# Modeling Distance and Bandwidth Dependency of  TOA-Based UWB Ranging Error for Positioning

Authors: Giovanni Bellusci; Gerard J. M. Janssen; Junlin Yan; Christian C. J. M. Tiberius
Journal: Research Letters in Communications
 (2009)

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## Abstract

A statistical model for the range error provided by TOA estimation
using UWB signals is given, based on UWB channel measurements
between 3.1 and 10.6 GHz. The range error has been modeled
as a Gaussian random variable for LOS and as a combination of a
Gaussian and an exponential random variable for NLOS. The distance
and bandwidth dependency of both the mean and the standard
deviation of the range error has been analyzed, and insight is
given in the different phenomena which affect the estimation
accuracy. A possible application of the model to weighted least
squares positioning is finally investigated. Noticeable
improvements compared to the traditional least squares method have
been obtained.


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## Body

## 1. Introduction

Time  of arrival
(TOA) estimation using ultra-wideband (UWB) signals appears the most suitable
ranging technique for indoor  positioning applications which require centimeter-
to decimeter-level accuracy [1]. For this reason, it is fundamental to provide
an in depth investigation and modeling of the accuracy which can be achieved
using this approach and its dependency on various system parameters. Some
papers investigated and sometimes modeled the ranging performance achieved
using UWB [2–6]; however, to the best of our knowledge, the related work
reported in literature still
lacks a complete and coherent understanding of the
various factors which affect the range  estimation accuracy, and of good
statistical error models.The contribution of this letter is to provide a better
understanding of the types of range error usually experienced in indoor
environments and to propose a novel statistical model for the error obtained by
TOA-based UWB range estimation. Unlike in the available literature, the
distance and the bandwidth dependency of both the bias of the range error, and
its random variations have been investigated and statistically modeled.
Finally, a possible application of the model to weighted least squares
positioning is analyzed and the improvements compared to the classical least
squares approach are evaluated.


## 2. Measurement Setup, Signal Processing,
and Range Estimation

The channel
impulse response (CIR) measurements used in this paper were collected using a
time domain technique and cover the bandwidth between 3.1 and 10.6 GHz allowed
by the FCC for UWB radio transmissions. Details of the system setup can be
found in [7]. The transmitter-receiver separation varied between 2 and 15 m for
LOS, and between 3 and 12 m for NLOS. In NLOS, there is a physical obstruction,
represented by walls, between the transmitter and the receiver. To allow
distance dependency modeling, a total of about 600 measurements, each corresponding
to different transmitter and receiver positions, were collected.The CIRh(t) is estimated by deconvolving the received
signal in the frequency domain using the inverse filtering technique: the
spectrum of the received signal is divided by the one of the reference signal,
measured at a distance of one meter, in the absence of reflections. To model the bandwidth dependency of the range error, the spectra of the received signal and of the reference one are filtered with a rectangular window centered at fc=(3.1+10.6)/2 GHz =6.85 GHz, and with a bandwidth B which varies from 0.5 GHz to 7.5 GHz, increased in steps of 0.5 GHz. The first path of the CIR is defined as
the first local maximum of the envelope of the estimated channel, with
amplitude within 20 dB from the strongest peak. The range error is
defined as ϵ=d^−d,
where d^ is the distance estimated by multiplying the
TOA of the first path by the speed of light in air, and d is the true distance between the transmitter
and receiver.


## 3. Range Error Results, and Modeling

The range error is expressed as the sum of a biasm,
plus a zero-mean random variable S which accounts for the random variations
around m.
The proposed model explicitly considers the distance and bandwidth dependency
of these two quantities:(1)ϵ(B,d)=m(B,d)+S(B,d).In LOS, the biasm(B,d) is due to multipath and reflects the fact that
nonresolvable paths due to reflections always arrive later than the direct
one. In NLOS, m(B,d) is due to both multipath, as for LOS, and to
the additional delay introduced by the propagation through dielectric
materials. By analyzing the data, m(B,d) has been modeled as a linearly increasing
function of d;
the slope of the linear fit increases with decreasing B:
(2)m(B,d)=mB(B)d+m0,
where
mB(B) accounts for its bandwidth dependency and it
has been modeled as mB(B)=m1exp(−B/m2)+m3.
The coefficients m0, m1, m2,
and m3 have been obtained by least squares fitting of
the data corresponding to different bandwidths, and are reported in Table 1. m(B,d) becomes larger for smaller B,
since in this case, the number of nonresolvable multipath components that
follow the direct path increases, resulting in more bias. At the same time, the
bandwidth dependency becomes more evident with larger d.
In fact in LOS, for increasing d,
the direct path tends to have less weight with respect to the total signal,
since the path loss exponent corresponding to the first path power is close to
2 [8], while that corresponding to the total power is usually between 1 and 2
[8], thus smaller; this means that multipath is more deteriorating for the
resolvability of the direct path for larger d.
Moreover, by increasing d,
the distance difference between the direct path and the reflected ones tends to
decrease, making the nonresolvable multipath more dense. For NLOS, the distance
and bandwidth dependency of m(B,d) can be explained with similar arguments; its
significantly larger values compared to LOS, even for larger B,
are due to the additional delay introduced by the propagation through
dielectric materials.Table 1
Parameters for the characterization ofm(B,d) and of S(B,d).
m(B,d):m0  (m)m1m2  (GHz)m3LOS00.01480.480NLOS0.0190.0270.470.013S(B,d):σS0 (m)αg1 (m−α)g2 (GHz)LOS0.0161.50.640.60NLOS0.0491.50.210.73The deviation from the mean,S(B,d),
has been modeled as a random variable with standard deviation σS(B,d) which depends on both B and d.
As for m(B,d),
also σS(B,d) becomes larger by decreasing B;
in addition, to reflect the fact that for the full bandwidth no distance
dependency of σS(B,d) has been observed, while for smaller
bandwidths, σS(B,d) significantly increases with d,
the following model for σS(B,d) is proposed:(3)σS(B,d)=σS0(gB(B)gd(d)+1),with gB(B)=g1exp(−B/g2) and gd(d)=dα.
This behavior for σS(B,d) can be justified observing that due to
multipath, the resolvability of the direct path becomes worse for smaller
bandwidths or larger distances, as previously motivated in the analysis of m(B,d).
The parameters σS0, α, g1,
and g2 have been obtained by least squares fitting of
the data; their values are also reported in Table 1. Figure 1 presents the
scatter plot of the range error versus distance both for (a) LOS and for (b)
NLOS, for B=7.5 and B=0.5 GHz for comparison.Range error
versusd for (a) LOS and for (b) NLOS, for B=7.5 GHz and B=0.5 GHz.
(a)
LOS(b)
NLOSFor the statistical characterization ofS(B,d),
a difference between the two propagation conditions arises. Therefore, we
introduce the new random variable(4)S0=S(B,d)gB(B)gd(d)+1,which represents S(B,d) normalized to the terms which model its
bandwidth and distance dependency. By analyzing the histogram of S0,
for LOS, it has been modeled as a zero mean Gaussian random variable with
standard deviation σS0.
For NLOS, we noticed an asymmetric distribution around the zero mean, for all
bandwidths. In fact, the tail corresponding to the positive values of the
histogram of S0 is longer than that corresponding to the
negative ones. The physical explanation of this aspect is the following: in
NLOS, with strong multipath conditions, there are situations in which several
reflected and nonresolvable paths may arrive closely after the direct one; this
can move the detected first path of the CIR to a delay which is significantly
larger than 1/B.
On the contrary, the negative errors due to multipath are always limited to
approximately 1/B.
Figure 2 explains this point. This phenomenon was not experienced in LOS due to
the less dense multipath. To account for this aspect, in NLOS, S0 has been modeled as the sum of two independent
random variables(5)S0=sNRN+sERE+s0,where RN is a standard Gaussian random variable and RE is a standard exponential random variable. The
coefficients sN=wσS0, sE=1−w2σS0,
and s0=−sE=−1−w2σS0 can be expressed as functions of only one
variable w  (0≤w≤1), and are necessary to weight the two
distributions with the constraints for the mean and the standard deviation of S0.
By fitting the data, we obtained w=0.6.Figure 2
NLOS range error behavior.From the described model, it is possible to derive the
mean and standard deviation of the range error obtained from the total set of
measurements, as a function ofB.
From (1) and (2), the global mean range error results in(6)mean{ϵ(B,d)}=mB(B)E[d]+m0,where E[⋅] represents the expectation operation. E[d]=6.95 m for LOS and E[d]=7.74 m for NLOS has been evaluated from the
measured data. For the standard deviation of the range error, using (1)–(4), ϵ(B,d) can be expressed as(7)ϵ(B,d)=mB(B)d+m0+S0(gB(B)dα+1),which is a nonlinear function of
two independent random variables, d and S0.
Using the variance propagation law applied to (7), linearized around the means
of d and S0,
and approximated up to its second order, we find(8)std{ϵ(B,d)}≅(mB(B)σd)2+(σS0(gB(B)(E[d])α+1))2,where σd=3.41 m for LOS, and σd=2.05 m for NLOS is the standard deviation of the
distribution of the distances, which has been evaluated from the measured data.Figure3 shows the mean and the standard deviation of
the range error directly obtained using the total set of data, and the model
proposed in (6)–(8) for comparison.Figure 3
Mean and standard deviation of the global range error versusB.


## 4. Weighted Least Squares Positioning

In this
section, a possible application of the proposed model to improve the classical
least squares positioning is investigated. From the total set of measurements,
a subset ofn of them is randomly chosen. A pair (θ,ϕ) is associated to each of the n selected ranges. In this way, a reference
point with spherical coordinates (d,θ,ϕ) in the three-dimensional space is associated
to each measurement, and a simulated positioning system is obtained from the n selected ranges. θ is the angle between the z-axis and the
reference point, ϕ is the angle between the x-axis and the
projection of the reference point on the x-y plane. The axis origin represents
the unknown position to be estimated using the set of the n corresponding range estimations d^.
The general system model can be written as d¯=A(x¯u)+ϵ¯,
where the underline indicates vector quantities. A(⋅) is a mapping from R3 to Rn, d¯ and ϵ¯∈Rn,
and x¯u=(0,0,0).
From the vector of the estimated distances, an estimate x^¯u of x¯u can be obtained. The performance of the
positioning algorithms is evaluated by calculating ϵx^¯u=E[∥x^¯u∥],
which represents the root mean square error in the position estimation.In traditional least squares, the problem is solved by
searching the minimum of the objective function(9)x^¯uLS=argminx¯u∥d¯−A(x¯u)∥2.If the error model is known, a
weighted and unbiased estimation of the position can be obtained(10)x^¯uWLS=argminx¯u∥d¯−m¯(B,d)−A(x¯u)∥Qdd−12,where Qdd is the variance matrix (a diagonal matrix
containing as the [j,j] element the variance of the jth measured distance). Since both m(B,d) and Qdd depend on the real distance which is unknown,
the estimated distance, instead of the unknown real distance, has been used for
their evaluation in (10).  Six different
scenarios have been simulated: LOS and n = 4, LOS and n = 8, NLOS and n = 4, NLOS and n = 8, n = 4 with 2 measurements in LOS and 2 in NLOS,
and n = 8 with 4 measurements in LOS, and 4 in NLOS.
Table 2 shows the obtained results, averaged over 10 000 repetitions, for the two methods proposed in (9), (10), and
for B=0.5 GHz and B=7.5 GHz, for comparison. Only for the case of LOS
and B=7.5 GHz, there is no improvement using the range
error model knowledge, since the range error in this case is unbiased and it
does not depend on the distance. For all the other scenarios, the range error
model knowledge always allows to improve the final position estimation accuracy
up to about 300% when combining together LOS and NLOS
measurements for B=7.5 GHz.Table 2
Final position estimation accuracy for different
scenarios.
LOSLOSNLOSNLOSLOS/NLOSLOS/NLOSn=4n=8n=4n=8n=4n=8B=7.5 GHzLS0.025 (m)0.017 (m)0.143 (m)0.093 (m)0.167 (m)0.064 (m)WLS0.025 (m)0.017 (m)0.091 (m)0.068 (m)0.054 (m)0.024 (m)B=0.5 GHzLS0.211 (m)0.147 (m)0.322 (m)0.210 (m)0.308 (m)0.180 (m)WLS0.180 (m)0.102 (m)0.235 (m)0.160 (m)0.207 (m)0.121 (m)


## 5. Conclusions

In this letter,
a statistical model for the range error obtained by TOA estimation using UWB
signals has been proposed. It is shown that both a decrease of the bandwidth
and an increase of the distance cause an increase in the mean and standard
deviation of the range error due to the more dense multipath which results in
these situations. The range error is modeled as a Gaussian random variable for
LOS, and as the combination of a Gaussian and an exponential random variable
for NLOS. The more complex characterization in NLOS describes the effects on
the range error of the dense multipath in this propagation condition, and
represents a generalization which includes also the LOS scenario. A possible
application of the model to weighted least squares positioning is finally
investigated. Improvements compared to traditional least squares are especially
evident when there is a significant redundancy or a combination of LOS and NLOS
measurements.



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*Source: 468597-2009-02-25.xml*