1258 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
Beamforming in Nonorthogonal AmplifyandForwardRelay Networks
Tung T. Pham, Ha H. Nguyen,
Senior Member, IEEE
, andHoang D. Tuan,
Member, IEEE
Abstract
—This paper considers wireless amplifyandforward (AF) relay networks in which the source communicates with the relays anddestination in the ﬁrst phase and the relays
simultaneously
forward signalsto the destination in the second phase over uncorrelated Rayleigh fadingchannels. We examined one scenario in which each relay only knows theperfect information of its source–relay channel, whereas the destinationknows the exact information of the relay–destination channels and thestatistics of the source–relay channels. Based on a combiner that wasdeveloped at the destination, we propose an efﬁcient beamforming schemeat the relays and develop its quantized version using Lloyd’s algorithmto work with a limitedrate feedback channel. Simulation results showthat the nonorthogonal relaying with the proposed beamforming schemeoutperforms the orthogonal relaying with power allocation in terms of the ergodic capacity. In terms of the signaltonoise ratio (SNR), thenonorthogonal scheme also becomes superior to the orthogonal schemewhen the number of quantization regions increases.
Index Terms
—Beamforming, nonorthogonal amplify and forward (AF),power allocation (PA), wireless relay networks.
I. I
NTRODUCTION
In recent years, designing transmission methods for wireless relaynetworks that can adapt to partial knowledge of channelstate information (CSI) has gained a signiﬁcant interest (see, e.g., [1]–[3]). Thisis because the tradeoff between a potential performance improvementoffered by a relayassisted transmission and a large amount of feedback overhead has been well recognized to be an important issue.To reduce the amount of feedback information from the destinationto the relays, a distributed suboptimal beamforming scheme is proposed in [4], whereas in [5] and [6], the relay selection scheme isrecommended to be a good choice. Although the beneﬁts in terms of implementation complexity and overhead information exchange havebeen demonstrated, the common limitation of the methods in [4]–[6]is that the instantaneous CSI of all involved channels in the system isrequired at the destination.In our previous work [7], a wireless amplifyandforward (AF)relay network model in which
orthogonal
transmissions are conductedbetween the relays and the destination in the second phase has beenconsidered.Itisassumedthateveryrelayknowsonlytheinstantaneouschannel from the source to itself, whereas the destination knows theinstantaneous channels from the source and all the relays to itself andthe ﬁrst and secondorder statistics of the channels from the source toalltherelays.Thispracticalassumptionisalsoconsideredin[1].Wheneach relay is assigned an orthogonal channel, interrelay interference
Manuscript received July 8, 2010; revised October 18, 2010 andJanuary 4, 2011; accepted January 5, 2011. Date of publication January 20,2011; date of current version March 21, 2011. This work was supported by aDiscovery Grant from Natural Sciences and Engineering Research Council of Canada. The review of this paper was coordinated by Dr. E. K. S. Au.T. T. Pham and H. H. Nguyen are with the Department of Electrical andComputer Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9,Canada (email: tung.pham@usask.ca, ha.nguyen@usask.ca).H. D. Tuan is with the School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, N.S.W. 2052, Australia(email: h.d.tuan@unsw.edu.au).Color versions of one or more of the ﬁgures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identiﬁer 10.1109/TVT.2011.2107039
is completely eliminated, and the processing task of the destinationbecomes easier. One major drawback of the orthogonal relayingscheme is that it requires more channel uses for each transmission(and, therefore, may not offer high bandwidth efﬁciency) comparedwith the
nonorthogonal
scheme. However, developing an appropriaterelaying scheme for the nonorthogonal case under such a partial CSIassumption has not been adequately investigated.This paper extends the model examined in [7] to the nonorthogonalAF relaying scheme. Using the proposed signal processing approach,we cophase the signal received at each relay to prevent the destructiveeffects that are propagated from the source–relay channels over therelay–destination channels. Under the assumption of uncorrelatedRayleigh fading channels, an
approximate
maximalratio combiner isemployed at the destination, which allows us to derive a beamformingscheme for the relays based on an approximate averaged signaltonoise ratio (SNR). Because the destination has to compute the optimalbeamforming scheme and then send this information back to all relays,vector quantization using Lloyd’s algorithm [8] will be implemented.With this scheme, the destination broadcasts only some bits that represent the index of the best beamforming vector in the codebook througha ﬁniterate feedback channel. A comparison in terms of the reliability(i.e.,theSNR)andbandwidthefﬁciency(withtheergodiccapacity[9])between the orthogonal and nonorthogonal relaying schemes is carriedout. The special case of relay selection is also examined and compared.The rest of this paper is organized as follows. Section II summarizesthe signal processing model proposed in [7], with some modiﬁcationsfor the
nonorthogonal
AF relaying scheme. Section III provides abeamforming scheme and its quantized version and discusses a relayselection scheme. A comparison of different relaying schemes in termsof the average SNR and ergodic capacity is presented in Section IV.Finally, some conclusions are given in Section V.
Notations:
Italic, bold lowercase, and bold uppercase letters denote scalars, vectors, and matrices, respectively. The superscripts
(
·
)
T
,
(
·
)
H
, and
E
{·}
stand for the transpose, Hermitian transpose,and statistical expectation operations, respectively. The notation
x
∼CN
(
µ
,
Σ
)
means that
x
is a vector of complex Gaussian random variables with mean vector
µ
and covariance matrix
Σ
. The notation
I
L
stands for an identity matrix of size
L
×
L
, whereas
diag(
x
1
,...,x
L
)
is a diagonal matrix with the diagonal elements
x
1
,...,x
L
.II. S
YSTEM
M
ODEL
This sectionsummarizes the signalmodel considered in[7,Sec. IV]and explains key differences when the nonorthogonal AF relaying scheme is employed. Fig. 1 shows a wireless relay systemin which the source terminal
S
communicates with the destination terminal
D
through
L
relay terminals
R
1
,...,R
L
. All terminals are equipped with one antenna and operate in a halfduplexmode. The transmission for every information symbol
s
happensin two phases. In the ﬁrst phase, the source transmits the signal to the destination through the direct channel
h
s
∼ CN
(0
,
Σ
(
s
)
)
and to the relays through the “uplink” (source–relay) channels
h
u
= [
h
u
1
,...,h
uL
]
T
∼ CN
(
0
,
Σ
(
u
)
)
, where
h
ul
is the instantaneous channel coefﬁcient between the source and the
l
th relay. In thesecond phase, the relays
simultaneously
forward their received signalsto the destination through the “downlink” (relay–destination) channels
h
d
= [
h
d
1
,...,h
dL
]
T
∼ CN
(
0
,
Σ
(
d
)
)
. All channels are assumed tobe uncorrelated Rayleigh fading (i.e.,
Σ
(
u
)
and
Σ
(
d
)
are diagonal).The signal received at the destination in this phase is
d
1
=
h
s
s
+
n
s
(1)
00189545/$26.00 © 2011 IEEE
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 1259
Fig. 1. System model of the wireless relay network under consideration.
where
n
s
∼ CN
(0
,σ
2
s
)
represents additive white Gaussian noise(AWGN) with variance
σ
2
s
. Similarly, the signals received at the relayscan collectively be written as
r
= [
r
1
,r
2
,...,r
L
]
T
=
h
u
s
+
n
u
(2)where
n
u
∼ CN
(
0
,σ
2
u
I
L
)
accounts for AWGN with the same variance
σ
2
u
at the relays.Assuming that the
l
th relay only knows
h
ul
, whereas the destinationknows
h
s
,
h
d
, and
Σ
(
u
)
, appropriate signal processing operationsneed to be carried out at both the relays and the destination. First,because every relay cannot compute the beamforming scheme, thedestination shall determine it and then send the result back to the relaysthrough some feedback channel.
1
Second, the random phase that wasintroduced by the uplink channel is corrected by setting the
l
th relaygain as
g
l
=
w
l
e
−
jθ
l
P
S

h
ul

2
+
σ
2
u
−
1
/
2
(3)where
P
S
=
E
{
s

2
}
is the average transmitted power of the source,and
θ
l
is the phase of the uplink channel coefﬁcient
h
ul
. With theaforementioned scaling factor, the transmitted power of the
l
th relayis limited to

w
l

2
. The received signal at the destination in the secondphase can be expressed as
2
d
2
=
L
l
=1
g
l
r
l
+
n
d
=
wH
d
F
u
¯
h
u
·
s
+
wH
d
F
u
¯
n
u
+
n
d
(4)where
w
= [
w
1
,...,w
L
]
is the beamforming vector that was usedby the
L
relays,
H
d
= diag(
h
d
1
,...,h
dL
)
,
F
u
= diag((
P
S

h
u
1

2
+
σ
2
u
)
−
1
/
2
,...,
(
P
S

h
uL

2
+
σ
2
u
)
−
1
/
2
)
,
¯
h
u
=[

h
u
1

,...,

h
uL

]
T
,
¯
n
u
=[
n
u
1
e
−
jθ
1
,...,n
uL
e
−
jθ
L
]
T
∼ CN
(
0
,σ
2
u
I
L
)
, and
n
d
∼ CN
(0
,σ
2
d
)
is the AWGN that the destination experienced in the secondphase.The received signals at the destination in two phases can be represented as the following standard Gaussian vector form:
d
= [
d
1
,d
2
]
T
= ¯
a
·
s
+ ¯
n
(5)where
¯
a
= [
h
s
,
wH
d
F
u
¯
h
u
]
T
, and
¯
n
= [
n
s
,
wH
d
F
u
¯
n
u
+
n
d
]
T
.
1
The feedback information may be in the form of analog feedback, i.e., thetrue beamforming vector, or in the form of ﬁniterate feedback, i.e., a quantizedversion of the true beamforming vector.
2
In [7], the destination receives
L
orthogonal signals from
L
relays in thesecond phase. Therefore, a power allocation scheme at the relays is sufﬁcientto help the destination constructively combine those signals. However, in thenonorthogonal case, power allocation alone is not the optimal scheme.
Last, because the instantaneous amplitudes of the uplinks
h
u
arenotknownatthedestination,an
approximate
maximalratiocombining(MRC) ﬁlter that combines the signals received after two phases isgiven as [7]
f
=
R
−
1
E
h
u
{
¯
a
}
(6)where
R
is the covariance matrix of the noise vector
¯
n
. It is computedas follows:
R
=
E
{
¯
n
¯
n
H
}
= diag
σ
2
s
,σ
2
u
wH
d
T
u
H
H d
w
H
+
σ
2
d
(7)where
T
u
= 1
σ
2
u
E
h
u
F
u
¯
n
u
¯
n
H u
F
H u
=
E
h
u
F
u
F
H u
= diag(
T
u
1
,...,T
uL
)
(8)
T
ul
=
E
h
ul
F
2
ul
= 2
ξ
l
σ
2
u
exp(2
ξ
l
)
E
1
(2
ξ
l
)
(9)where
ξ
l
=
σ
2
u
/
(2
P
S
Σ
(
u
)
ll
)
, and
E
1
(
x
) =
+
∞
x
(
e
−
t
/t
)
dt
,
x >
0
isthe exponential integral.Under uncorrelated Rayleigh fading uplink channels, the term
˜
a
=
E
h
u
{
¯
a
}
in (6) is found as
˜
a
=
h
s
,
wH
d
E
h
u
{
F
u
¯
h
u
}
T
= [
h
s
,
wH
d
q
u
]
T
(10)where
q
u
=[
q
u
1
,...,q
uL
]
T
q
ul
=
E
h
ul
{
F
ul

h
ul
}
= 1
√
P
S
ξ
l
exp(
ξ
l
)[
K
1
(
ξ
l
)
−
K
0
(
ξ
l
)]
(11)where
ξ
l
is deﬁned in (9), and
K
α
(
x
)
is the
α
thorder modiﬁed Besselfunction of the second kind.
Lemma 1: With the approximate MRC ﬁlter given in
(6),
the estimate of the data symbol
s
at the ﬁlter
’
s output is
ˆ
s
=
f
H
d
,
and theresulting average SNR can be shown to be
SNR
non
≈
P
S
˜
a
H
R
−
1
˜
a
=
P
S

h
s

2
σ
2
s
+
wH
d
q
u
q
H u
H
H d
w
H
σ
2
u
wH
d
T
u
H
H d
w
H
+
σ
2
d
.
(12)The proof can be similarly carried out as shown in [7] and is omittedhere for brevity. Note that, unlike the orthogonal relaying scheme withpower allocation (PA) in [7], the average SNR that was obtained inthe nonorthogonal relaying scheme has a special form. This form isexploited in the next section to compute an efﬁcient beamformingscheme.III. R
ELAY
B
EAMFORMING
S
CHEMES
A. Proposed Relay Beamforming
Based on the average SNR expression in (12), a general beamforming scheme can numerically be obtained under the total andindividual power constraints at the relays through secondorder coneprogramming [10]. For the case that only the total transmit powerat all the relays is limited, a closedform beamforming scheme canbe derived using the following lemma (the proof directly follows byapplying the Rayleigh–Ritz theorem [11]).
1260 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
Lemma 2: Based on
let
A
be an
n
×
n
Hermitian matrix
and
B
be an
n
×
n
positive deﬁnite Hermitian
.
Furthermore
,
let
B
bedecomposed as
B
=
LL
H
.
Then
x
H
Axx
H
Bx
≤
λ
max
for all
x
∈
C
n
(13)
where
λ
max
is the largest eigenvalue of
L
−
1
A
(
L
H
)
−
1
.
The equalityholds if
x
=
c
(
L
H
)
−
1
u
max
,
where
c
is any nonzero constant
,
and
u
max
is the eigenvector of
L
−
1
A
(
L
H
)
−
1
that corresponds to
λ
max
.With the total relay power constraint
ww
H
=
P
R
, the average SNRcan be rewritten as
SNR
non
≈
P
S

h
s

2
σ
2
s
+
wH
d
q
u
q
H u
H
H d
w
H
w
σ
2
u
H
d
T
u
H
H d
+
σ
2
d
P
R
I
L
w
H
.
(14)Because the second term in (14) is exactly the special case (withequality) that was considered in Lemma 2, we can readily ﬁnd thefollowing beamforming vector
3
to maximize the average SNR, i.e.,
SNR
non
:
w
opt
=
c
˜
w
=
c
q
H u
H
H d
V
(15)where
V
= (
σ
2
u
H
d
T
u
H
H d
+ (
σ
2
d
/P
R
)
I
L
)
−
1
, and
c
=
P
R
/
(˜
w
˜
w
H
)
. It follows that
SNR
non
≈
P
S
σ
2
s

h
s

2
+
L
l
=1
P
S
σ
2
u
q
2
ulP
R
σ
2
d

h
dl

2
T
ulP
R
σ
2
d

h
dl

2
+
1
σ
2
u
.
(16)With the average SNR given in (16), one interesting SNR comparison between the nonorthogonal and orthogonal relaying schemes isstated in the following corollary.
Corollary 1: With the analog feedback
,
the average SNR that wasachieved by the nonorthogonal relaying transmission with the optimalbeamforming scheme is always higher than the average SNR that wasachieved by the orthogonal relaying transmission with optimal PA.
The proof of Corollary 1 is given in Appendix A. Note that theaforementioned SNR comparison is also true for the case with perfectand full CSI assumption, in which the instantaneous SNRs of bothschemes have the same form as given in [1, eq. (10)]. For the systemmodel with a ﬁniterate feedback, further analysis and comparison arecarried out in Section IV.Given the beamforming scheme (15), the ergodic capacity of thetransmission model under consideration can be calculated as
I
= 12
E
h
s
,
h
u
,
h
d
log
2
1 +
S N
(17)where the received signal power is
S
=
P
S

h
s

2
σ
2
s
+
q
H u
H
H d
VH
d
F
u
¯
h
u
2
(18)and the averaged noise power is
N
=

h
s

2
σ
2
s
+
q
H u
H
H d
V
2
×
σ
2
u
H
d
F
u
F
H u
H
H d
+
σ
2
d
P
R
I
L
H
d
q
u
.
(19)
3
Note that a beamforming scheme usually includes the following two components: 1) the phase compensation and 2) the power allocation (see, e.g., [1],[10], and [12]). The beamforming scheme (15) is also in this form.
Obtaining a closedform expression of the ergodic capacity asdeﬁned in (17) appears to be very difﬁcult. Nevertheless, a tight upperbound of the ergodic capacity can be found as follows. Because thefunction
log
2
(1 +
x
)
is concave in
x >
0
, applying Jensen’s inequality to (17) yields
I ≤
12 log
2
1 +
E
h
s
,
h
u
,
h
d
S N
≈
12 log
2
(1 +
SNR)
(20)where
SNR =
E
h
s
,
h
d
{
SNR
non
}≈
P
S
σ
2
s
Σ
(
s
)
+
L
l
=1
P
S
q
2
ul
σ
2
u
T
ul
[1
−
χ
l
exp(
χ
l
)
E
1
(
χ
l
)]
(21)and
χ
l
=
σ
2
d
/
(
σ
2
u
P
R
T
ul
Σ
(
d
)
ll
)
. The simulation results in Section IVwill validate the tightness of the aforementioned upper bound.
B. Quantized Relay Beamforming
The beamforming scheme in (15) is computed at the destinationand is sent back to the relays through some feedback channel. Forpractical implementation, we sketch in this section the vector quantization procedure based on Lloyd’s algorithm to design the quantizedrelay beamforming codebook. This procedure can also be appliedfor the PA codebook design in the orthogonal relaying scheme (seeAppendix B). Let
¯
γ
(
w

h
d
) =
wH
d
q
u
q
H u
H
H d
w
H
w
σ
2
u
H
d
T
u
H
H d
+
σ
2
d
P
R
I
L
w
H
.
(22)Given a codebook of beamforming vectors
W
=
{
w
1
,
w
2
,...,
w
Z
,
} ∈
C
L
(23)where
Z
is the number of quantization regions, and
⌈
log
2
Z
⌉
, in which
⌈
x
⌉
is the smallest integer that is greater than or equal to
x
, is thenumber of feedback bits, the average distortion with respect to theoptimal beamforming vector can be deﬁned as
δ
(
W
) =
E
h
d
¯
γ
(
w
opt

h
d
)
−
max
1
≤
z
≤
Z
{
¯
γ
(
w
z

h
d
)
}
.
(24)The codebook design using Lloyd’s algorithm vector quantizationcan be summarized as follows [8].1) Randomly generate a codebook
W
(0)
=
{
w
(0)1
,
w
(0)2
,...,
w
(0)
Z
,
}
. Set
t
= 1
.2) Generate a set of
N
test channel realization vectors
{
h
(1)
d
,
h
(2)
d
,...,
h
(
N
)
d
}
. Divide this set into
Z
quantization regions, with the
z
th region deﬁned as
C
z
=
h
(
n
)
d

¯
γ
w
(
t
−
1)
i

h
(
n
)
d
≤
¯
γ
w
(
t
−
1)
z

h
(
n
)
d
(25)for all
i
=
z,
1
≤
n
≤
N
.3) Construct a new codebook
W
(
t
)
, with the
z
th codewordgiven as
w
(
t
)
z
= argmax
w
E
h
(
n
)
d
∈C
z
¯
γ
w

h
(
n
)
d
(26)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 1261
where
E
h
(
n
)
d
∈C
z
¯
γ
w

h
(
n
)
d
=
E
h
(
n
)
d
∈C
z
wH
(
n
)
d
q
u
q
H u
H
(
n
)
d
H
w
H
w
σ
2
u
H
(
n
)
d
T
u
H
(
n
)
d
H
+
σ
2
d
P
R
I
L
w
H
≈
w
E
h
(
n
)
d
∈C
z
H
(
n
)
d
q
u
q
H u
H
(
n
)
d
H
w
H
w
σ
2
u
E
h
(
n
)
d
∈C
z
H
(
n
)
d
T
u
H
(
n
)
d
H
+
σ
2
d
P
R
I
L
w
H
and
E
h
(
n
)
d
∈C
z
{
¯
γ
(
w

h
(
n
)
d
)
}
is approximated by the ﬁrst term of its Taylor series [1].
4
4) If
δ
(
W
(
t
−
1)
)
−
δ
(
W
(
t
)
)
>ǫ
, set
t
←
t
+1
, and go back to step 2.Otherwise, set
W
=
W
(
t
)
and stop.Depending on the initial codebook in step 1 and the test set of channel vectors that were generated in step 2, Lloyd’s iterative procedure can result in local maxima. Therefore, to obtain a more reliablesolution, a large test set is preferred. Although the ofﬂine designof the codebook appears to be computationally intensive, codewordselection is really simple. Because the destination and all relays knowthe codebook, the destination only needs to broadcast the index of theoptimal codeword to all the relays. This approach signiﬁcantly reducesthe number of feedback bits compared with the optimal beamformingscheme with
analog
feedback.
C. Relay Selection
Relay selection can be considered a special scheme of both beamforming and PA. When full CSI is available at the destination, relayselection has been proven an efﬁcient scheme with a simple feedback strategy [13]. In our scenario, the selection algorithm can only beimplementedbasedontheinstantaneousdownlinkchannelcoefﬁcientsand the statistics of the uplink channels. Based on (14), it can beinferred that the
l
⋆
th relay is selected to forward the signal in thesecond phase if
l
⋆
= argmax
l
∈
[1
,L
]
P
R

h
dl

2
q
2
ul
σ
2
u
P
R

h
dl

2
T
ul
+
σ
2
d
(27)and the feedback channel needs
⌈
log
2
L
⌉
bits to inform all the relaysthrough a broadcast channel.IV. N
UMERICAL
R
ESULTS AND
D
ISCUSSION
All the uplink and downlink channels are assumed to be underuncorrelated Rayleigh fading but with different variances, i.e.,
Σ
(
u
)11
=
−
10
dB,
Σ
(
u
)22
= 0
dB,
Σ
(
u
)33
= 10
dB,
Σ
(
u
)44
= 20
dB,
Σ
(
d
)11
= 20
dB,
Σ
(
d
)22
= 10
dB,
Σ
(
d
)33
= 0
dB, and
Σ
(
d
)44
=
−
10
dB. Similarly, the directchannel is assumed to be
h
s
∼ CN
(0
,
1)
. Binary phaseshift keyingmodulation is used at the source. Assuming that
σ
2
s
=
σ
2
u
=
σ
2
d
= 1
,
4
Note that, if
C
z

= 0
, then
w
is randomly regenerated. If
C
z

= 1
, usethe special case in Lemma 2 to ﬁnd the closedform solution; otherwise, usethe general solution that was given in Lemma 2. In addition, using a higherorder Taylorseries approximation for
E
h
(
n
)
d
∈C
z
{
¯
γ
(
w

h
(
n
)
d
)
}
may be moredesirable but appears to be complicated.Fig. 2. SER of the nonorthogonal relaying model with the optimal beamforming scheme and the orthogonal relaying scheme with the power allocationscheme in [7]. The number of relays is given as
L
= 2
,
3
,
4
.
the average channel signaltonoise ratio (CSNR) is simply deﬁnedas
P
S
/σ
2
s
=
P
S
/σ
2
u
=
P
S
/σ
2
d
=
P
S
. The total transmit power at therelays is chosen to be
P
R
=
P
S
.Fig. 2 compares the symbol error rate (SER) performances achievedwith the nonorthogonal relaying with optimal beamforming schemeand the orthogonal relaying with PA scheme in [7]. The numberof relays is given as
L
= 2
,
3
,
4
. It is shown in the ﬁgure thatthe performance improvement (from the “
equivalent
” coding gain)with nonorthogonal relaying is signiﬁcant: about 3 dB/relay at highSNR. However, because the destination does not know full CSI, thediversity order does not increase with the number of relays. Thiscondition also agrees with the results of the orthogonal case in [7]and of the nonorthogonal case in [1], where the diversity order isproven to be only 2. As a reference, the SER that was obtained bydirect transmission with a total transmit power of
P
=
P
S
+
P
R
isalso shown in Fig. 2. Even with such a large transmit power, theperformance of the direct transmission is signiﬁcantly worse than therelayassisted transmission. Note that, in general, the performanceimprovement of the relayassisted transmission strongly depends onthe channel conditions among the source, relays, and the destination.The performance curves shown in Fig. 2 should be interpreted for thespeciﬁc channel conditions under consideration.Next, we compare the nonorthogonal relaying scheme under consideration with the orthogonal relaying scheme considered in [7]under the same CSI assumption. A broadcasting feedback channelthat accommodates 2 or 6 feedback bits is assumed. To adapt withthese limited feedback rates, the quantized version of the beamformingvector (given in Section IIIB) and of the PA scheme used in [7] (givenin Appendix B) are employed. In the procedure, the number of testchannel vectors is
N
=
20000, and the termination criterion is chosenas
ǫ
= 10
−
4
δ
(
W
(
t
−
1)
)
.As predicted from Corollary 1, the simulation results in Fig. 3conﬁrm that, when inﬁniterate (i.e., analog) feedback is available, thebeamforming scheme in the nonorthogonal case is superior in termsof the average SNR compared with the PA scheme in the orthogonalcase. This is because the phases of both uplink and downlink channelscan perfectly be matched at each relay, resulting in
a constructivesuperposition
(
basically the addition of signal amplitudes
)
of all thecophased signals at the destination
. Consequently, the energy of thecombined signal is higher than the orthogonal case, where
energies of
1262 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
Fig. 3. Average SNRs of the nonorthogonal relaying scheme with beamforming and the orthogonal relaying scheme with the power allocation scheme in[7]. The number of relays is given as
L
= 4
.Fig. 4. Ergodic capacity of the nonorthogonal relaying scheme with beamforming and the orthogonal relaying scheme with the power allocation schemein [7]. The number of relays is given as
L
= 4
.
different signals, and not their amplitudes, are accumulated
. When nofeedback is available, it is expected that the nonorthogonal relayingscheme is inferior to the orthogonal scheme. For the case with only2 feedback bits, the performance of the orthogonal scheme is stillbetter. This condition also reﬂects the sensitivity of the nonorthogonalrelaying scheme to the quality of the phase estimates available atthe relays. However, the gap is very small, and it can practically beremoved and even reversed with more feedback bits (e.g., with 6feedback bits or more in our simulation setup). In addition, note that,under the assumed channel model, the performance of the PA schemedoes not improve much as the number of feedback bits increases.Because the average SNRs of the two relaying schemes are notmuch different, their bandwidth efﬁciencies mostly depend on thenumber of channel uses (which is also the number of relays) requiredin the second phase. As shown in Fig. 4, the ergodic capacity of thenonorthogonal scheme is about 2.5 times larger than the ergodic capacity of the orthogonal scheme at any SNR region (i.e., approximately
(
L
+ 1)
/
2
times for the cases with 2 or 6 feedback bits). Withoutfeedback information, the nonorthogonal scheme performs a little bitworse at low and mediumSNR regions. For benchmark comparison,the capacity of the model in [1] under the assumption of full CSIavailable at the destination is also plotted. As expected, a decrease incapacity can clearly be observed when less CSI is available at both thetransmitters and receiver.Figs. 3 and 4 also plot curves (marked with ﬁlled circles) to validatethe approximate
SNR
given in (21) and the upper bound of the ergodiccapacity given in (20), respectively. It is observed that both
SNR
andthe upper bound of the ergodic capacity are very close to the actualvalues. Other performance curves in Figs. 3 and 4 also show thatthe relay selection scheme in (27) does not perform as well as thebeamforming scheme with the 2bit feedback channel. However, it issuperior to the PA scheme (even with analog feedback case). Thus,relay selection is a suitable solution when we need to balance betweenthe implementation complexity and performance gain.V. C
ONCLUSION
Extending our previously proposed orthogonal relaying model, theproblem of joint design of signal processing at the relays and destination has been considered for wireless
nonorthogonal
relay networks.With a highly accurate approximation of the SNR, a closedformbeamforming scheme for the relays has been proposed. Although nodiversity gain can be obtained as the number of relays increases, theerrorperformanceofthesystemcanstillbesigniﬁcantlyimproved.Fora practical implementation with limited feedback from the destination,the quantized beamforming scheme is also developed. Simulationresults have shown that the proposed nonorthogonal relaying withbeamforming scheme can outperform the orthogonal relaying with PAin terms of the SNR and, consequently, the error performance, as wellas the bandwidth efﬁciency.A
PPENDIX
AP
ROOF OF
C
OROLLARY
1Using the same notations in [7], let
W
= diag(
w
1
,w
2
,...,w
L
)
,
˜
a
= [
h
s
,
(
WH
d
q
u
)
T
]
T
, and
R
= diag(
σ
2
s
,σ
2
u
WH
d
T
u
H
H d
W
H
+
σ
2
d
I
L
)
. The average SNR for the orthogonal relaying transmissiongiven in [7, eq. (41)] can be rewritten as
SNR
ortho
≈
P
S
˜
a
H
R
−
1
˜
a
=
P
S

h
s

2
σ
2
s
+
q
H u
H
H d
W
H
×
σ
2
u
WH
d
T
u
H
H d
W
H
+
σ
2
d
I
L
−
1
WH
d
q
u
=
P
S
σ
2
s

h
s

2
+
L
l
=1
P
S
σ
2
u
q
2
ul

w
l

2
σ
2
d

h
dl

2
T
ul

w
l

2
σ
2
d

h
dl

2
+
1
σ
2
u
.
(28)For the nonorthogonal relaying transmission with our proposed beamforming scheme, the average SNR is given in (16). Both
SNR
non
given in (16) and
SNR
ortho
given in (28) have the same form,which are increasing functions of
P
R
and

w
l

2
, respectively. Because
Ll
=1

w
l

2
=
P
R
, i.e.,

w
l

2
≤
P
R
∀
l
, it is obvious that substituting

w
l

2
in (28) by
P
R
leads to
SNR
ortho
≤
SNR
non
. The equality holdswhen
L
= 1
.