Digital Signal Processing Reference
In-Depth Information
or simply amplify and forward the original signal. To reduce the effect of error or noise
propagation, the reliability of the signal received at each node can be defined based on
different criteria, such as the received SNR or the probability of error. The mechan-
ics of this system emulate the chanting of
Olé!
in a football stadium, where the signal
first starts from a single source and propagates to the entire stadium as more and more
people follow.
Suppose that there are
N
nodes in the network. Let us start by considering a symbol-
by-symbol relaying scheme described as follows. At the beginning of the network broad-
cast operation, the source node first transmits an
M
-ary symbol represented by one of
the waveforms
p
m
(
t
),
m
∈{1, …,
M
}, which has average energy normalized to 1,
M
1
∑
∫
2
ptdt
m
()
=
1
.
M
m
=
1
In the case where all the intermediate nodes are eventually able to correctly decode the
message, but at different instants in time, the signal received at user
i
can be written as
N
∑
1
yt
()
=
ht
()
⋅ ⋅ − +=
ε
pt
(
τ
( ))
t
nt
()
s
,
+
()
tnt
(),
(11.16)
i
in
,
nm
in
,
i
iim
i
n
=
where
n
i
(
t
) is the additive white Gaussian noise (AWGN) at the
i
t h
node with variance
N
0
, τ
i,n
(
t
) is the time that user
n
transmits plus the propagation delay between the
i
t h
node and the
n
th
node, ε
n
is the energy of the pulse transmitted by user
n
, and
h
i,n
(
t
)
is the channel fading coefficient between user
i
and user
n
. The relaying nodes act as
active scatters that form the multipath signal
s
i,m
(
t
). Interestingly, this signal is unique
for each user
i
due to the different channel gains and propagation delays experienced by
the incoming signals. We refer to this as the
network signature
of user
i
given that the
m
th
symbol was transmitted.
Assume that
h
i,n
(
t
) and τ
i,n
(
t
) are constant over the symbol duration
T
s
. Notice that
T
s
is proportional to the maximum delay spread of the signals
s
i,m
(
t
), for all
i
, which is
denoted by Δτ. Specifically, we have
∫
∞
(
t
−|
)
2
s
()
t
|
2
t
τ
i
im
,
∆=
τ
max
,
(11.17)
−∞
i
∫
∞
2
dt
|
st
()
|
im
,
−∞
where
∫
∫
∞
2
ts
()
t t
im
,
τ
i
=
−∞
(11.18)
∞
2
st
()
dt
im
,
−∞
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