distances from the BSs; in other words, following our
notation, such measurements have to be considered sam-
ples of P([d
i
- d
i
*
] ? w
los
) instead of the conditional
probability P([d
i
- d
i
*
] ? w
los
|d
i
*
), through which an
observation equation is defined (as a consequence, if a
characterization of the pdf of the multipath error were
pursued on the basis of this data, it would be implicitly
assumed that the multipath lag is independent on the LOS
distance).
We also argue that in order to achieve a more realistic
model of NLOS errors, the autocorrelation of the signal
w
nlos
ðÞ should be taken into account;
1
this will not be
addressed here.
In this work, we refer to a IS-95 CDMA network, where
all BSs use the same carrier frequency for the pilot channel,
and are synchronized to a common CDMA system time,
which is derived from a precise clock reference supplied by
GPS. TDOA (absolute) values can be estimated using a
well known cross-correlation based technique, by looking
for a peak in the PN code acquisition matrix [7, 27]. In this
case TDOA readings are constituted by the terms |d
i
- d
j
|,
assuming no prior information available about which
multipath distance is shorter, corrupted by system mea-
surement noise, that is:
z
ij
¼ z
ji
¼jd
i
d
j
jþw
ij
: ð4Þ
In case of n - 1 TDOA measurements, n BS, with BS
1
the common reference, denote the readings by Z
1
¼
fz
21
; ...; z
n1
g, the sight condition by s ¼½s
1
; ...; s
n
2f0; 1g
n
, define by F
n-1
(s) the pdf of P(Z
1
|p
*
,s) when
the various w
ij
are neglected. For a single measurement,
hence 2 BS:
F
1
ðsÞ¼
Z
1
0
f
1;s
1
ðyÞf
2;s
2
ðy þz
21
Þdy þ
Z
1
0
f
1;s
1
ðy
þ z
21
Þf
2;s
2
ðyÞdy; ð5Þ
where the two integrals arise considering the sign of d
1
- d
2
.
For multiple readings, by the correlation existing among
the measurements due to the common reference base,
obviously PðZ
1
jp
Þ 6¼
Q
i
Pðz
i1
jp
Þ (see e.g., [9]), and a
computation similar to (5) should be followed, in this case
accounting for the 2
n-1
combinations of signs of the
absolute values, thus computing 2
n-1
integrals. For n = 3
we have (omitting the reference to the s
i
):
F
2
ðsÞ¼
Z
1
0
f
1
ðyÞf
2
ðy þz
21
Þf
3
ðy þz
31
Þdy
þ
Z
1
0
f
1
ðy þz
31
Þf
2
ðy þ z
21
þ z
31
Þf
3
ðyÞdy
þ
Z
1
0
f
1
ðy þz
21
Þf
2
ðyÞf
3
ðy þz
21
þ z
31
Þdy
þ
Z
1
0
f
1
ðy þz
21
Þf
2
ðyÞf
3
ðy þz
21
z
31
Þdy:
ð6Þ
The four terms in the above equation are relative to the fol-
lowing combinations of the sign of d
2
- d
1
and d
3
d
1
:
hþ; þi; hþ; i; h; þi; h; i (hence, in particular, the
first integral is relative to the case |d
2
- d
1
| = d
2
- d
1
and |d
3
- d
1
| = d
3
- d
1
; notice that the fourth term
could be equivalently expressed as
R
1
0
f
1
ðy þz
31
Þf
2
ðyþ
z
31
z
21
Þf
3
ðyÞdy).
The pdf’s F
n-1
can be particularized according to any
standing assumption regarding NLOS error modeling, and
also considering all the possible (2
n
) sight conditions. This
would lead to the computation of 2
n-1
2
n
integrals. Now,
considering also system measurement noise, i.e., the vari-
ous terms w
ij
, we have that P(Z
1
|p*, s) results from the
convolution of F
n-1
(s) with the joint distribution of the w
ij
(for which we may have, e.g., a multivariate Gaussian
distribution with zero mean and diagonal covariance
matrix, independently by state s). To proceed further,
several working assumptions can be made to tackle the
complexity of this model, both due to the exponential
complexity of the combination of the signs, and due to the
need to perform the convolution between F
n-1
and the pdf
characterizing the w
ij
.
2.2 Approximated measurement model
First, if one considers only position estimation, letting
LOS/NLOS detection, as in this work, marginalization over
the variable s is not mandatory, and one could adopt a
measurement model relative to the worst case scenario,
namely s
i
= 1, V i (denoted s = 1). In this case, the pre-
dominant contribution in P(Z
1
|p*, 1), in terms of variance,
is that relative to the terms |d
i
- d
j
|; thus, neglecting w
ij
we
stand with a slightly less accurate but simpler measurement
model, P(Z
1
|p*, 1) % F
n-1
(1), (see the comparison for a
single TDOA reading in Fig. 2). It is notable, however, that
considering a gaussian additive term w
ij
to |d
i
- d
j
| could
be not satisfactory as this makes non zero the probability to
get negative readings, which is not realistic, in our context.
1
For example, it is reasonable to expect that at one given position the
multipath distance of the sensed signal is likely to remain approx-
imately the same, and in this case a MS which, for instance,
temporarily remains still would take measures affected by similar
NLOS errors—see also the considerations made in [6, 15].
Wireless Netw (2012) 18:579–589 581
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