)

(9)

W(i) D

where W(i) and D are the supports of W(i) and D, respectively [29]. The mutual information
between the approximations in (6) and D are given in Fig. 3, versus L. From Fig. 3, it is evident

14

that the mutual information between D and the approximation W(L−1) is the highest across all
numbers of participating anchors shown.
4) Investigating the High Mutual Information of D and W(L−1) : To explore the reasoning
behind this result, we examine the effect that W(L−1) has on the total value of D. We begin by
rewriting D as follows:
Proposition 1. The random variable D from Definition 5 can be equivalently expressed as
!
j+i−1
L−2 Õ
L−i
L
Õ
Õ
Õ

sin2 ∠(k) +
sin2
∠k .
D=
i=2 j=1

k=1

k= j



Proof. See Appendix A.

By separating the sine-squared terms of the internodal angle order statistics from the total
sum, Proposition 1 makes it clearer as to how our approximations from (6) may affect the total
value of D. To reveal the effects of W(L−1) in particular, we present the following lemma along
with its corollaries:
Lemma 2. The cdf of the second largest order statistic of the internodal angles, ∠(L−1) , conditioned on L, is given by
F∠(L−1)

 
X
Õ
nϕ
n−1 L
ϕ|L =
(−1)
(n − 1) 1 −
2π
n
n=0


! L−1
,

(10)


where X = min L, b2π/ϕc and the support is 0 ≤ ∠(L−1) ≤ π. (Note L ≥ 2, since if L < 2,
∠(L−1) would not exist.)
Proof. We refer the reader to the conference version of this paper, i.e., Appendix B of [1].



Corollary 2.1. Given a finite L, the expected value of the second largest order statistic of the
internodal angles, ∠(L−1) , conditioned on L, is given by
!
 
L
Õ
L
2π(n
−
1)
E[∠(L−1) |L] =
(−1)n
n
nL
n=2
Proof. See Appendix B.
Corollary 2.2. Given a finite L, the variance of ∠(L−1) , conditioned on L, is given by
!"
#
 
L
2
Õ
4π
L n−1
2
c
VAR[∠(L−1) |L] =
(−1)n
+ ,
L n=2
n
n
n(L − 1) L

(11)


(12)

15

where c =




m+1 L m−1
m=2 (−1)
m
m

ÍL



.


2
2
2
Proof. Note: VAR[∠(L−1) |L] = E[∠(L−1)
|L] − E[∠(L−1) |L] , where the derivation of E[∠(L−1)
|L]
is analogous to that of E[∠(L−1) |L] in the proof of Corollary 2.1.



Next, we plot Corollary 2.1, plus/minus one and two standard deviations of ∠(L−1) . This is
given in Fig. 4, versus L. Here, we can see that the second largest internodal angle is centered
and concentrated around π/2, suggesting that W(L−1) = sin2 (∠(L−1) ) will be concentrated about
its maximum of one. This implies that, for the majority of anchor node placements, sin2 (∠(L−1) )
will be a dominant term in the expression for D in Prop. 1. Thus, the sin2 (∠(L−1) ) term will tend
to contribute the most, that a given sin2 (·) term could contribute, to the total value of D. This
is especially true for small values of L, which is our focus.
Also for low L, ∠(L−1) is intuitively the dominant angle in that it places the greatest constraints
on the remaining angles. That is, once ∠(L−1) is determined, it restricts both ∠(L) and ∠(L−2) ,
and thus gives the greatest sense of the total setup of anchors. Note, when considering order
statistics other than ∠(L−1) , the constraints placed on the remaining angles are not as pronounced.
Furthermore, by examining different realizations of anchors (Fig. 2 as an example), along with
Prop. 1, one can see that when W(L−1) is small (∠(L−1) ≈ 0 or π), than so is D, and when W(L−1)
is large (∠(L−1) ≈ π/2), a large value of D follows. Thus, W(L−1) ’s consistency as a dominant
term in Prop. 1, along with its intuitive correlation with D, offer supporting evidence as to why
I(D; W(L−1) |L = `) is higher than both I(D; W(L) |L = `) and I(D; W(L−2) |L = `) for low L.
In summary, mutual information has proved its utility by revealing that W(L−1) is perhaps the
best approximation of D, for the desirable lower values of L. Since W(L−1) possesses the two
desirable traits for an approximation, discussed at the beginning of this section, we henceforth
use W(L−1) in our approximation of D.
5) Completing the Approximation: To complete the approximation of D, and consequently
S, all that now remains is to ensure that D and W(L−1) have the same range of possible values,
i.e., the same support. This will ensure that our ultimate approximation for S will produce the
same range of values as the true S. In order to accomplish this, we approximate D with a scaled
version of W(L−1) , i.e., D ≈ k W(L−1) , and thus search for the value of the constant k so that
kW(L−1) yields the desired support. Since D = [0, dmax ] and W(L−1) = [0, 1] (which follows from
the support of ∠(L−1) , Lemma 2), then in order to have the support of kW(L−1) equal that of D,
we simply need to set k = dmax . The value of dmax is presented in the following lemma:

16

1

π
E[6
E[6
E[6

7 π /8

(L−1)
(L−1)
(L−1)

| L]
| L] ± σ
| L] ± 2σ

0.9
0.8

P[S ≤ abscissa | L]

0.7
5 π /8
π /2

3 π /8

E[6

(L−1) |L]

(Radians)

6 π /8

Increasing L
(L = 3, 5, 7, 9)

0.6
0.5
0.4
0.3

2 π /8
0.2
π /8

Solid: True
Dashed: Theorem 4

0.1
0
3

4

5

6

7

8

9

0
10

20

30

Number of Participating Anchor Nodes (L)

40

50

60

70

Meters

Fig. 4. T HE S ECOND L ARGEST I NTERNODAL A NGLE O R -

Fig. 5. ACCURACY OF T HEOREM 4. The true conditional cdf

S TATISTIC . This figure gives a sense of the concentration

of S given L was generated using a Monte Carlo simulation of

of the distribution of ∠(L−1) around π/2. E[∠(L−1) |L] is given
q
by Cor. 2.1 and σ = VAR[∠(L−1) |L] is from Cor. 2.2.

(4) over 1 million random setup realizations of the internodal

DER

angles. Note, σr = 20 m.

Lemma 3. Let L be finite. If D is given as in (5), then its maximum value is dmax = L 2 /4.
Proof. See Appendix C.



Thus, we now have D ≈ (L 2 /4) · W(L−1) , which completes our approximation of D. Lastly,
substituting this approximation for D into the expression for S in (4) finally yields:
Approximation 1. The localization performance benchmark, S, can be approximated by
r
4
1
S ≈ σr ·
·
,
L sin(∠(L−1) )

(13)

where 0 ≤ ∠(L−1) ≤ π, as stated in Lemma 2.

D. The Conditional CRLB Distribution
Theorem 4. If the localization performance benchmark, S, is given by Approximation 1, then
the cdf of S conditioned on L is
! L−1 X
! L−1
 
 
X2
1
Õ
Õ
L
nϕ
L
nϕ
2
1
FS (s | L, σr ) =
(−1)n−1
(n − 1) 1 −
−
(−1)n−1
(n − 1) 1 −
, (14)
n
2π
n
2π
n=0
n=0
p

where ϕ1 = sin−1 (a/s), ϕ2 = π − sin−1 (a/s), a = σr · 4/L, X1 = min L, b2π/ϕ1 c , X2 =

min L, b2π/ϕ2 c , and the support is S ∈ [a, ∞).

17

Proof. See Appendix D.



Remark. Although Theorem 4 is the conditional distribution of our approximation of S, it
provides two clear advantages over the true conditional distribution presented in [16] and over
the approximate conditional distribution presented in [14]. First, Theorem 4 offers a simple,
closed-form, algebraic expression involving only finite sums, as opposed to the rather complex
expression in [16] involving an improper integral of products of scaled Bessel functions. Second,
Theorem 4 is remarkably accurate for lower numbers of participating anchor nodes, see Fig. 5.
This comes in contrast to the approximate distribution presented in [14], which was derived
asymptotically and therefore only accurate for higher numbers of participating anchors. This
selective accuracy of Theorem 4 is desirable since a device is more likely to hear lower numbers
of participating anchors, especially in infrastructure-based, terrestrial wireless networks.4
E. The Distribution of the Number of Participating Anchors
The next step needed to achieve our goal is to find the distribution of the number of participating anchors, fL (`). In this section, we build upon the localizability results from [17] in order
to obtain this distribution. Towards this end, we present the relevant theorems from this work
and modify them for our use here. Finally, we conclude with a discussion on the applicability
of these results.
1) Overview of Localizability Work: Recall from Section I-A that the goal of [17] was to
derive an expression for the probability that a mobile can hear at least ` base stations for
participation in a localization procedure in a cellular network, i.e., P[L ≥ `]. To derive this
expression, the authors assumed that base stations were placed according to a homogeneous
PPP, and then examined the SIRs of the base station signals received at a “typical user” placed
at the origin.5 Specifically, they examined the SIR of the ` th base station (denoted SIR` ), since
this was used directly to determine P[L ≥ `].
Since SIR` depends on the locations of interfering base stations, then the base stations’
placement according to a PPP implies that SIR` becomes a random variable. Consequently, its
distribution also becomes a function of the PPP density, λ. Additionally, the authors incorporate
4 By

infrastructure-based wireless networks, we mean any wireless network setup with mobile devices, fixed access points,

and separate uplink/downlink channels.
5 SIR

= Signal to Interference Ratio. Noise is ignored since [17] assumes an interference-limited network.

18

a network loading parameter, q, into SIR` , where 0 ≤ q ≤ 1. This means that any given base
station can be considered active (i.e., interfering with base station `’s signal) with probability
q. Furthermore, SIR` is also a function of pathloss, α, where α > 2, and the distances of base
stations to the target.
With SIR` statistically characterized, the authors were able to determine P[L ≥ `] by noting
that P[L ≥ `] = P[SIR` ≥ β/γ], where β/γ is the pre-processing SIR threshold for detection
of a signal.6 Here, γ is the processing gain at the mobile (assumed to also average out the
effect of small scale fading), and β is the post-processing threshold. Thus, since P[L ≥ `] =
P[SIR` ≥ β/γ], then P[L ≥ `] must also depend on all of the network parameters described
above. We denote this dependency by P[L ≥ ` | α, λ, q, γ, β].
Before continuing to the localizability results, we mention one last caveat regarding the PPP
density. That is, when shadowing is present, it can easily be incorporated into the PPP network
model through small displacements of the base station locations. This results in a new PPP
density, which accounts for this effect of shadowing. This new shadowing-transformed density
is given by λ̃ = λ E[Sz2/α ], where Sz is assumed to be a log-normal random variable representing
the effect of shadowing on the signal from base station z to the origin [31].7 Thus, by using λ̃,
we incorporate shadowing under the log-normal model presented in Section II of [31].
2) The Localizability Results: In this section, we present the main theorem which will enable
us to obtain fL (`).
Lemma 5. (Theorem 2, [17]) The probability that a mobile device can hear at least ` base
stations for participation in a localization procedure is given by
P[L ≥ ` | α, λ̃, q, γ, β] =
`−1
© Õ
α−2
­1− e− 2qβ/γ
­
n=0
«



α−2
2qβ/γ

n!

n


∫ ∞∫ r` 
`−1
−α

` Õ
ª
r
β
4(
λ̃π)


`
® fΩ (0) +
1
≥
f
(ω)


Ω
2−α −r 2−α
®
r


(` − 1)! ω=1
γ
2πq
λ̃
2(ω−1)
2−α
`
1
0 0  r −α +
· r 2 −r 2 + α−2 r`

2−α
1
¬


`
1
× r1 (r`2 − r12 )ω−1r`2(`−ω)−1 ωe−λ̃πr` dr1 dr`,
2

where ` ∈ {1, 2, . . . }, and Ω is a random variable denoting the number of active participating
6 Note

that for q < 1, this equality holds for all but a few rare corner cases. However, the probability of these cases occurring

is vanishingly small and thus has little to no impact on the accuracy of the subsequent localizability results [17].
7 The

Sz are assumed to be i.i.d. ∀z. Sz ’s log-normal behavior implies it is distributed normally when expressed in dB [31].

19

base stations interfering with the ` th . Note, Ω ∼ Binomial(` − 1, q) = fΩ (ω). Additionally, for the
trivial case of ` = 0, we define P[L ≥ 0] = 1. (Note: γ and β are in linear terms, not dB.)
Remark. This theorem was derived under the following assumptions: (1) a dominant interferer
and (2) interference-limited networks. We refer the reader to Section III-D of [17] for further
details regarding these assumptions and the consequent derivation of this theorem.
3) The Distribution of L: With these localizability results, we now finally present the distribution of the number of participating anchors.
Theorem 6. The pdf of L is given by
fL (` | α, λ̃, q, γ, β) = P[L ≥ ` | α, λ̃, q, γ, β] − P[L ≥ ` + 1 | α, λ̃, q, γ, β],
where the support is L ∈ {0, 1, 2, . . . } and the probabilities are given by Lemma 5.
4) The Distribution of L with Frequency Reuse: Using Theorem 6, we may obtain another
expression for fL (`) which incorporates a frequency reuse parameter, K. This parameter models
the ability of base stations to transmit on K separate frequency bands, thereby limiting interference to a per-band basis. This can easily be incorporated into the model by considering K
independent PPPs whose densities are that of the original PPP divided by K. Thus, if nk is
the number of participating base stations in band k, then the total number of participating base
Í
stations is given by L = K
k=1 n k . Thus, to find P[L = `] under frequency reuse, we simply need
to account for all of the per-band combinations of participating base stations such that their sum
equals `. This is given in the following corollary, which is a modification of Theorem 3 of [17].
Corollary 6.1. The pdf of L, given a frequency reuse factor of K, is
!
K
Õ Ö
λ̃
fL (` | α, λ̃, q, γ, β, K) =
fL nk α, , q, γ, β ,
K
k=1
{nÍ1,...,n K }
i ni =`

where the multiplicands are given by Thm. 6, K ∈ {1, 2, . . . }, and the support is L ∈ {0, 1, 2, . . . }.
Remark. When K = 1, this corollary reduces to Theorem 6. Further, this corollary may be
evaluated numerically through the use of a recursive function.
5) Applicability of the Results: Now that we have obtained the pdf of L, we conclude
with a brief discussion regarding its applicability. We begin by taking note of the support of
fL (`). Whereas up until this section we have proceeded under the assumption that the target is

20

localizable, the support of fL (`) now allows us to consider cases where the target is unlocalizable,
i.e., L = 0, 1, 2. Thus, as will be addressed in the following section, we may use these cases to
determine the percentage of the network where a target is unlocalizable.
Lastly, we note that while the localizability results in [17] (Lemma 5) were presented in
the context of cellular networks, these results are actually applicable to any infrastructure-based
wireless network using downlink measurements, so long as the distribution parameters are altered
accordingly. This implies that the distribution for L (Corollary 6.1) also has this applicability,
since it was derived using Lemma 5. Thus, since we use Corollary 6.1, along with a modified
Theorem 4, to derive the marginal distribution of S, then this final distribution will also be
applicable to any infrastructure-based wireless network employing a TOA localization strategy.
F. The Marginal CRLB Distribution
In this section, we modify Theorem 4 and combine this with Corollary 6.1 to obtain the
marginal distribution of S. First, we state one last network assumption often used in practice:
Assumption 5. For a given localization procedure, only a finite number of anchor nodes, N
(≥ 3), are ever tasked to transmit localization signals.
Remark. Just because N anchors are tasked, does not mean that all N signals are necessarily
heard.
Under this assumption, and further considering scenarios where the target is unlocalizable, we
modify Theorem 4 as follows:





 FS (s | L = N, σr )
0
FS (s | L, σr ) = FS (s | L, σr )



 u(s − M)


L≥N
L = 3, . . . , N − 1

(15)

L = 0, 1, 2

where FS0 is the new modified conditional distribution of S, FS is the previous conditional
distribution given in Theorem 4, and M ∈ R+ is a predetermined localization error value used
to account for unlocalizable scenarios (described in more detail below).
Remark. While Theorem 4 only accounted for scenarios in which the target was localizable,
this modified form, however, now accounts for unlocalizable scenarios. For these scenarios, this
modified conditional distribution yields a step function, which is a valid cdf and corresponds to
a deterministic value for the localization error, i.e., P[S = M | L] = 1. Thus, we account for

21

cases where the target is unlocalizable by assigning an arbitrary localization error value for M,
which is chosen to represent cases where there is ambiguity in a target’s position estimate.8
Remark. It is possible that a mobile may hear more anchors than are tasked to perform the
localization procedure, i.e., L > N. In this case, the N participating anchors are likely those with
the highest received SIR at the target, if connectivity information is known a priori. Therefore,
in this scenario, localization performance will only be based on the N anchors tasked. This is
clearly reflected in the modified conditional distribution of S in (15).
Using this modified Theorem 4 in (15), along with Corollary 6.1, we may now obtain the
distribution of localization error for an entire wireless network:
Theorem 7. The marginal cdf of the localization performance benchmark, S, is
h
i
FS (s | σr , α, λ̃, q, γ, β, K, M, N) = FS (s | ` = N, σr ) 1 − P[L ≤ N − 1|α, λ̃, q, γ, β, K]
+

N−1
Õ

FS (s | `, σr ) fL (` | α, λ̃, q, γ, β, K)

`=3

+ u(s − M) P[L ≤ 2|α, λ̃, q, γ, β, K],
where FS (s | L = `, σr ) is given by Theorem 4, fL (` | α, λ̃, q, γ, β, K) is given by Corollary 6.1,
Íx
and P[L ≤ x|α, λ̃, q, γ, β, K] = `=0
fL (` | α, λ̃, q, γ, β, K).
Proof. Multiplying the modified conditional distribution of S, given in (15), by the marginal
distribution of L from Corollary 6.1 gives the joint distribution of S and L. Then, setting L
equal to a particular realization, `, and summing over all realizations, gives us the marginal cdf
of S, as desired.



Remark. First, recall that the conditional distribution of our approximation of S, FS (s | L = `, σr )
given by Theorem 4, is accurate for lower ` values. Next, note that: (1) fL (`) declines rapidly
as ` increases (an intuitive result, since the probability of hearing many anchor nodes should
be small in infrastructure-based wireless networks), and (2) only a maximum of N nodes are
tasked to perform a localization procedure (Assumption 5). Thus, these two facts validate the
use of our approximation, since the cases where our approximation is less than ideal (i.e., for
8 In

this paper, we choose a value for M that applies for all L < 3. Thus, a large enough M allows for a clear distinction

between the localizable and unlocalizable portions of the network through a quick examination of the cdf of S. Note however,
that one could account for the L = 0, 1, 2 scenarios separately. For example, if L = 1 then in a cellular network one may want
to choose M to be the cell radius, since the user equipment typically knows in which cell it is located [32].

22

large L) will now either be multiplied by fL (`) ≈ 0, or considered invalid in a realistic network
under Assumption 5. As a consequence, Theorem 7 will also retain this accuracy.
Remark. We conclude by noting that this distribution accounts for localization error over all
setups of anchor nodes, numbers of participating anchors, and placements of a target anywhere in
the network. Hence, this distribution completely characterizes localization performance throughout an entire wireless network and represents the main contribution of this work.
IV. N UMERICAL A NALYSIS
In this section, we examine the accuracy of Theorem 7 and investigate how changing network
parameters affects localization performance throughout the network.
A. Description of Simulation Setup
Here, we discuss the parameters that are fixed across all simulations and include a description
of how the simulations were conducted.
1) Fixed Parameter Choices and their Effect on Model Assumptions: For these simulations,
we consider the case of a cellular network, and thus, we place anchor nodes such that the
PPP density matches that of a ubiquitous hexagonal grid with 500m intersite distances (i.e.,
√

λ = 2/ 3 · 5002 m2 ) [17]. Furthermore, we choose a shadowing standard deviation of 8 dB,
which defines our shadowing-transformed density parameter, λ̃.
Next, we set our pathloss exponent, α ≈ 4, which was chosen to represent a pathloss similar to
that seen in a typical cellular network. Note that this pathloss value is indicative of NLOS range
measurements, which are inherently a part of localization in cellular networks. Recall however,
that Assumption 3 implied the use of Line-of-Sight (LOS) measurements. Thus, we attempt to
mimic NLOS in our simulations by selecting a range error to account for a reasonable delay
spread under NLOS conditions. This is described further in the following section. We note that
a subject of future work will be to seek a refinement of this model by incorporating NLOS
directly into the range measurements.
The last parameter that remains fixed across simulations is M. This parameter was chosen to
be large enough such that an examination of the cdf of S will reveal which percentage of the
network the target is unlocalizable. Towards this end, it was sufficient to choose M ≈ 200 m for
our simulations here. Again, the choice of this value is arbitrary and left up to network designers
and how they would like to treat the unlocalizable cases.

23

1

1
Increasing
Frequency Reuse
(K = 1, 2, 3, 6)

0.9
0.8

0.9
0.8
0.7

P[S ≤ abscissa]

P[S ≤ abscissa]

0.7
0.6
0.5
0.4
0.3

0.6
Decreasing q
(q = 1, 0.75, 0.50, 0.25)

0.5
0.4
0.3

0.2

0.2
Solid: True
Dashed: Theorem 7

0.1

Solid: True
Dashed: Theorem 7

0.1

0

0
0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

Meters

Fig. 6.

T HE E FFECT

OF

F REQUENCY R EUSE. This result

25

30

35

40

45

50

Meters

Fig. 7.

T HE I MPACT

OF

D ECREASING N ETWORK L OAD.

exposes the large impact that frequency reuse has on local-

This plot demonstrates the improvement in localization perfor-

ization performance throughout the network. The parameters

mance due to a decrease in network loading. The parameters

are: N = 10, β = 10 dB, γ = 20 dB, q = 1, and σr = 20 m.

were chosen as follows: N = 10, β = 10 dB, γ = 20 dB,

The plots’ “piece-wise” appearance is due to L being discrete.

K = 2, and σr = 20 m.

2) Conducting the Simulations: The true marginal cdf of S was generated through a simulation
over 100,000 positioning scenarios. Each scenario consisted of an average placement of 1,000
anchor nodes, placed according to a homogeneous PPP, with the target located at the origin. Next,
the anchor nodes whose SIRs surpassed the detection threshold were deemed to participate in
the localization procedure, and their corresponding coordinates were used to calculate the true
value of S given by Definition 4. If more than N anchor nodes had signals above the threshold,
then the N anchors with highest SIRs were used to calculate S.
B. The Effect of Frequency Reuse on the Network-Wide Distribution of Localization Error
In this section, we explore how frequency reuse impacts localization performance throughout
the network. This simulation, as well as all subsequent simulations, compares the true (simulated)
distribution of S with our analytical model from Theorem 7. All parameters were fixed at the
levels stated in Fig. 6, while the only parameter varied was the frequency reuse factor, K. The
range error standard deviation, σr , was chosen according to the detection threshold, β, and
the CRLB of a range estimate (see [6], equ. 3.40), assuming a 10MHz channel bandwidth.
Approximately 10m was added to account for a reasonable delay spread in NLOS conditions.
From Fig. 6, the most notable impact that frequency reuse has on localization performance is

24

that of localizability. That is, with just a small increase in frequency reuse from K = 1 to K = 2,
the portion of the network with which a target is localizable increases from only ≈ 25% to an
astonishing ≈ 85%. Furthermore, localization error is also reduced, although the improvement
is not as drastic as the increase in localizability. Additionally, as frequency reuse increases, the
gains in localizability stop after K = 3, with the gains in localization error also declining after
K = 3 as well. Thus, we can conclude that an increase in frequency reuse is strongly advisable if
once desires an increase in localization performance within a network, a result which coincides
with what has been seen in practice, viz. 3GPP. Lastly, we note the excellent match between
the true simulated distribution and our analytically derived distribution given by Theorem 7. We
will see that this accuracy of Theorem 7 is retained across all of our results in this section.
C. Examining the Effects of Network Loading
Here we examine the effect that network loading has on localization performance throughout
the network. This is accomplished by varying the percentage of the network, q, actively transmitting (interfering) during a localization procedure. All parameter values other than q were
fixed and σr was chosen in the same manner as in the frequency reuse case.
From the distributions plotted in Fig. 7, we can see that a decrease in network load leads to
an improvement in localizability, as well as an improvement in localization error. However, the
improvement is not as pronounced as in the frequency reuse case. Further, examining the 80th
percentile for example, it is evident that the rate of improvement in localization error declines
as the network load declines as well. Thus, since low network traffic is usually never desirable,
a network designer looking to optimize localization performance may find solace in the fact that
gains in performance begin to decline as network loading decreases also.
D. The Impact of Processing Gain
In this section, we examine the effects of changing the processing gain, since it is perhaps the
easiest parameter for a network designer to change in practice. (Note that we choose σr here as in
the previous simulations.) From Fig. 8, it is evident that as the processing gain increases, there is
a corresponding improvement in localizability across the network, as well as an improvement in
localization error. As a consequence, there exists a clear trade-off between sacrificing processing
time for gains in localization performance. However, it appears that these improvements begin
to level off at a processing gain of ≈25 dB. This is promising, as processing gains higher than

1

1

0.9

0.9

0.8

0.8

0.7

0.7

P[S ≤ abscissa]

P[S ≤ abscissa]

25

0.6
Increasing Processing Gain
(γ = 15 dB, 20 dB, 25 dB)

0.5
0.4
0.3

Increasing σr
(σr = 20m, 40m, 60m, 80m)

0.6
0.5
0.4
0.3

0.2

0.2
Solid: True
Dashed: Theorem 7

0.1
0

Solid: True
Dashed: Theorem 7

0.1
0

0

10

20

30

40

50

60

70

0

Meters

Fig. 8. T HE E FFECT

OF I NCREASING

20

40

60

80

100

120

140

160

Meters

P ROCESSING G AIN.

Fig. 9. T HE I MPACT OF I NCREASING R ANGE E RROR. Inject-

This figure highlights the trade-off that exists between pro-

ing range error results in a predictable effect on localization

cessing time and localization performance. The distribution

performance throughout the network, yet has no effect on

parameter values are: N = 10, β = 5 dB, q = 1, K = 2, and

localizability. The parameter values are: N = 10, β = 10 dB,

σr = 30 m.

γ = 20 dB, K = 3, and q = 1.

this can quickly become impractical. Examining the 80th percentile, we can see that a 10 dB
increase in processing gain can lead to about a 30m improvement in localization error throughout
the network. Thus, increasing the processing gain can be an easily implementable solution for
achieving moderate gains in localization performance.
E. Range Error and its Impact on Localization Performance within the Network
With this last result, we attempt to mimic the effect of an increasing NLOS bias by injecting
additional range error into the measurements. In examining Fig. 9, we note that the first σr value,
i.e., σr = 20 m, was chosen as in the frequency reuse simulation. Therefore, the subsequent
choices represent that of mimicking the impact of increasing NLOS bias. From Fig. 9, we can
see that increasing range error has no effect on localizability within the network. This should
be clear from the analytical model, since σr does not appear as a parameter in Corollary 6.1.
Additionally, injecting more range error into the measurements results in a predictable effect
on the distribution of localization performance. This is also evidenced by examining Theorem
4, where σr appears as a scale parameter. Thus, mimicking the effects of NLOS measurements
results in a scaling of the distribution of S, implying a predictable reduction in localization
performance.

26

V. C ONCLUSION
This paper presents a novel parameterized distribution of localization error, applicable throughout an entire wireless network. Invoking a PPP network model, as well as common assumptions
for TOA localization, has enabled this distribution to simultaneously account for all possible positioning scenarios within a network. Deriving this result involved two main steps: (1) a derivation
of an approximate distribution of our localization performance benchmark, S, conditioned on
the number of hearable anchor nodes, L, which yielded a conditional distribution with desirable
accuracy and tractability properties, and (2) a modification of results from [17] to attain the
distribution of L. Thus, using these two distributions, we arrived at the final, marginal distribution
of S, which also retained the same parameterizations as its two component distributions.
What followed was a numerical analysis of this network-wide distribution of localization
performance. This analysis revealed that our distribution can offer an accurate, baseline tool for
network designers, in that it can be used to get a sense of localization performance within a
wireless network, while also providing insight into which network parameters to change in order
to meet localization requirements.
Since this marginal distribution of S is a distribution of the TOA-CRLB throughout the
network, it consequently provides a benchmark for describing localization performance in any
network employing an unbiased, efficient, TOA-based localization algorithm. The amount of
insights that this network-wide distribution of localization error reveals are numerous, and the
results presented in this paper have only begun to explore this new paradigm. It is our hope
that this work spawns additional research into this new concept, and that future work, such
as incorporating NLOS measurements, accounting for collaboration, adding other localization
strategies (TDOA/AOA), etc., can further refine the model presented here.
In closing, this work presents an initial attempt to provide network designers with a tool for
analyzing localization performance throughout a network, freeing them from lengthly simulations
by offering an accurate, analytical solution.

27

A PPENDIX A
P ROOF OF P ROPOSITION 1
It first helps to visualize the sin2 (·) terms of the sum in (5) on a 2-D grid, where the i’s
represent the rows and the j’s represent the columns. As an example, the case of L = 4 gives
j=1 j=2
i=1

j=3

j=4

∠1 + ∠2 ∠1 + ∠2 + ∠3

∠1

i=2

∠2 + ∠3

∠2

i=3

∠3

where just the arguments of the sin2 (·) terms are represented in the grid for clarity. From this
arrangement, it is evident that the sum in (5) represents the process of summing each row
sequentially, starting at i = 1.
Now, however, we choose to sum the terms diagonally, starting with the lowest diagonal and
working our way upward. This yields
D=

L−1 Õ
L−i
Õ
i=1 j=1

sin2

j+i−1
Õ

!
∠k .

(16)

k= j

Considering the cases i = 1 and i = L − 1 separately, we may rewrite (16) as
!
!
j+i−1
L−1
L−1
L−2 Õ
L−i
Õ
Õ
Õ
 Õ
2
2
2
∠ k + sin
∠k .
D=
sin ∠ k +
sin
i=2 j=1

k=1

k= j

k=1

Next, note that
2

sin

L−1
Õ

!
∠ k = sin

2

2π −

L−1
Õ

!


∠ k = sin2 2π − (Θ(L) − Θ(1) ) = sin2 ∠ L ,

(17)

k=1

k=1

where the last two equalities follow from Definition 3. Hence,
D=

L
Õ
k=1

sin2 ∠ k +


L−2 Õ
L−i
Õ

sin2

i=2 j=1

j+i−1
Õ

!
∠k .

k= j

Lastly, we complete the proof by noting that the first sum may be equivalently expressed by
replacing the internodal angles with their order statistics.
A PPENDIX B
P ROOF OF C OROLLARY 2.1
Using Lemma 2 and our assumption of a finite L, the pdf of ∠(L−1) conditioned on L is
!
! L−2
 
X
Õ
d
L
n(n
−
1)(L
−
1)
nϕ
f∠(L−1) (ϕ|L) =
F∠ (ϕ|L) =
(−1)n
1−
,
dϕ (L−1)
n
2π
2π
n=0

28


where X = min L, b2π/ϕc and the support is 0 ≤ ∠(L−1) ≤ π, as in Lemma 2.
Next, note that the lower summation limit may be rewritten as n = 2 and that the upper
summation limit, X, can be simplified to just L by appending an indicator function to the
summand. This gives the logically equivalent expression:
!
! L−2 "
#
 
L
Õ
n(n − 1)(L − 1)
nϕ
2π
n L
f∠(L−1) (ϕ|L) =
(−1)
1−
1 ϕ≤
.
n
2π
2π
n
n=2
Using the above expression for the pdf, the expectation is derived as follows:
∫ π
E[∠(L−1) |L] =
ϕ f∠(L−1) (ϕ|L) dϕ
0

!
! L−2
 
L n(n − 1)(L − 1)
nϕ
ϕ
(−1)
1−
dϕ
=
n
2π
2π
0
n=2
! ∫ 2π
! L−2
 
L
Õ
n
nϕ
n(n − 1)(L − 1)
n L
=
(−1)
ϕ 1−
dϕ
n
2π
2π
0
n=2
!
 
L
Õ
L
2π(n
−
1)
=
(−1)n
,
nL
n
n=2
∫

2π
n

L
Õ

n

(18)

(19)

(20)

where (18) follows from absorbing the indicator function into the integration limits since n ≥ 2,
(19) follows from our assumption of a finite L, and (20) is derived through integration by parts.
A PPENDIX C
P ROOF OF L EMMA 3
This is a straightforward application of the lowest Geometric Dilution Of Precision (GDOP)
presented in [30]. First, since L is finite, then D is a continuous real function defined on a
compact subset of R L . Thus, its maximum must exist. Call it dmax .
Next, under our Assumptions 2, 3, and 4, GDOP, presented in [27], can be written as GDOP =
p

L/D, where D follows from the lemma assumption. Further, under our assumptions, [30] asserts
√
that the lowest GDOP is given by GDOPmin = 2/ L. Since GDOPmin must occur when D is at
p
√
its maximum, we have that 2/ L = L/dmax , and the lemma follows.
A PPENDIX D
P ROOF OF T HEOREM 4
Under Approximation 1, we have S = a/sin(∠(L−1) ), where a is defined in Theorem 4. To
determine the support of S conditioned on L, we know from Lemma 2 that if 0 ≤ ∠(L−1) ≤ π,

29

then the support for the RV sin(∠(L−1) ) must be 0 ≤ sin(∠(L−1) ) ≤ 1. From here, we see that
0 ≤ sin(∠(L−1) ) ≤ 1

⇒

1≤

1
sin(∠(L−1) )

⇒

a≤

a
sin(∠(L−1) )

⇒

a ≤ S,

and hence, S ∈ [a, ∞).
Next, to find the cdf of S conditioned on L, consider the following
FS (s | L, σr ) = P[S ≤ s | L, σr ]
"
a
≤s
=P
sin(∠(L−1) )
"
= P ∠(L−1) ≤ π−sin−1

#
(21)

L
a
s

!

#

"

L − P ∠(L−1) ≤ sin−1

a
s

!

#
L

= P[∠(L−1) ≤ ϕ2 | L] − P[∠(L−1) ≤ ϕ1 | L]


= F∠(L−1) ϕ2 | L − F∠(L−1) ϕ1 | L ,

(22)
(23)

where (21) follows from Approximation 1 and the fact that we may drop the parameter, σr , as
a condition since the dependency is now explicit. Further, ϕ1 and ϕ2 from (22) are defined in
Theorem 4, and (23), through use of Lemma 2, gives (14) from Theorem 4, as desired.
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