Presented at IEEE Communication Theory Workshop (CTW), Cancun, Mexico, May 10-12, 2010. PROCEEDINGS ARE NOT AVAILABL

Caceres Duran, Mauricio Andres

Penna, Federico

Wymeersch, H.

English

PORTO Publications Open Repository TOrino

1Cramér-Rao Bound for Hybrid Peer-to-Peer
Positioning
Henk Wymeersch1, Federico Penna2, Mauricio Caceres2
1Chalmers University of Technology - Dept. of Signals and Systems, Gothenburg, Sweden
2Politecnico di Torino - Dept. of Electronics, Turin, Italy
Email: henkw@chalmers.se, {federico.penna, mauricio.caceresduran}@polito.it
I. INTRODUCTION
Novel cooperative positioning methods have been proposed
to operate in GPS-challenged environments. However, such
cooperative schemes can be also used in combination with
GPS, so as to improve positioning accuracy in cases where
GPS measurements are available (i) intermittently; or (ii) from
a limited number of satellites; or (iii) are strongly affected by
noise or multi-path. “Hybrid cooperative positioning” schemes
can thus be designed to fuse information from peers and
from GPS satellites. This contribution provides a theoretical
characterization of achievable performance of hybrid cooper-
ative positioning, by expressing the Cramér-Rao lower bound
(CRLB) for the aforementioned scenario. Our results extend
[?], by including the unknown clock bias, and [?], by taking
into account satellites in addition to terrestrial devices.
II. PROBLEM FORMULATION
Given a heterogeneous network (Fig. ??), comprising satel-
lites with known clock bias and known position, anchor nodes
with known position but unknown clock bias, and agents with
unknown clock bias and unknown position. Let M be the
set of agents, S the set of satellites, A the set of anchors;
denote by Sm the set of satellites that agent m can see, by
Am the set of anchors that agent m can communicate with,
and by Mm the set of peers it can communicate with. The
position of a satellite s ∈ S , an anchor a ∈ A, and an
agent m ∈ M, are indicated respectively by xs, xa, xm.
Our focus will be on 2-dimensional positioning, from which
the extension to 3-dimensional case is straightforward. The
variable bm represents the clock bias of agent m, expressed
in distance units.
Two types of measurements are available to agent m: rn→m
is the measured distance between agent m and node n ∈
Am ∪Mm, with rn→m = ‖xn − xm‖+ vn→m, where vn→m
is measurement noise; ρs→m is a pseudorange measurement
between node m and satellite s ∈ Sm, ρs→m = ‖xs − xm‖+
bm + vs→m, where vs→m is measurement noise.We assume
that: (i) all measurement noise is zero-mean Gaussian; (ii)
for peer-to-peer measurements, the link variance is symmetric:
σ2n→m = σ
2
m→n.
Our goal is to compute the CRLB of the deterministic
unknown [X,b], where X = {xm∈M} and b = {bm∈M}, as
The authors thank the European Space Agency (ESA), in particular Jaron
Samson, and Prof. Roberto Garello for their support to this work in the
framework of the “Peer to Peer Positioning” project.
1
2 3
4
5
6
Sat. 2
Sat. 1
Sat. 3 Sat. 4
Sat. 5
Sat. 6Sat. 7
Figure 1. Example network topology. Agents’ positions (in m): 1: [-5 1]; 2:
[0 3]; 3: 5 4]; 4: [3 -2]; 5: [-3 -4]; 6: [0 0]. Satellites’ positions (in m): 1: [-50
-10]; 2: [-50 20]; 3: [-20 50; 4: [12 50; 5: [50 0]; 6: [10 -50]; 7: [-10 -50].
Measurement noise: σs→m = 3m ∀m ∈ M, s ∈ Sm; σn→m = 0.10m
∀m ∈M, n ∈Mm.
a function of the (range and pseudorange) measurement noise
variances σ2a→m, σ
2
n→m, σ
2
s→m, and of the network geometry.
III. FISHER INFORMATION MATRIX
The CRLB of any unbiased estimator of [X,b] is obtained
by inverting the corresponding Fisher information matrix
(FIM). Let F the FIM for our hybrid scenario. We will first
compute the FIM under a non-cooperative setting, and then
extend this result to the cooperative case.
A. Non-cooperative Case
We focus on a single agent, say m. Then the log-likelihood
function of its measurements with respect to anchors and
satellites is
log p
({ra→m}a∈Am , {ρs→m}s∈Sm |xm, bm )
=
∑
a∈Am
log p (ra→m |xm ) +
∑
s∈Sm
log p (ρs→m |xm, bm )
.
= Λm (xm, bm) .
The Fisher information matrix is given by
Fm = −E { Hm (Λm (xm, bm))} ,
where the expectation is with respect to the measurements,
and Hm(·) is the Hessian operator containing the second-order
21 2 3 4 5 6
0
2
4
6
8
10
12
14
16
...
Inf
agent index
CR
LB
 m
et
ric
 [m
]
 
 
position Non−Coop
position Coop
bias Non−Coop
bias Coop
Figure 2. Comparison of position- and bias-CRLB: cooperative vs. non-
cooperative setting.
partial derivatives with respect to each element of [xm, bm].
In case of 2D positioning F is a 3 by 3 matrix:
Fm =
[
Fxm fxm,bm
fTxm,bm Fbm
]
 0, (1)
where
Fxm =
∑
a∈Am
1
σ2a→m
qamq
T
am +
∑
s∈Sm
1
σ2s→m
qsmq
T
sm
Fbm =
∑
s∈Sm
1
σ2s→m
fxm,bm =
∑
s∈Sm
− 1
σ2s→m
qsm,
in which qim = xi−xm‖xi−xm‖ is a unit-length column vector
between xm and xi. Considering all M agents, the global
non-cooperative FIM is a block-diagonal matrix as
Fnon−coop =
 F1 . . .
FM
 . (2)
B. Cooperative Case
The log-likelihood function is now
log p
({rn→m}n∈Am∪Mm , {ρs→m}s∈Sm |xm, bm )
=
∑
m∈M
Λm (xm, bm) +
∑
m∈M
∑
n∈Mm
log p (rn→m |xm ) .︸ ︷︷ ︸
.
=Λcoop(X)
The Fisher information matrix is of the form
F = Fnon−coop + Fcoop (3)
and has dimension 3M × 3M , for M agents. The first term
Fnon−coop, representing the non-cooperative contribution, is
given in (??), while the cooperative part Fcoop can be ex-
pressed as
Fcoop = −E {H(Λcoop(X))} ,
where H(·) is again the Hessian operator. Notice first of
all that Λcoop (X) does not depend on the bias. Under the
hypothesis of Gaussian measurement noise in peer-to-peer
communication, the result is a block matrix of the form
Fcoop =

F′1 0 K12 0 . . . K1M 0
0T 0 0T 0 0T 0
K21 0 F
′
2 0
0T 0 0T 0
...
. . .
KM1 0 F
′
M 0
0T 0 0T 0

 0,
(4)
where
F′m =
∑
n∈Mm
1
σ2n→m
qnmq
T
nm
Kmn =
{ − 1σ2n→mqnmqTnm, if n ∈Mm
0 otherwise.
The above results allow to compute F for a given network
configuration and, by inverting (??), to express the CRLB.
IV. NUMERICAL RESULTS
We consider a representative indoor GPS-challenged sce-
nario with 6 agents, no anchors and 7 satellites, where only
agents close to windows can see satellites (Fig. ??). Observe
that agents 1 and 4 see enough satellites for positioning
without cooperation, while agent 6 sees no satellites at all.
In Fig. ??, we show for a non-cooperative and cooperative
setting, (expressed in meters)
√
Tr(Jxm) and
√
Jbm , where
Jxm and Jbm are, respectively, the m-th position- and bias-
related blocks of J, that is the CRLB matrix obtained by
inversion of Fnon−coop (??) or F (??) after removing rows
and columns corresponding to non-estimable variables1. As
expected, cooperation allows a significant performance im-
provement in positioning accuracy: without cooperation, only
node 1 and node 4 have a finite CRLB; in the cooperative case,
the CRLB is finite for all nodes, with lower values for nodes
1 and 4 compared to the previous ones. Notice that at least
one satellite connection is necessary for estimating the bias.
This explains why the bias-CRLB of node 6 is still infinite
even with cooperation, while it is finite for node 5 thanks to
its connection with satellite 7.
V. CONCLUSION AND FURTHER WORK
The results derived in this paper give insight into the gains
in hybrid cooperative positioning, node placement strategies,
and network blind spots. We plan a detailed investigation of
hybrid networks for realistic operational scenarios.
1Non-estimable variables are: positions and biases, for nodes whose total
number of connections is less than 3; biases, for nodes connected to no
satellites. These variables generate matrix singularities, hence CRLB →∞.
3REFERENCES
[1] Larsson, E. G, “Cramér-Rao Bound Analysis of Distributed Positioning
in Sensor Networks,” IEEE Signal Processing Letters, 2004, vol. 11, part
3, pp. 334-337
[2] Y. Shen, H. Wymeersch, and M. Z. Win, “Fundamental Limits of Wide-
band Cooperative Localization via Fisher Information”, IEEE Wireless
Communications and Networking Conference (WCNC), pp.3951-3955,
March 2007


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