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where the pulse-energy-to-noise ratio is repre-
sented by
E
N
Generally, the CRLB provides a loose bound
on the TOA estimate which is not realizable in
multipath environments (Dardari, Chong et al.
2006). Another bound that provides more accurate
results, and is suitable for multipath environments
is the Ziv-Zakai lower bound (ZZLB). The ZZLB
for the coherent detection of binary signaling is
as given by (Dardari, Chong et al. 2006):
p
, and
β
is the second moment of
0
the spectrum
P(f)
of the pulse shape used
p(t)
defined by (Dardari, Chong et al. 2006):
∞
∫
f P f
2
2
( )
df
β
2
=
−∞
(5)
E
p
T
a
=
1
(
)
∫
Z
ZLB
z T
−
z P z dz
( )
(9)
a
imin(
T
a
0
Assuming a Gaussian pulse defined in terms
of the pulse width
T
p
and
τ
p
= 0.5 *
as
T
where,
P
min(s)
is the minimum attainable probability
of error expressed as (Sangyoub 2002):
(Dardari, Chong et al. 2006):
(
)
−
(
)
p t
( ) =
exp
2
t
2
/
2
π
τ
(6)
0
p
E
N
p
P z
imin(
( ) =
Q
(1
−
ρ
( ))
z
(10)
pp
0
The
n
-th order Gaussian pulse has the form
(Sangyoub 2002):
and the pulse autocorrelation
ρ
pp
( )
normalized
by the pulse-energy
E
p
is (Dardari, Chong et al.
2006):
2
2
t
−
2
π
( )
n
d
dt
τ
p
p t
( ) =
e
(7)
n
n
∞
∫
( ) =
1
ρ
pp
z
p t p t
( ) (
−
z dt
)
(11)
E
p
−∞
The mean square estimation error (MSE) is as
(Dardari, Chong et al. 2006):
This bound transforms the estimation problem
into a binary detection problem, which simplifies
the bound estimation in multipath environments.
The derivation of
P
imin(
(z)
depends on the receiver
a-
priori
knowledge about the multipath phenomena
(Dardari, Chong et al. 2006). However, the evalu-
ation of the estimator in complex channel models
is not analytically tractable (Dardari, Chong et al.
2006). As a result, the ZZLB is typically evaluated
using experimentally measured channel impulse
responses or Monte-Carlo simulations (Dardari,
Chong et al. 2006):
∞
∫
=
1
2
z
{ }
2
E
ε
zP
ε
≥
dz
(8)
τ
ˆ
τ
ˆ
2
0
where the expectation is with respect to
τ
, and
P
τ
( )
is the probability density function (pdf) of
the TOA in the absence of any information is as-
sumed to be uniformly distributed in the interval
z
[0,
T
a
].
P
ε
ˆ
≥
is equivalent to the probabil-
2
ity of a binary detection scheme with equally-
probable hypothesis, where
T
a
is the observation
window (Dardari, Chong et al. 2006).
N
ch
1
SNR
d
∑
2
P z
imin(
( )
≅
Q
( )
z
(12)
k i
, (*)
N
2
ch
k
=1
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