Information Technology Reference
In-Depth Information
7.3
Beyond the UDG Model
A simple extension to the UDG is to replace the Heaviside step function behavior
of the probability of radio packet reception by a continuous spatial distribution. The
straightforward method for achieving this is as follows.
The probability for receiving correctly a packet of length
bit for which at most
' bits can be forward error-corrected, can be expressed in terms of the bit-error
ratio,
p
e
, as
¢
æö
λ
-
λ
ppr
=
p
(1
-
p
)
ç
èø
e
e
λ
λ
=
0
For this work, we assume packets of 128 bits length and use a slightly more
complex (7,3) Reed-Muller block coding [
10
].
The bit-error ratio,
p
e
, can be determined by the received signal-to-noise ratio
and the modulation scheme employed. For example, for quadrature phase shift keying
(QPSK), the bit-error ratio is given by [
10
],
æ
ö
æ
ö
E
1
E
2
p
=
erfc
ç
b
÷
-
erfc
ç
b
÷
ç
÷
ç
÷
e
N
4
N
è
ø
è
ø
0
0
¥
= -
ò
is the complementary error function,
N
o
is the
receiver (one-sided) noise spectral density and
E
is the received energy per bit.
Inverting the above formula (only possible numerically), yields the signal-
to-noise threshold
2
where
2
erfc( )
z
exp{
t dt
}
z
b
EN
corresponding to a given value of
p
, which is used in
computing the outage probability integral [
11
] which uses the path-loss distribution
and the probability density function for
x
σ
. The computation of such outage
integrals is beyond the scope of this chapter and the reader is referred to standard
textbooks in the literature [
12
].
As a consequence, the probability of packet reception can be plotted as a
function of the separation of the transmitting and receiving nodes, d, by integrating
over all possible values of
x
σ
(weighted by its probability density function in the
integrand) to yield a symmetric coverage model as shown in Fig.
7.1
.
In the trivial case when
/
0
σ
, there is no shadow fading and the probability of
packet reception can be computed in closed analytic form. We plot this in Fig.
7.1
for the Konstantinou [
4
] propagation model, for
0
dB
h
==
,
f
=
2,100 MHz
,
1.5 m
tx
rx
G
==
and a dense urban environment with the published transition
model [
4
] for LOS to non-LOS path-loss ratio, as shown in Fig.
7.2
. The propagation
model used to plot Fig.
7.2
is summarized here for ease of reference,
2.16 dBi
tx
rx
Ld
( )
=
4.62 20 log
+
4
π
λ
-
2.24
h
-
4.9
h
+
29.6
log
d
LOS
10
t
r
10
Ld
( )
=
20 log
4
π
λ
-+
2
h
40lg
o
d
NLOS
10
10
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