1
Estimation Bounds for Localization
Cheng Chang,Anant Sahai
Electrical Engineering and Computer Science,University of California,Berkeley,CA94720
Email:{cchang,sahai}@eecs.berkeley.edu
Abstract—1The localization problem is fundamentally impor-tant for nsor networks.This paper studies the Cram´e r-Rao lower bound(CRB)for two kinds of localization bad on noisy range measurements.Thefirst is Anchored Localization in which the estimated positions of at least3nodes are known in global coordinates.We show some basic invariances of the CRB in this ca and derive lower and upper bounds on the CRB which can be computed using only local information.The cond is Anchor-free Localization where no absolute positions are known. Although the Fisher Information Matrix is singular,a CRB-like bound exists on the total estimation variance.Finally,for both cas we discuss how the bounds scale to large networks under different models of wireless signal propagation.
Index Terms—Cram´e r-Rao bound,localization,estimation bounds,ranging information,nsor networks.
大兴安岭在哪里I.I NTRODUCTION
嘉靖帝In wireless nsor networks,the positions of the nsors play a vital role.Position information can be exploited within the network stack at all levels from improved physical layer communication[2]to routing[3]and on to the application level where positions are needed to meaningfully interpret any physical measurements the nsors may take.Becau it is so important,this problem of localization has been studied extensively.Most of the studies assume the existence of a group of“anchor nodes”that have a-priori known positions. There are three major categories of localization schemes that differ in what kind of geometric information they u to estimate locations.Many,such as tho of[4],[5],[6],[7],[8], u only the connectivity information reflecting whether node i can directly communicate with node j,or anchor k.Such approaches are attractive becau connectivity information is accessible at the network layer due to its u in multi-hop routing.
The cond category us both ranging and angular infor-mation for localization.Such schemes are studied in[9],[10], [11].The are uful when there is a line of sight and antenna arrays are available at the nsor nodes so that beamforming is possible to determine the angles.
The third category is localization bad solely on ranging measurements among nodes and between nodes and anchors. In[12],[13],the schemes for estimating ranges are discusd.
[14],[15]estimate the positions directly bad on such node to anchor ranging estimates.In contrast,[16],[17]first estimate positions in an anchor-free coordinate system and then embed it into the coordinate system defined by the anchors.In this 1An earlier version of this paper was prented at the First IEEE Commu-nications Society Conference on Sensor and Ad Hoc Communications and Networks[1].paper we also focus on localization using ranging information alone.
The Cram´e r-Rao lower bound(CRB)[18]is widely ud to evaluate the fundamental hardness of an estimation problem. The CRB for anchored localization using ranging information has been studied in[19],[20],[21].The expression for the CRB was derived in[19].In[21],a comparison of the CRB with the simpler Bayesian Bound has been studied.In[20], simulation is ud to study the impact of the density of the anchors and the size of the nsor network on the CRB.
As far as anchored localization goes,our additional con-tribution is giving a geometric interpretation of the CRB and deriving local lower and upper bounds on the CRB. The lower bounds imply that local geometry is critical for localization accuracy.The corresponding upper bounds show through si
mulation that the errors are not a lot wor if only the nearby anchors or nodes are involved in the position estimation of a particularly node.The results show that distributed localization schemes are promising.
For anchor-free localization,as mentioned in[10],the Fisher Information Matrix(FIM)is singular and so the standard CRB analysis fails[22].The CRB on anchor-free localization has not been thoroughly studied.In this paper,we give a geometric interpretation on a modified CRB and derive some properties of it.Furthermore,we show by example that anchor-free localization sometimes has a lower total estimation variance bound than anchored localization.
如何写
A.Outline of the paper
After reviewing some basics in this introduction,Section II studies bounds for anchored localization.Assuming the rang-ing errors are iid Gaussian,we give an explicit expression for the FIM solely bad on the geometry of the nsor network and show that the CRB is esntialy invariant under zooming, translation,and rotation.Using matrix theory,we give a lower bound on the CRB that is determined by only local geometry. This converges to the CRB as the local area is expanded.We also give a corresponding local upper bound on the localization CRB.Finally we study the wireless sit
uation in which the noi variance on the range measurements depends on the inter-nsor distance.Simulation results validate our intuition that the faster the signal decays,the less the CRB benefits from faraway information.A heuristic argument reveals the basic scaling laws involved.
Section III studies the bound for anchor-free localization. The rank of the FIM for M nodes is shown to be at most 2M−3.The corresponding modified CRB is interpreted as a bound on the sum of the estimation variances.We obrve that
2 the per node bound in simulations appears to be proportional
to the average number of neighbors and conjecture that the
total estimation variance scales with the total received signal
energy.
B.Cram´e r-Rao bound on ranging
Since range is our basic input,wefirst review the CRB
for wireless ranging.The distance between two nodes is ct d,
where c is the speed of light and t d is the time of arrival(TOA).
TOA estimation is extensively studied in the radar literature.If
T is the obrvation duration,A(t)is the pul2,and N0is the不完全燃烧
noi power spectral density,then for any unbiad estimate
of t d[23]:
E[(ˆt d−t d)2]≥
N0
T
[∂A(t)
∂t
]2dt
Notice that T
(∂A(t)
∂t
)2dt is proportional to the energy in
the signal with the proportionality constant depending on the pul shape.Becau of the derivative,we know that having a pul with a wide bandwidth is beneficial.Calling that proportionalityτ2r we have:
E[(ˆt d−t d)2]≥
τ2r
SNR
(1)
The CRB on ranging is a fundamental bound coming only from the Gaussian thermal noi in the rec
eived signal.In reality,there are other sources of small ranging errors in-cluding interference,multipath spreading,unpredictable clock drifts,operating system latencies,etc.The can cau the ranging error to be non-Gaussian even near the mean.More significantly,the ranging errors do not scale with SNR.We ignore all the other sources of error in this paper.
C.Models of localization
We idealize the localization problem by assuming all the nsors arefixed on a2-D plane.We have a t S of M nsors with unknown positions,together with a t F of N nsors(anchors)with known positions.Becau the size of each nsor is assumed to be very small,it is treated as a point.
Each nsor generates limited-energy wireless signals that enable node i to measure the distance to some nearby nsors in the t adj(i).We assume j∈adj(i)iff i∈adj(j)for symmetry.Throughout,we also assume high SNR3and so are free to assume that the distance measurements are only corrupted by independent zero mean Gaussian errors.
2Notice that ranging estimates can be obtained from any pul who shape is known at the receiver.This includes data carrying packets that have been successfully decoded as long as we know the time they were suppod to have been transmitted.In a wireless nsor network,we are thu
s not restricted to u a dedicated radio for ranging.
3Suppo that we are estimating the propagation time by looking for a peak in a matchedfilter.By high SNR we mean that the peak wefind is in the near neighborhood of the true peak.At low SNR,it is possible to become confud due to fal peaks arising entirely from the noi.
1)Anchored localization:If there are at least three nodes with positions known in global coordinates(|F|≥3),then it is possible to estimate such global coordinates for each node using obrvations D and position knowledge P F.
D={ˆd i,j|i∈S∪F,j∈adj(i)}(2)
P F={(x i,y i)T|i∈F}(3) Our goal is to estimate the t
P S={(ˆx i,ˆy i)T|i∈S}(4) (x i,y i)is the position of
nsor i.The measured distance between nsor i and j isˆd i,j=
(x i−x j)2+(y i−y j)2+ i,j,where i,j’s are modeled as independent Gaussian errors ∼N(0,σ2ij).
Fig.1.A nsor network,solid dots are anchors,circles are nodes with unknown positions.The rangeˆd i,j is estimated for nsor pairs i,d i,j≤R visible.
2)Anchor-free localization:If|F|=0,no nodes have known positions.This is an appropriate model whenever either we do not care about absolute positions,or if whatever global positions we do have are far more impreci than the quality of measurements available within the nsor network.However, local coordinates are not unique.If P S={(ˆx i,ˆy i)T|i∈S}is
a position estimate,then P
S
={R(α)(±ˆx i,ˆy i)T+(a,b)T|i∈S}is equivalent to P S where the±reprents reflecting the entire network about the y axis and R(α)is a rotation matrix:
R(α)=
cos(α)−sin(α)
sin(α)cos(α)
(5)
Thus,the performance measure for anchor-free localization should not be
i
(x i−ˆx i)2+(y i−ˆy i)2.The distance between equivalence class should be ud instead.Since the FIM for anchor-free localization is singular[10],the bound will be developed using the tools provided in[22].
II.E STIMATION BOUNDS FOR ANCHORED LOCALIZATION The Cram´e r-Rao bound(CRB)can be derived from the FIM.
3
A.The anchored localization FIM
In[19],[20],[21],expressions for the localization FIM were
derived.The derivations are repeated below for completeness
and furthermore,we obrve that the FIM for localization
is a function of the angles between nodes and anchors.As
illustrated in Fig.2,the angleαij∈[0,2π)from node i to j
is defined as:
cos(αij)=
x j −x i
(x j−x i)2+(y j−y i)2
=
x j−x i
d ij
sin(αij)=
y j−y i
(x j−x i)2+(y j−y i)2
=
y j−y i
d ij
(6)
Fig.2.αij illustrated
Let x i,y i be the2i−1’th and2i’th parameters to be estimated respectively,i=1,2,...,M.The FIM is J2M×2M. Theorem1:(FIM for Anchored Localization)∀i= 1,...,M
J2i−1,2i−1=
j∈ad j(i)cos2(αij)
σ2
ij
(7)
J2i,2i=
j∈ad j(i)sin2(αij)
σ2
ij
(8)
J2i−1,2i=J2i,2i−1=
j∈ad j(i)cos(αij)sin(αij)
σ2
ij
(9)
For nondiagonal entries j=i,if j∈adj(i):
J2i−1,2j−1=J2j−1,2i−1=−
1
σ2
ij
cos2(αij)(10)
J2i,2j=J2j,2i=−
1
σ2
ij
sin2(αij)(11)
J2i−1,2j=J2j,2i−1=J2i,2j−1=J2j−1,2i
=−
1
σ2
ij
sin(αij)cos(αij)=−
1
2σ2
ij
sin(2αij)
If j/∈adj(i),the entries are all zero.
Proof::We have the conditional pdf4:
p( d|x M1,y M1)=
i<j,j∈ad j(i)e
−(ˆd ij−d ij)2
2σ2ij
2πσ2
ij
4 d={ˆd i,j|i<j,j∈adj(i)}is the obrvation vector. x M1=(x1,x2,...,x M),similarly for y M1.
The Log-likelihood is ln(p( d|x M1,y M1))=C−
i<j,j∈ad j(i)
(ˆd i,j−d i,j)2
2σ2
ij
and so:
J2i−1,2i−1=E(
∂2ln(p( d|x M1,y M1))
∂x2
i
)
=
j∈ad j(i)
1
σ2
ij
(
x j−x i
(x j−x i)2+(y j−y i)2
)2
=
j∈ad j(i)
cos2(αij)
σ2
ij
and similarly for other entries of J.
B.Properties of the anchored localization CRB
Given the FIM,the CRB for any unbiad estimator is5:
E((ˆx i−x i)2)≥J−1
2i−1,2i−1
E((ˆy i−y i)2)≥J−1
2i,2i
Corollary1:(The FIM is invariant under zooming and translation)J({(x i,y i)})=J({(ax i+c,ay i+d)})for a=0.
Proof::The anglesαij and noiσij are unchanged and so the result follows immediately. Corollary2:The CRB for a single node is invariant un-der rotation and reflection:Let A=J({(x i,y i)}),B= J({R(x i,y i)}),where R is a2×2matrix,with RR T= I2×2.Then A−12i−1,2i−1+A−12i,2i=B−12i−1,2i−1+B−12i,2i,∀i= ,M.
Proof::Going through the derivation of the FIM,wefind that B=QAQ T,where Q is a2M×2M matrix with the following form:
Q2i−1,2i−1Q2i−1,2i
Q2i,2i−1Q2i,2i
=R(12)
with all other entries of Q being0.Obviously Q T Q=QQ T= I2M×2M and so B−1=QA−1Q T.Write
A(i)=
A−1
2i−1,2i−1
A−1
2i−1,2i
A−1
2i,2i−1
A−1
2i,2i
(13)
and similarly for B(i).Then B(i)=RA(i)R T.Since
T r(XY)=T r(Y X),we have:B−1
2i−1,2i−1
+B−1
2i,2i
= T r(B(i))=T r(RA(i)R T)=T r(R T RA(i))=T r(A(i))=
A−1
2i−1,2i−1
+A−1
2i,2i
.
C.A lower bound to the anchored localization CRB
In order to invert the FIM and thereby evaluate the CRB, we need to take the geometry of the whole nsor network into account.In this ction,we derive a performance bound for node l that depends only on the local geometry around it. This has the potential to be valuable to“local”algorithms that try to do localization without performing all the computations in one center.
First we review a lemma for estimation variance:
5We write(A−1)i,j as A−1
i,j
for a non-singular matrix A.
4
Lemma 1:(Submatrix bound)Let the row vector θ=(θ1,θ2,...,θN )∈R N ,∀M,1≤M <N ,write θ∗=(θN −M +1,...,θN ),then for any unbiad estimator for θ,
E ((θ∗−ˆθ
∗)T (θ∗−ˆθ∗))≥C −1(14)
Where C is the (N −M )×(N −M )matrix :
J (θ)=
A B
B T C
(15)
where J (θ)is the non-singular,and hence positive definite,FIM for θ.
Proof::Write the inver of J (θ)as :
J (θ)−1= A B
B T
C (16)
J (θ)is positive definite,then Theorem 5in the appendix guarantees:
C ≥C −1
(17)
The CRB theorem then gives E ((θ∗−ˆθ
∗)T (θ∗−ˆθ∗))≥C ≥C −1. Notice that for any subt of M nodes,we can always reorder them to get indices N −M +1,...,N .By directly applying Lemma 1we get:
Theorem 2:(A lower bound on the CRB)Write θl =(x l ,y l )T and write
J l =1σ
J (θ)2l −1,2l −1J (θ)2l −1,2l
J (θ)2l,2l −1J (θ)2l,2l (18)
Then for any unbiad estimator ˆθ
.E ((ˆθl −θl )(ˆθl −θl )T )≥J −1l .
This means we can give a bound on the estimation of (x l ,y l )using only the local geometry around nsor l
Corollary 3:J l only depends on (x l ,y l )and (x i ,y i ),i ∈adj (l )
Proof::J l in Eqn.7only depends on (αlj ,σlj ),j ∈adj (l ).The only depend on (x l ,y l )and (x i ,y i ). As
sume that the ranging errors are iid Gaussian with zero mean and common variance σ2and define the normalized FIM K =σ2J .This is similar to the Geometric Dilution of Precision (GDOP)in radar[24]since K is dimensionless and only depends on the angles αij ’s.Let W =|adj (l )|with nsors ∈adj (l )being l (1),...,l (k ),...,l (W ).Using elementary trigonometry and writing αk =αl,l (k ):
J l =
1σ
W 2+ W
k =1cos(2αk )2 W k =1
sin(2αk )2 W
k =1sin(2αk )2
W 2
−
W
k =1cos(2αk )
2
The sum of the estimation variance
E ((x l −ˆx i )2+(y l −ˆy i )2)≥J −1l 11+J −1l 22=
4W σ2
W 2−( W k =1cos(2αk ))2−( W
k =1sin(2αk ))2
≥4σ2W (19)
with equality when W k =1sin(2αk )=0, W
k =1cos(2αk )=0.This happens if the centroid of the unit vectors
(cos(2αk ),sin(2αk ))is the origin.A special ca is when the angles 2αk ’s are uniformly distributed in [0,2π).
Above,we ud one-hop geometric information around node i to get a lower bound on the CRB.This bound can be interpreted as the CRB given perfect knowledge of the positions of all other nodes 6.We can u more information to tighten the bound.The lower bound using 2-hop information is the CRB given the positions of all nodes j ,j /∈adj (i ),and similarly for multiple-hops.The larger the local region we u to calculate the CRB,the tighter it is.We define the CRB on such an estimation problem as the N −hop bound for that particular node.Obviously,the N −hop bound is non-decreasing with N ,and the ∞−hop bound is the same as the CRB for the original estimation problem.
In our simulation,we have 200nodes and 10anchors all uniformly randomly distributed inside the unit circle,j ∈adj (i ),if and only if d i.j ≤0.3.In Figure.3,we plot the bounds for 20randomly chon nodes.
Fig.3.Dot:CRB,Cross:2-hop,Square:1-hop,Curve:4ad j (l )
The nodes are indexed with decreasing adj (i )
D.An upper bound to the anchored localization CRB The CRB in Theorem 1gives us the best perfor
mance an unbiad estimator can achieve given all information from the nsor network,including the positions of all anchors and
all the available ranging information ˆd
i,j .This bounds the performance of a centralized localization algorithm where a central computer first collects all the information and then estimate the positions of the nodes.
In a nsor network,distributed localization is often pre-ferred.In this “local”estimation problem only a subt of the anchors F l ⊆F and a neighborhood of the nodes l ∈S l ⊆S may be taken into account.The CRB V (x l )and V (y l )of this local estimation problem computed from the 2|S l |×2|S l |FIM is an upper bound on the CRB for the original problem becau strictly less information is ud for estimation.7In this ction,the two bounds are compared through simulation.
6It’s
equivalent to know the positions of all the neighbors.
7In [19],a rigorous proof is given for the equivalent proposition that the localization CRB for a node is non-increasing as more nodes or anchors are introduced into the nsor network.
5 The wireless nsor network is shown in Fig.4.Anchors
are on the integer lattice points in a7×7square region.There
are20nodes with unknown positions uniformly randomly
distributed inside each grid square.Sensors i and j can e
each other only if they are parated by a distance less than
0.5.
Fig.4.The tup of the nsor network
Anchors are shown as squares,nodes are shown as dots,nodes inside the
central grid are shown as black dots.
We compute the normalized CRBs(V i=V x i+V y
i ,i=
1,2,...,20)for localization of the nodes inside the central grid A1A2A3A4in4different cas corresponding to in-formation from within the squares:A1A2A3A4,B1B2B3B4, C1C2C3C4,and the whole nsor network.As shown in Fig.5, V i(A)≥V i(B)≥V i(C)≥V i(ALL),i=1,2,...,20.We obrve that V i(C)(squares in Fig.5)is extremely clo to V i(ALL)(the curve in Fig.5).More surprisingly,we obrve that V i(B)is much smaller than V i(A).
To explore further,we gradually increa the size of the square region and compute the average CRB for A1A2A3A4. As shown in Fig.6,the average CRB decreas as the network size increas.Afterfirst dropping significantly,the upper bound levels off once we have included all the nodes directly adjacent to our neighborhood.This bodes well for doing distributed localization—distant anchors and ranging infor-mation do not significantly improve the estimation accuracy.
E.CRB under different propagation models
In the previous discussion,the ranging information was assumed to be corrupted by iid Gaussian errors.The ranging CRB,Eqn.1,implies that the varianceσ2i,j of the additive noi on the distance measurement should depend on the distance d i,j between two nodes i,j,becau the received wireless signal A(t)attenuates as a function of d.We
assume Fig.5.Cram´e r-Rao bounds
Circle:estimation bounds using the information inside A1A2A3A4.
Dot:estimation bounds using the information inside B1B2B3B4. Square:estimation bounds using the information inside C1C2C3C4. Curve:estimation bounds using all the information.
1 1.
2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Size of the nsor network
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Fig.6.Circle:CRB using information from local network.
Line:CRB using whole network.
σ2i,j=σ2d a i,j,whereσ2is the noi variance when d=1.8
Furthermore,we assume a range estimate is available between all nsors,though it may be bad if they are far apart. Interference is ignored.This is reasonable only when there is no bandwidth constraint for the system as a whole,or if the data rates of communication are so low that all nodes can u signaling orthogonal to each other.
Define K=σ2J to be the normalized FIM.Just as in the ca where a=0,translations of the whole nsor network do not change the FIM.Rotation does not change the CRB on
any node K−1
2i−1,2i−1
+K−1
2i,2i
.However,zooming does have an effect on the FIM.
Corollary4:(The normalized FIM K is scaled under zooming)If the propagation model is d a,a≥0,and the whole nsor network is zoomed by a zooming factor c>0.
K({c(x i,y i)})=1
c a
K({(x i,y i)}),c=0.
Proof::Zooming does not change the anglesαi,j be-
8Earlier,we had a hybrid model with a=0locally and a=∞at a great distance since the range is only available for nsor pair i,j,if d i,j< R visible.
6
tween nsors.If the zooming factor is c ,then the decaying factor changes to (cd i,j )a =c a d a i,j ,Substitute the new decaying factors into the FIM as in Theorem 1,we get:K ({c (x i ,y i )})=1c a K ({(x i ,y i )}).
The CRB σ2K −1i,i changes proportional to c a
,if the whole nsor network is zoomed up by a factor c .书法作品简介
Next,we have a simulation in which we fix the node density and examine the average CRB for different a ’s as we vary the size of the nsor network.The nsor network is the same as in Fig.4and the sizes are taken at 1×1,3×3,...,13×13.We calculate the average CRB inside the central square and plot the average estimation bound in 10log 10scale in Fig.7.The average CRB decreas as the size of the nsor network increas.This is expected since there is more infor-mation available and no interference by assumption.Asymp-totically,the CRB decreas at a faster rate for smaller a since the noi variance increas more slowly with range.
Fig.7.Average CRB in the central grid for different a
Circle:a =1,Dot:a =2,Cross:a =3
Heuristically,the localization accuracy for node i is mainly determined by the total energy received by it.Suppo that the distance between nodes is ≥r m ,and the nodes are uniformly distributed.We approximate the total received energy P R coming from nsors within distance R as:
辩论词格式P R =β 2π0 R r m ρ−a
ρdρdθ=2βπ R
r m ρ1−a dρ
=
2βπ2−a (R 2−a −r 2−a
m )if a =22βπ(ln(R )−ln(r m ))if a =2.When a <2,P R behaves like R 2−a which grows unboundedly as the network grows and similarly for a =2where P R behaves like ln(R ).In such non-physical cas,it would be possible to save each node’s transmitter power by going to a larger network and then turning down the transmit power in such a way as to keep the position accuracy fixed.But in the
physically relevant ca of a >2,P R converges to 2βπa −2r 2−a
m and local measurements should be good enough.This heuristic explanation is a qualitative fit with simulations as illustrated in Fig.7.
III.E STIMATION B OUNDS FOR A NCHOR -F REE
L OCALIZATION
For anchor-free localization,only the inter-node distance measurements are available.The nature of anchor-free local-
ization is very different from anchored localization,in that the absolute positions of the nodes cannot be determined.We first review the singularity of the FIM using the treatment from [18].
Lemma 2:(Rank of the FIM)Let d
be the obrvation vector,and θbe the n dimensional parameter to be estimated.
Write the log likelihood function as l ( d
|θ)=ln(p ( d |θ)).The rank of the FIM J is n −k ,k ≥0,if and only if the expectation
of the square of directional derivative of l ( d
|θ)at θis zero for k independent vectors b 1,...,b k ∈R n .
Proof::The directional derivative of l ( d
|θ)at θ,along direction b i is :τ(b i )=(∂l/∂θ1,∂l/∂θ2,...,∂l/∂θn )b i .
E (τ(b i )2)
=E (b T i (∂l/∂θ1,...,∂l/∂θn )T
(∂l/∂θ1,...,∂l/∂θn )b i )=b T i Jb i
(20)If k independent vectors b 1,...,b k make b T i Jb i =0,the rank of J is n −k ,since J is an n ×n symmetric matrix. The FIM for anchor-free localization is given in Theorem 1,just with no anchors.With the above lemma,we can prove that the rank of this FIM is deficient by at least 3.This is intuitively clear since there are 3degrees of freedom coming from rotation and translation.
Theorem 3:For the anchor-free localization problem,with M nodes,the FIM J (θ)is of rank 2M −3
Proof::The log-likelihood function of this estimation problem is :
l ( d
|θ)=ln(p ({ˆd i,j ,1≤i,j ≤M,j ∈adj (i )}|{
(x i −x j )2+(y i −y j )2,1≤i,j ≤M,j ∈adj (i )})
=
1≤i,j ≤M,j ∈ad j (i )
ln(p (ˆd i,j | (x i −x j )2+(y i −y j )2))
The last equality comes from the independence of the mea-surement errors.The directional derivative of each term in the sum is 0along the vectors b 1, b 2, b 3∈R 2M . b 1=(1,0,1,0,...,1,0)T , b 2=(0,1,0,1,...,0,1)T , b 3=(y 1,−x 1,y 2,−x 2,...,y M ,−x M )T where b 1and b 2span the
2-D space in R 2M
corresponding to translations and b 3is the instantaneous direction when the whole nsor networks rotates. Since the FIM is not full rank,we cannot apply the standard CRB argument becau J −1does not exist.Instead,the CRB is the Moore-Penro pudo-inver J †.[22]A.What does J †mean:the total estimation bound
When the FIM is singular,we cannot properly define the parameter estimation problem in R n .However,we can estimate the parameters in the local subspace spanned by all k orthonormal eigenvectors v 1,..., v k corresponding non-zero eigenvalues of J .In that subspace,the FIM Q is full
rank.Write V =(v 1,...,v k ),V is an n ×k matrix and V T V =I k ,then Q =V T JV ,and Q −1=V T J †V ,thus J †is the intrinsic CRB matrix for the estimation problem.The total estimation bound for the estimation problem in the k
