Digital Signal Processing Reference
In-Depth Information
where
α
kP
+
−
1
α
2
1
1
(
−
−
kP
)
2
2
R
=
log
1
+
c
s
log
11
+−
(
ρ
)
c
s
DF
1
ds
ds
2
α
2
α
and
R
DF
2
=
C
ub
2
. The achievable rate with CF is [24]
R
≤
max{
R
(
α
, +
kR
)
(
α
, ,
k
)}
(12 .11)
CF
CF
1
CF
2
0
≤≤,≤≤
α
01
k
where
2
c P
1
2
kP
2
rs
s
Rk
(
αα
, =
)
log
1
+
c
s
+
CF
1
ds
2
α ασ
ω
(
1
+
)
and
,
1
2
(
1
1
−
−
kP
)
2
R
(
α
, =−
k
)
(
1
α
)
log
1
+
c
s
CF
2
ds
α
with σ
ω
2
being the WZC noise [13] given by
2
2
α
+ +
c
c
kP
rs
ds
s
2
σ
=
.
ω
1
α
α
−
2
2
2
1
+
cP
1
− +
α
c
(
1
−
kkP
)
1
α
+
c P
dr
r
ds
s
ds
s
DF and CF give the best-known results on the achievable rates for the half-duplex
relay channel (however, a hybrid approach may give a higher rate). Depending on the
parameters, either DF or CF can be superior. Indeed, DF outperforms CF when the
link between the source and relay is better than that between the relay and destination
(e.g., when the relay is close to the source). On the other hand, CF provides higher rates
when the link between the relay and destination is clean (e.g., when the relay is close to
the destination). See
Figure 12.5
. We show in
Figure 12.6
for one setup the rate bounds
in (12.10) and (12.11), achievable with DF and CF, respectively, together with the upper
bound given by (12.9) and the rate bound with multihop transmission, which is given
by the minimum between the capacity at the source-relay link,
2
log(1 + |
c
rs
|
2
P
s
), and the
capacity at the relay-destination link,
2
log(1 + |
c
dr
|
2
P
r
). We plot the rate gain relative
to direct transmission (i.e., no relaying) as a function of |
c
rs
|
2
. The increase in |
c
rs
|
2
can
be construed as the result of decreased distance between the source and the relay. It is
seen from Figure 12.6 that CF outperforms DF for low |
c
rs
|
2
. When |
c
rs
|
2
< |
c
ds
|
2
= 0 dB,
DF is worse than direct transmission. On the other hand, CF always outperforms direct
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