r
2
. The total transmit SNR at the BS is defined as
q ¼
D
P=r
2
, and the transmit SNR at each relay is q=N.In
this paper, term ‘‘capacity’’ means ‘‘channel capacity’’.
In our SC-relaying scheme, the transmission of source
data is completed within two equal-length time slots. In the
first time slot, the BS (s) broadcasts signal x
1
to all the
relays (r) and the user (d), each relay tries to decode the
received signal, and the user only restores the received
signal instead of decoding it. In the second time slot, the
BS broadcasts signal x
2
, the relays which have successfully
decoded signal x
1
in the first time slot forward x
1
to the
user, other relays which have failed in decoding signal x
1
remain silent. The user performs a joint decoding on signal
x
1
and x
2
based on the received signals within the two time
slots.
Next, we use mathematical expressions to explain the
system model. In the first time slot, the BS broadcasts
signal x
1
to all the relays and the user with transmit power
kP (0 k 1). We assume there are L (1 L N) relays
which can successfully decode signal x
1
in the first time
slot, then the received signals at relay i (1 i L) and at
the user are:
y
ri
¼
ffiffiffiffiffiffi
kP
p
h
sri
x
1
þ z
ri
; 1 i L;
ð3Þ
y
d 1ðÞ
¼
ffiffiffiffiffiffi
kP
p
h
sd
x
1
þ z
d 1ðÞ
;
ð4Þ
where z
ri
is the additive white Gaussian noise at relay
i; z
d 1ðÞ
is the additive white Gaussian noise at the user in the
first time slot.
In the second time slot, the BS broadcasts signal x
2
with
transmit power ð1 kÞP, each of the L relays which has
successfully decoded signal x
1
forwards x
1
to the user with
a transmit power of P=N. As we know, the maximum
mutual information for DF relaying transmission is (Please
refer to Eq. (15)in[7])
I
DF
¼
1
2
min log
2
1 þ SNR o
s;r
2
; log
2
1 þ SNR o
s;d
2
n
þSNR o
r; d
2
o
;
ð5Þ
where SNR represents the transmit signal-to-noise ratio,
o
s;d
2
; o
s;r
2
and o
r; d
2
represent the channel gain from
the source to the destination, the channel gain from the
source to the relay, the channel gain from the relay to the
destination in a 3-node system model, respect ively. The
first term in (5) represents the maximum rate at which the
relay can successfully decode the source message, while
the second term in (5) represents the maximum rate at
which the destination can successfully decode the source
message given repeated transmissions from the source and
the relay. Requiring both the relay and the destination to
decode the source message without error results in the
minimum of the two mutual information in (5). Therefore,
assuming that the signals from separate transmitt ers can be
perfectly synchronized at the user, if the channel capacity
between the BS and the L-th relay is greater than the sum
of the channel capacity between the BS and the user, and
the total channe l capacities between the selected L relays
and the user, then the maximum mutual information
between input signal x
1
at the BS and output signal at the
user is restricted by the latter (i.e., the sum of the channel
capacity between the BS and the user, and the total channel
capacities between the selected L relays and the user), then
in the second time slot the received signal at the user is
y
d 2ðÞ
¼
ffiffiffiffi
P
N
r
h
rd1
þþh
rdL
ðÞx
1
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 kðÞP
p
h
sd
x
2
þz
d 2ðÞ
;
ð6Þ
where z
d 2ðÞ
is the additive white Gaussian noise at the user
in the second time slot. The noise power at all the nodes is
r
2
.
So after two time slots, the observations at the user can
be expressed as
y ¼ D
1
x
1
þ D
2
x
2
þ z; ð7Þ
where y ¼ y
d 1ðÞ
;y
d 2ðÞ
T
;D
1
¼
ffiffiffiffiffiffi
kP
p
h
sd
;
ffiffiffi
P
N
q
h
rd1
þþð
h
h
rdL
Þ
i
T
;D
2
¼ 0;
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1kðÞP
p
h
sd
T
;z ¼ z
d 1ðÞ
;z
d 2ðÞ
T
. There-
fore, within two time slots, the average maximum mutual
information between input signal x
1
at the BS and output
signal y at the user is given by
I
1
x
1
;yðÞ¼
1
2
log
2
det I þD
1
D
H
1
K
1
n
¼
1
2
log
2
1 þkq h
sd
jj
2
þ
q
N
h
rd1
jj
2
þþ h
rdL
jj
2
1 þ 1 kðÞq h
sd
jj
2
0
@
1
A
;
ð8Þ
where K
n
¼ diag 1; 1 þ 1 kðÞq h
sd
jj
2
no
.
If the channel capacity between the BS and the L-th
relay is less than the sum of the channel capacity between
the BS and the user, and the total channel capacities
between the selected L relays and the user, then the max-
imal mutu al information between input signal x
1
at the BS
and output signal y at the user is restricted by the former
(i.e., the channel capacity between the BS and the L-th
relay), and is given by
I
2
x
1
; yðÞ¼
1
2
log
2
1 þkq h
srL
jj
2
:
ð9Þ
In summary, within two time slots, the average maximal
mutual information between input signal x
1
at the BS and
output signal y at the user can be expressed as
1804 Wireless Netw (2015) 21:1801–1817
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