Information Technology Reference
In-Depth Information
where
p
γ
is the symbol error probability of the forward channel. Since the return packet
also has an access code of 72 bits, the probability for the event D has the same form of event
A,
(
)
f
72
∑
−
=
T
⎛
⎞
72
[
] [
]
,
k
72
−
k
P
[
D
]
=
⋅
p
(
γ
)
⋅
1
−
p
(
γ
)
⎝
⎠
r
r
k
k
0
where
p
γ is the symbol error probability of the reverse channel. The forward channel is
used to send data packets and the reverse channel indicates the success or not of the
transmission of a packet (for unidirectional transmission). The events B or E occur if any of
the eight triples of the repetition code (3,1) were incorrectly decoded,
(
)
r
{
}
18
P
[
B
]
=
3
p
(
γ
)[
1
−
p
(
γ
)]
2
+
[
−
p
(
γ
)]
3
f
f
f
{
}
18
2
3
P
[
E
]
=
3
p
(
γ
)[
1
−
p
(
γ
)]
+
[
−
p
(
γ
)]
r
r
r
The most probable error is that defined by event C. For DHx, AUX1 and AUX2 packets
it occurs when any of the data bytes were received with error:
[
]
b
P
[
C
]
=
1
−
p
(
γ
)
,
f
where
b
is the size of the payload in bits. For DMx and HAMx packets the data are protected
by a Hamming code, where B is the number of blocks with 10 bits. The probability of event C
for the DMx and HAMx packets is:
[
]
B
14
15
P
[
C
]
=
15
p
(
γ
)[
1
−
p
(
γ
)]
+
[
−
p
(
γ
)]
f
f
f
The BCHx packets contain a (232, 156) binary BCH code that can correct up to
t
=10
errors. Then, for BCHx packets the probability of event C is:
t
[
] [
]
⎛
⎞
232
∑
=
k
232
−
k
P
[
C
]
=
⋅
p
(
γ
)
⋅
1
−
p
(
γ
)
⎝
⎠
f
f
k
k
0
Bluetooth basic rate packets uses GFSK modulation with time-bandwidth product
BT=0.5
and modulation index
i
between 0.28 and 0.35. In this work
i=0.32
, the same value
used in [7, 13]. When the modulation index
i
is less than 0.5, the signal correlation ρ is given
by [10]:
sin
(
2
π
i
)
ρ
=
2
π
i