1
Network Localization Cramér–Rao Bounds for General Measurement Models
Panos N. Alevizos, Student Member, IEEE, and Aggelos Bletsas, Senior Member, IEEE.
Abstract—A closed-form Cramér-Rao bound (CRB) for general multi-modal measurements is derived for D-dimensional
network localization (D ∈ {2, 3}). Links are asymmetric and
the measurements among neighboring nodes are non-reciprocal
depending on an arbitrary differentiable function of their position
difference, subject to additive Gaussian noise (with variance
that may depend on distance). The provided bound incorporates
network connectivity and could be applied for a wide range
of typical, unimodal ranging measurement methods (e.g., angleof-arrival or time-of-arrival or signal strength with directional
antennas) or multi-modal methods (e.g., simultaneous use of
unimodal ranging measurements). It was interesting to see that
for specific network connectivity, MSE performance of various
different ranging measurement methods coincide, while performance of network localization algorithms is clearly sensitive on
network connectivity.
Index Terms—Estimation theory, Cramer-Rao bounds.
I. I NTRODUCTION
In network localization [1], [2], agents with unknown location exchange measurements with reference anchors (noncooperative localization) and other neighboring agents (cooperative localization). Measurements of relative location and
distance can be quantified by a variety of metrics, such as
received signal strength (RSS) or time-of-arrival (ToA) [1],
[3], angle-of-arrival (AoA) [4], [5] or combination [6].
Performance of any localization algorithm usually employs
the mean squared-error (MSE) metric, which for unbiased
deterministic estimators, is lower-bounded by the CramérRao bound (CRB) [7]. However, prior art has derived CRB
formulas for specific measurement setups. For instance, work
in [1] offered CRB for two-dimensional (2D) cooperative
localization with ToA [8] or RSS-based measurements. Related work [9] has offered 2D non-cooperative localization
CRB with hybrid ToA and RSS, while work in [10] offered
performance bounds for 2D noncooperative localization with
ToA and AoA. It is also worth mentioning that reciprocity
in distance-based measurements is typically assumed, which
may not hold in practice, given that measuring apparatus at
different nodes exhibit independent noise levels. CRB including multi-modal measurements—where more than one type
of measurements may be simultaneously offered—is typically
not covered in the existing network localization prior art, apart
from special cases. Seminal work in [11] offered MSE lower
bound expressions, derived directly from exchanged signal
This research has been co-financed by the European Union (European
Social Fund-ESF) and Greek national funds through the Operational Program
Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) - Research Funding Program: Thales. Investing in knowledge
society through the European Social Fund.
The authors are with the Telecom Laboratory, School of Electronic and
Computer Engineering, Technical University of Crete, Chania 73100, Greece,
(e-mail: palevizos@isc.tuc.gr and aggelos@telecom.tuc.gr).
Copyright (c) 2016 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
waveforms and other physical layer-related information, such
as multi-path profile or parameters relevant to the channel
statistics; extension to cooperative case is given in [12].
Related applications can be found in [13] and references
therein.
This work offers closed-form CRB for D-dimensional network (cooperative or not) localization (D ∈ {2, 3}), covering
multi-modal ranging measurements, asymmetric links, nonreciprocal measurements, and studies the impact of network
connectivity on localization. Distance dependence in the variance of ranging error measurements can be also modeled.
Differentiable functions, modeling a large class of ranging
methods are employed. Notation: 1 A denotes the indicator
function of statement A, which is one if A is true and zero,
otherwise; tr(·) is the trace operator.
II. S YSTEM M ODEL
The network consists of N agents with unknown locations,
indexed by set Ng , {1, 2, . . . , N } and L anchors with a
priori known coordinates (across the entire network), indexed
by set Nan , {N + 1, N + 2, . . . , N + L}. The set of all
network nodes is denoted by H , Ng ∪ Nan and for Ddimensional localization, the coordinates of node i ∈ H are
represented by vector xi = x i,1 x i,2 . . . x i, D ⊤ ∈ R D .
Connectivity matrix A is defined with elements Ai, j = 1
(Ai, j = 0) if node i can (cannot) receive measurements
from node j, ∀i, j ∈ H and Ai,i = 0, ∀i ∈ H . In other
words, the ith row of A indicates the nodes that can send
measurements to node i; the set of all these nodes can be
written as H (i) = { j ∈ H : Ai, j = 1}. For more flexible
modeling, Ai, j , A j,i in general, i.e., connectivity between
terminals may be asymmetric; i may receive measurements
from j, while j may not receive measurements from i. The
set of directed (measurement connectivity)
edges is defined)
(
as G , {(i, j) : i ∈ H , j ∈ H (i)} = (i, j) ∈ H 2 : Ai, j = 1
and notice that (i, j) ∈ G does not necessarily mean that
( j,i) ∈ G. Thus, any type of network connectivity can be
explicitly modeled. It is further assumed that all node positions
are distinct.
In network localization, agents exchange measurements
with other neighboring nodes (agents or anchors) for estimation of the unknown (agent)
coordinates {x n } n ∈Ng , )using
( (m)
set of measurements Y , yi←j : (i, j) ∈ G, m ∈ M . Set
M = {1, 2, . . . , M} denotes the type of measurements for the
pair (i, j), e.g., ToA (m = 1), RSS (m = 2), AoA (m = 3) or
(m)
of
other. Specifically, node i ∈ H conducts measurement yi←j
type m by receiving signal transmitted from neighboring node
j ∈ H (i), resulting to ranging measurement given by:
(m)
q
(m)
(m)
(1)
di j wi←j ,
xi , x j + hi(m)
, pi←j
yi←j
j
(m)
(m)
with function pi←j
xi , x j and
di j + vi←j
xi , x j , gi(m)
j
(m)
(m)
di j , kxi − x j k; it is emphasized that yi←j , y j←i in general,
2
even for symmetric connectivity, e.g.,
due to independent noise)
( (m)
: (i, j) ∈ G, m ∈ M
levels at different receivers; thus, wi←j
are independent, non-identically distributed (i.n.i.d.),
(m) 2 zeromean Gaussian random variables of variance σi←j
. For
(m)
all (i, j) ∈ G, m ∈ M, functions gi j : R+ −→ R and
: R+ −→ R+ are assumed differentiable over positive
hi(m)
j
(m)
reals and depend solely on di j , while vi←j
: R2D −→ R
(m)
(m)
depends solely on xi − x j (i.e., vi←j (xi , x j ) = H
vi←j
(xi − x j )
(m)
: R D −→ R) and is also
for some suitable function H
vi←j
assumed differentiable over a neighborhood of (xi , x j ). It can
(m)
(m)
(xi , x j ). For each
(xi , x j ) = −∇x j vi←j
be shown that ∇xi vi←j
(directed) link (i, j) ∈ G and measurement type m ∈ M, one
available measurement is assumed. Specific examples from
practice follow.
Examples: For ToA ranging measurements, hi(1)
j (·) = 1,
d
(1)
ij
vi←j
(·) = 0, gi(1)
j (di j ) = c + bi j , c denotes the signal
propagation velocity and bi j models a bias term accounting
for possible non-light-of-sight (NLOS) effects.
For RSS ranging measurements with
omnidirectional
an
di j
(2)
tennas, gi j (di j ) = Pi j − 10νi j log10 d0, i j , where νi j is the
path-loss exponent (PLE) between nodes i and j and Pi j is
the known received power at a short reference distance d 0,i j .
For RSS ranging measurements with directional anten(2)
(2)
nas, function vi←j
is also added in pi←j
, modeling the
Tx
gain G j of node’s j transmit antenna and gain GRx
i
of node’s
i receive antenna
(in dB). Specifically
for 2D,
(2)
vi←j
(φi←j ) GRx
xi , x j = 10log10 GTx
i (φ j←i ) , where phase
j
−1 x i,2 −x j,2
φi←j , tan2 x i,1 −x j,1 when φi←j ∈ [0, π), and φi←j ,
x i,2 −x j,2
2π + tan−1
when φi←j ∈ [π, 2π), with function
2 x i,1 −x j,1
−1 y
tan2 x , atan2(y, x), and atan2(y, x) ∈ [−π, π] as defined in [14, Eq. (5.22)]; such definition offers φi←j ranging
in [0, 2π) (as opposed to classic atan, which is limited
to [−π/2,
π/2])
and offers
a differentiable φi←j . For 3D,
(2)
(φi←j , θ i←j ) GRx
(φ j←i , θ j←i ) ,
vi←j
xi , x j = 10log10 GTx
j
i
x −x
where θ i←j = cos−1 kxi,3i −x jj,3k ∈ (0, π).
As a simple 2D example, consider dipole antennas placed
parallel to the x-axis [15].
Due
to symmetry of the dipole
(2)
directivity pattern, vi←j
xi , x j is given by:
f
π
3π
3

, 2π

2
 20log10 1.67cos φi←j , φi←j ∈ 0, 2 ∪
(2)
 20log −1.67cos3 φi←j , φi←j ∈ π , 3π .
10
2 2

For a simple 3D example, consider the 3D directivity pattern
of dipoles vertical to x-y plane that depends
solely on θ i←j
(2)
[15]. Due to symmetry, function vi←j xi , x j can be simplified
to:
(2)
(3)
xi , x j = 20log10 1.67sin3 θ i←j ,
vi←j
with θ i←j ∈ (0, π).1 It is worth noting that in RSS ranging
measurements, standard deviation (in dB) may depend on
distance and thus, hi(2)
j (·) , 1 could be also adopted.
(2)
that due to the antenna reciprocity theorem, v(2)
i← j ( ·) = v j←i (·);
the adopted notation assists clarity regarding which is the transmitting and
which is the receiving antenna and network node.
1 Notice
Finally, the 2D AoA measurement model
is given by
(3)
(3)
(3)
hi(3)
j (·) = 1 and pi←j (·) = vi←j (·) with vi←j xi , x j = φ i←j .
III. C RAM ÉR -R AO B OUND
f
g
⊤ ⊤ ∈ R D(N +L) is defined, where
Vector x = x⊤
g xan
xg = {xn } n ∈Ng and xan = {xl } l ∈Nan are associated with agent
and anchor positions, respectively. From Eq. (1), measurement
at node i (due to transmission from neighboring node j) is
distributed according to:
(m) 2 (m)
(m)
,
(4)
yi←j
; xi , x j ∼ N pi←j
di j σi←j
xi , x j , hi(m)
j
and( is independent
from the rest of the measurements
)
(m)
Y\ yi←j
. Thus, the joint log-likelihood distribution of measurements Y is expressed as:
g
X X X f (m)
(5)
ln f (Y; x) =
ln f yi←j ; xi , x j .
{z
}
i ∈H j ∈H (i) m ∈M |
(m)
λ i←
j
The Fisher information matrix (FIM) depends on connectivity and location of all nodes (agents or anchors) and is denoted
as J(xg , xan , G); it is associated with f (Y; x) and unknown
parameters xg , and is given by:
f
g
. (6)
J(xg , xan , G) = EY;x ∇xg ln f (Y; x) ∇⊤
xg ln f (Y; x)
FIM in (6) can be equivalently written as [7]:
 J1,1

J ≡ J(xg , xan , G) =  ...

J N,1
···
..
.
···


 ,
J N, N 
J1, N
..
.
where for n, k ∈ Ng ,
f
g
Jn,k = EY;x ∇xn ln f (Y; x) ∇⊤
xk ln f (Y; x)
(7)
(8)
is a D × D matrix. The directed node ID pairs in the two
outermost summations of Eq. (5) involving an agent n ∈ Ng ,
are given by the following set:
[ (
) (
)
A(n) ,
(n, j) : An, j = 1 ∪ ( j, n) : A j, n = 1 . (9)
j ∈H
The set above incorporates all (directed) links in the network
where agent n is involved either as transmitter or receiver,
during ranging measurements.
(m)
Regularity conditions of f yi←j
; xi , x j for each (i, j) ∈ G
will be utilized [7], i.e., for any n ∈ Ng and any (i, j) ∈ A(n),
f
g
(m)
= 0 D , for a.e. xn ∈ R D .
(11)
EY;x ∇xn λ i←j
(m)
Applying ∇xn in (5) and eliminating the terms λ i←j
that do
not depend on xn offers:
X
(9) X
(m)
∇xn ln f (Y; x) =
∇xnλ i←j
.
(12)
m ∈M (i, j ) ∈A (n)
For any n ∈ Ng and any (i, j) ∈ A(n), (i ′ , j ′ ) ∈ A(n) and
′)
(m)
and yi(m
m, m ′ ∈ M, measurements yi←j
′ ←j ′ are independent,
unless m = m ′ , i = i ′ , and f j = j ′. Thus, by′ the
g regularity
(m) ⊤ (m )
∇xn λ i ′ ←j ′ is non-zero
conditions of Eq. (11), EY;x ∇xn λ i←j
only if m = m ′ , i = i ′ , j = j ′ for any (i, j) ∈ A(n). Using the
3
f
g2


X * X * ḣi(m)
(kxi − x j k) (xi − x j )(xi − x j ) ⊤

j


..
..


g2
f (m)




(kx
−
x
k)
2
h
kxi − x j k 2
m ∈M (i, j ) ∈A (n)
i
j

ij

,
,

!
!
f (m)
g
f
g



ġi j ( kxi −x j k) (xi −x j )⊤
ġi(m)

(m)
j ( kxi −x j k) (xi −x j )

⊤ v (m) (x , x )

(x
,
x
)
+
∇
v
+
∇

xn i←j i j
xn i←j i j ++
kxi −x j k
kxi −x j k


////



+

//// , n = k

2

(m)
(m)


σ
h
(kx
−
x
k)
i
j

i←j
ij



-

Jn,k = 
f (m)
g2

⊤

X *. X

*. ḣi j (kxi − x j k) (xi − x j )(xi − x j )


.

−
1(i, j ) ∈G .

f
g2
..




2 hi(m)
(kxi − x j k) kxi − x j k 2
(i, j ) ∈
m ∈M

j

,

n)

, (n,k)∪(k,


! f (m)
!
f (m)
g
g


ġi j ( kxi −x j k) (xi −x j )
ġi j ( kxi −x j k) (xi −x j )⊤

(m)

⊤ v (m) (x , x )

+
∇
v
(x
,
x
)
+
∇

xn i←j i j
xn i←j i j ++

kxi −x j k
kxi −x j k

////



+

//// , n , k.

2

(m)


σi←j
hi(m)
(kx
−
x
k)

i
j
j

-above and substituting Eq. (12) in (8) for k = n, the diagonal
blocks in (7) can be calculated as:
X
X
f
g
(m) ⊤ (m)
∇xn λ i←j .
(13)
EY;x ∇xn λ i←j
Jn, n =
m ∈M (i, j ) ∈A (n)
Similarly, due fto the regularity conditions,
for any k, n ∈ Ng ,
g
(m) ⊤ (m ′ )
with k , n, EY;x ∇xn λ i←j ∇xk λ i ′ ←j ′ is a nonzero matrix only
if k ∈ H (n), m = m ′ , i = i ′ = n, j = j ′ = k, or only if
n ∈ H (k), m = m ′ , i = i ′ = k, j = j ′ = n. Thus, after some
algebra, the non-diagonal blocks in Eq. (7) can be expressed
as:
X X
f
g
(m) ⊤ (m)
∇xk λ i←j . (14)
1(i, j ) ∈G EY;x ∇xn λ i←j
Jn,k =
m ∈M (i, j ) ∈
(n,k)∪(k, n)
+
−1
where J−1
n, n is the nth diagonal D × D matrix block of J .
A. FIM Evaluation for Different Measurement Models
To see the utility of Eq. (16), some application examples
follow:
(1)
(1)
1
• ToA: ġi j (di j ) = c and ∇xi vi←j xi , x j = 0 D .
•
(m)
∇xn h(m)
(dnk )∇⊤
xn h nk (d nk )
nk
.
f
g2
2 h(m)
(dnk )
nk
(15)
Equality (a) holds due to ∇xn λ (m)
= −∇xk λ (m)
; the latn←k
n←k
(m)
ter holds because λ n←k is a function that depends on the
difference xn − xk , stemming directly from Eq. (1) and the
assumptions of Sec. II. Equality (b) stems from the fact that
all functions are differentiable near (xn , xk ) and [7, Eq. (3.31)].
Substituting Eq. (15) in (13) and (14), we obtain the
expression in (16) at the top of the page. In (16), we also
k
employ the chain rule of differentiation and ∇xn dnk = kxxnn −x
−xk k ,
which is properly defined due to the distinct node positions
assumption. Notation ġ(m)
(dnk ) and ḣ(m)
(dnk ) denotes the
nk
nk
(m)
(m)
first derivative of gnk (·) and hnk (·), respectively, evaluated
at point dnk = kxn − xk k.
The MSE of any unbiased deterministic estimator D
xn , of the
nth agent position is lower bounded by:
g
f
xn − xn 2 ≥ tr J−1 ,
EY;x D
(17)
n, n
2D RSS, according to Eq. (2): ġi(2)
j (di j ) =
(2)
∇xi vi←j
•
It is also noted that for any n ∈ Ng and k ∈ H (n),
g
f
g (a)
f
(m)
⊤ (m)
⊤ (m)
=
−E
∇
λ
∇
λ
EY;x ∇xn λ (m)
∇
λ
Y;x
x
x
x
n
n
n←k
n←k (m)
n←k k n←k
(m)
⊤ p
∇
p
(x
,
x
)
∇
(x
,
x
)
x
n
k
n
k
n n←k
xn n←k
(b)
=
+
(m) 2
(m)
hnk (dnk ) σ n←k
(16)
xi , x j


= 


60(x i,2 −x j,2 ) 2
ln(10) d2i j (x i,1 −x j,1 )
−60(x i,2 −x j,2 )
ln(10) d2i j
3D RSS, according to Eq. (3): ġi(2)
j (di j ) =
(2)
∇xi vi←j
xi , x j =
•
60
ln(10) d2i j






xi , x j


 .

−10ν i j
ln(10)di j
(x i,3 −x j,3 )2 (x i,1 −x j,1 )
d2i j −(x i,3 −x j,3 )2
2
(x i,3 −x j,3 ) (x i,2 −x j,2 )
d2i j −(x i,3 −x j,3 )2
−(x i,3 − x j,3 )
2D AoA: ġi(3)
j (di j ) = 0 and
(3)
∇xi vi←j
−10ν i j
ln(10)di j


= 


−(x i,2 −x j,2 )
d2i j
(x i,1 −x j,1 )
d2i j


 .

and
(18)
and


 .



(m)
(xi , x j )
With the help of the above and ∇xi vi←j
(m)
−∇x j vi←j (xi , x j ), matrix J is directly calculated.
(19)
(20)
=
IV. N UMERICAL R ESULTS
Numerical results for 2D and 3D localization are presented,
as a function of measurement noise variance, type of ranging
measurements (or combination of methods) and connectivity
radius (assuming—for conciseness—common connectivity radius r among all terminals). A common measurement noise
(m)
= σ (m) , ∀(i, j) ∈ G and
variance is assumed, i.e., σi←j
(m)
the values for σ
are taken from real experimental testbeds
[1], [2]. In addition, PLE is assumed known i.e., νi j = ν =
0
0
50
X [m]
100
2
0
15
Y 10
[m 5
]
101
2
0
15
00
5
15
10
X [m]
10
[m 5
]
0 0
5
15
10
X [m]
Fig. 1. Left: N = 117 agents and L = 4 anchors. Middle and right: N = 40
agents and L = 4 anchors for r = 7 and r = 12, respectively. Connectivity
for two nodes is also depicted in middle and right, for illustration purposes.
2.3, ∀(i, j) ∈ G in RSS measurements [2]. The final CRB
averaged across all agents is given by CRB = N1 tr J−1 .
Fig. 2 illustrates the CRB performance for the 2D topology
of Fig. 1-left, as a function of communication radius r, across
different ranging measurement methods. The RSS measurement model without directionality (line with crosses) has the
worst MSE performance. When network nodes are equipped
with dipole antennas parallel to the x-axis, the MSE (line with
circles) can be further reduced. AoA measurement model (line
with x’s) further reduces MSE compared to classic RSS for
the specific topology, while ToA (diamonds) offers the best
MSE across all measurement methods above. Interestingly,
exploitation of 2 types of measurements significantly improves
MSE performance, as shown in Fig. 2; AoA and RSS could
outperform ToA for specific network connectivity, while joint
ToA and RSS with dipole antennas can significantly reduce
MSE. It is also interesting to see that for specific network
connectivity, MSE performance of various different ranging
measurement methods coincide. Closed-form FIM calculation
of this framework allows simple performance comparisons
across different ranging methods and network topologies.
Fig. 3 offers results for the 3D topology of Fig. 1-middle
and right. RSS offers similar MSE compared to ToA measurements, while RSS with directivity outperforms ToA. That
is due to the small distances involved, and thus, the terms
not depending on distances dominate in FIM. The proposed
CRB was also used to benchmark state-of-the-art network
localization algorithms, e.g., MDS-MAP [16], which is a
refined version of classic multi-dimensional scaling (MDS),
originally designed for full connectivity scenarios, i.e., for
large values of r. Fig. 3 shows that for r ≥ 18, MDS-MAP
with ToA reaches CRB. Additionally, it is shown that performance comparison clearly depends on network connectivity.
Hopefully, the proposed bound will be used for benchmarking
in network localization research.
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