Research
Research
Research
Research on
on
on
on Bayesian
Bayesian
Bayesian
Bayesian E
E
E
E stimation
stimation
stimation
stimation of
of
of
of T
T
T
T ime
ime
ime
ime -
-
-
- varying
varying
varying
varying D
D
D
D elay
elay
elay
elay
Meng Wang
1
Ying Liu
2
Ji-wang Zhang
3
1 ,2,3 Department of Electrical Information Engineering, University of Beijing Transportation , Beijing , C hina
( 301074005 @ qq.com )
Abstract -
Time
Time
Time
Time delay
delay
delay
delay estimation
estimation
estimation
estimation is
is
is
is one
one
one
one of
of
of
of key
key
key
key techniques
techniques
techniques
techniques for
for
for
for a
a
a
a rray
rray
rray
rray s
s
s
s ignal
ignal
ignal
ignal p
p
p
p rocessing
rocessing
rocessing
rocessing ,
,
,
, and
and
and
and it
it
it
it has
has
has
has already
already
already
already had
had
had
had several
several
several
several
mature
mature
mature
mature algorithm
algorithm
algorithm
algorithm s.
s.
s.
s. A
A
A
A ccording
ccording
ccording
ccording to
to
to
to its
its
its
its different
different
different
different scenes,
scenes,
scenes,
scenes, time
time
time
time delay
delay
delay
delay estimation
estimation
estimation
estimation can
can
can
can be
be
be
be transferred
transferred
transferred
transferred t
t
t
t o
o
o
o the
the
the
the estimation
estimation
estimation
estimation of
of
of
of
coefficients
coefficients
coefficients
coefficients of
of
of
of a
a
a
a daptive
daptive
daptive
daptive f
f
f
f ilte
ilte
ilte
ilte
r,
r,
r,
r,
which
which
which
which is
is
is
is on
on
on
on the
the
the
the basis
basis
basis
basis of
of
of
of parameter
parameter
parameter
parameter model
model
model
model of
of
of
of a
a
a
a daptive
daptive
daptive
daptive f
f
f
f ilte
ilte
ilte
ilte
r.
r.
r.
r.
The
The
The
The simulation
simulation
simulation
simulation s
s
s
s of
of
of
of
Bayesian
Bayesian
Bayesian
Bayesian methods
methods
methods
methods including
including
including
including Extended
Extended
Extended
Extended Kalman
Kalman
Kalman
Kalman Filter,
Filter,
Filter,
Filter, Unscented
Unscented
Unscented
Unscented Kalman
Kalman
Kalman
Kalman Filter
Filter
Filter
Filter and
and
and
and Bootstrap
Bootstrap
Bootstrap
Bootstrap Particle
Particle
Particle
Particle Filter
Filter
Filter
Filter show
show
show
show
that
that
that
that under
under
under
under Gaussian
Gaussian
Gaussian
Gaussian nonlinear
nonlinear
nonlinear
nonlinear system,
system,
system,
system, EKF
EKF
EKF
EKF and
and
and
and UKF
UKF
UKF
UKF can
can
can
can estimate
estimate
estimate
estimate time-varying
time-varying
time-varying
time-varying delay
delay
delay
delay effectively.
effectively.
effectively.
effectively. B
B
B
B esides,
esides,
esides,
esides,
algorithm
algorithm
algorithm
algorithm s
s
s
s of
of
of
of UKF
UKF
UKF
UKF perform
perform
perform
perform better
better
better
better than
than
than
than that
that
that
that of
of
of
of EKF,
EKF,
EKF,
EKF, which
which
which
which are
are
are
are only
only
only
only subject
subject
subject
subject to
to
to
to Gaussian
Gaussian
Gaussian
Gaussian system.
system.
system.
system. In
In
In
In the
the
the
the nonlinear
nonlinear
nonlinear
nonlinear
non-Gaussian
non-Gaussian
non-Gaussian
non-Gaussian system
system
system
system ,
,
,
, BSPF
BSPF
BSPF
BSPF is
is
is
is able
able
able
able to
to
to
to estimate
estimate
estimate
estimate time
time
time
time delay
delay
delay
delay exactly.
exactly.
exactly.
exactly.
Key words:
T
T
T
T ime
ime
ime
ime delay
delay
delay
delay estimation;
estimation;
estimation;
estimation; Extended
Extended
Extended
Extended Kalman
Kalman
Kalman
Kalman Filter;
Filter;
Filter;
Filter; Unscented
Unscented
Unscented
Unscented Kalman
Kalman
Kalman
Kalman Filter;
Filter;
Filter;
Filter; Bootstrap
Bootstrap
Bootstrap
Bootstrap Particle
Particle
Particle
Particle Filter
Filter
Filter
Filter .
.
.
.
I. Introduction
Time delay , which is resulted by different
transmission distance of signals, refers to the time
difference accepted by different homologous receivers.
T he earliest method of time delay estimation is
Generalized Cross Correlation (GCC) algorithm put
forward by Knapp and Carter in 1976
[1]
.
Traditional methods of time delay estimation like
GCC algorithm and higher order cumulant approach c an
effectively estimate fixed delay under certain
circumstance s
[2]
. N evertheless, Extended Kalman Filter
(EKF) and Unscented Kalman Filter( UKF ) that adopts
Unscented Transformation (UT) are widely used in
location and tracking in nonlinear dynamic system. EKF
achieves fil tering by first-order linearization (Taylor
series expansions), which inevitably results in extra error
and leads to divergence in strong nonlinear system
[3]
.
UKF applies u nscented transformation so as to transfer
mean and covariance nonlinear ly and substitute s Jacobian
matrix of EKF with simple mathematics
[4]
.
UKF algorithm is of high precision , but it can only
be used in the system that noise obeys Gaussian
distribution . A s a sub-optimal estimation algorithm ,
particle filter is commonly applied in nonlinear and
non-Gaussian system s. T he thesis simulates EKF, UKF
and particle filter and analyzes their performance
according to different scenes, producing relatively good
estimation.
Ⅱ . Signal model of time delay estimation
A ssuming that s(t) represents signals from the same
mobile transmitter , then at t, the signals received by two
independent base stations can be formulated a s follows:
( ) ( ) ( )
( ) ( )( ) ( )
⎩
⎨
⎧
+−=
+=
tvttAstr
tvtstr
22
11
τ
(1)
____________________
Grant No. 61172130
I n order to facilitate analysis, the said formula is
simplified . A is amplitude ratio ;
stand s fo r
delayed signal;
denotes time-varying delay; v1(t) and
v2(t) refer to noise interference of tow signals, which is
assumed to be independent Gaussian white noise.
B y using parameter model of a daptive f ilte
r,
the
problem of time delay estimation can be resolved by the
procedure presented in figure 1
[5]
. r
1
(k) and r
2
(k) are
samples of r
1
( t ) and r
2
( t ) . If s ampling time T= Δδ , output
of FIR filter is:
( ) ( )
∑
−=
−=
p
pi
i
ikrkz
1
ξ
∞→
p
(2)
The m inimum quadratic sum of error e(k)=z(k)-r2(k )
can be achieved by adjustment of ξ . I f r
1
(t) and r
2
(t) are
expressed as formula (1), then in according with sampling
theorem ,e xpression is as follows:
( )( )
( )( )
( )( )
ki
ki
AkicA
i
τπ
τπ
τξ
−
−
=−=
sin
sin
(3)
FIR
-
+
Fig .1. B lock diagram for adaptive estimation of two time delays
I n practice, with tolerance for certain truncation error ,
error can be basically ignored , as long as p is larger than
maximum time delay
, for
instance
,the error can be basically
ignored. B y this way, process of estimation becomes less
complicated without consider ing signal s
’
waveform .