Information Technology Reference
InDepth Information
Two constants
a
and
b
can be defined:
(
)
(
)
γ
γ
2
2
a
=
1
−
1
−
ρ
,
b
=
1
+
1
−
ρ
2
2
where γ
is the instantaneous signaltonoise ratio (SNR). The error symbol probability
p
(γ
)
for the GFSK modulation must be applied and is given by [10]:
1
2
2
(
a
+
b
)
/
2
p
(
γ
)
=
Q
(
a
,
b
)
−
e
I
(
ab
)
1
o
2
where
Q
1
(a,b)
is the QMarcum function and
I
o
is the modified Bessel function of first kind.
Thus, the packet error probability of the forward channel,
PER
f
, and reverse,
PER
r
, can be
defined as:
∫
∞
PER
=
1
−
f
(
γ
)
P[
A]
P[
B]
P[
C]
d
γ
f
f
f
0
∫
∞
PER
=
1
−
f
(
γ
)
P[
D]
P[
E]
d
γ
r
r
r
0
where
f(
γ
f
)
and
f(
γ
r
)
are the probability density functions and γ
f
and γ
r
are SNR of the forward
and reverse channels, respectively.
The packet retransmission probability then given by
∞
∞
∫
∫
P
(
γ
,
γ
)
=
1
−
f
(
γ
)
P
[
A
]
P
[
B
]
P
[
C
]
d
γ
⋅
f
(
γ
)
P
[
D
]
P
[
E
]
d
γ
r
f
r
f
f
r
r
0
0
The data rate
R
is given by
K
R
=
,

6
D
⋅
N
⋅
625
⋅
10
where
D
is the number of occupied slots per transmission including the reverse packet,
N
is a
random variable representing the total number of times a particular packet must be
transmitted and
K
is the number of data bits in the packet (DM1:136, DM3: 968, DM5: 1792,
DH1: 216, DH3: 1464, DH5:2712, 2DH1: 432, 2DH3: 2936, 2DH5: 5432, 3DH1: 664, 3
DH3: 4416, 3DH5: 8168).
The average throughput is the expected value of
R
with respect to
N
,
∞
=
{}
[
] [
]
N
−
⋅
1
R
=
E
R
=
1
−
P
(
γ
,
γ
)
⋅
P
(
γ
,
γ
)
R
r
f
r
r
f
r
N
1