Image Processing Reference
In-Depth Information
Due to the constant signal speed in homogenous media, the difference of both distances ∆
d
=
d
−
d
is proportional to the time difference ∆
t
of the incoming signals.
∆
d
=
∆
t
⋅
c
=
d
−
d
(.)
Dividing
d
−
d
(Equation .) by ∆
d
results in the hyperbolic function of the TDoA method.
√
(
√
(
x
−
x
)
+(
y
−
y
)
x
−
x
)
+(
y
−
y
)
=
−
(.)
∆
d
∆
d
According to Equation ., a sensor node is located on one of the arms of the hyperbola with the
foci
B
and
B
.hecorrectarmdependsonthetimestampoftheincomingsignal.Asshownin
Figure .a, the first incoming signal of sender
B
implies that the distance
B
S
is shorter than
B
S
(
d
.
To detect the correct position among the hyperbola, another pair of beacons must be evaluated to
define a second hyperbola
h
<
d
). herefore,
S
can only be located on
h
(
B
;
B
)
)
defines the position of
S
(Figure .b). In some cases, both hyperbolas cut twice and deliver two
possible positions. hen, a third hyperbola must be validated to prevent the ambiguities.
(
B
;
B
)
. The interception of both hyperbolas,
h
(
B
;
B
)
and
h
(
B
;
B
6.2.3 Round Trip Time
The distance estimation based on the introduced ToF methods requires highly synchronized senders
or receivers and very precise clocks. herefore, the hardware effort increases compared to other tech-
niques. hus, ToF methods are mostly contradictory to the design objective of developing small and
resource-aware sensor networks.
A method to prevent synchronization is used by measuring the round trip time (RTT). At the
beginning, a sensor node
S
transmits a message to a target node
S
(Figure .a). his message con-
tains the time stamp at the beginning of the transmission. he target replies to the received message
immediately with an acknowledgment (ACK) after a reaction time
t
R
.
After reception of the ACK at
S
, the time difference of the round trip is equal to ∆
t
=
∆
t
F
+
∆
t
F
∆
t
F
represents the ToF between sender and receiver. Additionally, ∆
t
contains the reaction time of
S
and the time ∆
t
CS
i
representing the clock skew as well as the clock
drift. Disregarding the usual unknown ∆
t
CS
i
,adistancebetween
S
and
S
can be estimated with
Equation ..
+
t
R
,where∆
t
F
+
∆
t
−
∆
t
R
d
=
c
⋅
(.)
1. Measurement
2. Measurement
Clock
1
Clock
1
S
2
S
1
Clock
2
Clock
2
S
2
S
2
Δ
t
F
12
Δ
t
R
2
Δ
t
F
21
Δ
t
F
12
Δ
t
R
2
Δ
t
F
21
Δ
t
F
21
Δ
t
R
1
Δ
t
F
12
t
t
Δ
t
CS
1
Δ
t
CS
2
Δ
t
CS
1
Δ
t
CS
2
Δ
t
CS
3
Δ
t
CS
4
(a)
(b)
FIGURE .
(a) Round trip time RTT and (b) symmetric double-sided two-way ranging SDS-TWR.
