It is widely observed that in practical wireless cooperative communication systems, different links may experience different fading characteristics. In this paper, we investigate into the outage probability and channel capacity of twoway amplifyandforward (TWAF) relaying systems operating over a mixed asymmetric Rician and Rayleigh fading scenario, with different amplification policies (AP) adopted at the relay, respectively. As TWAF relay network carries concurrent traffics towards two opposite directions, both endtoend and overall performance metrics were considered. In detail, both uniform exact expressions and simplified asymptotic expressions for the endtoend outage probability (OP) were presented, based on which the system overall OP was studied under the condition of the two source nodes having nonidentical traffic requirements. Furthermore, exact expressions for tight lower bounds as well as high SNR approximations of channel capacity of the considered scenario were presented. For both OP and channel capacity, with different APs, effective power allocation (PA) schemes under different constraints were given to optimize the system performance. Extensive simulations were carried out to verify the analytical results and to demonstrate the impact of channel asymmetry on the system performance.
1. Introduction
R
ecently, cooperative relay networks have drawn much attention from research community as it provides simple solutions to extend radio coverage and to improve link quality. Of particular interest is the twoway (TW) relay network, compared with conventional oneway (OW) relay network, it brings considerable spectral efficiency improvement as completing one round of bidirectional information exchange in two time slots
[1]
between two halfduplex terminals. Different transmission protocols employing different signal processing technique can be utilized in cooperative relay networks such as amplifyandforward (AF), decodeandforward (DF), compressandforward (CF) and so on, among which the AF protocol is a popular concern because it is lowcost and easy to implement while providing satisfactory performance. The relay node in an AF cooperative network may adopt different amplification policies (AP), such as variable gain (VG) policy which requires instantaneous channel state information (CSI) of two incoming links, fixed gain (FG) policy where only statistical channel distribution information (CDI) is needed, and mixed policies where full CSI of one link and CDI of the other link is needed at the relay
[2]
.
During the last few years, much effort has been devoted to evaluate the performance of dual hop relay networks. The authors in
[3]
investigated into the symbol error probability (SER) for higher order modulation schemes in TW relay network under Rayleigh fading channels when physical layer network coding (PLNC) technique is adopted at the relay. For twoway amplifyandforward (TWAF) relay network with VG policy under Rayleigh fading channels, when two source terminals have nonidentical traffic requirements, the overall system outage probability (OP) was analyzed and then optimized subject to different constraints in
[4]
[5]
. By applying a geometric method, the OP of a TWAF VG relay network under Rician fading channels was presented in
[6]
. As for Nakagami
m
fading channels, endtoend performance including OP, channel capacity and SER was analyzed in
[7]
[8]
for relay networks with VG and FG polices, respectively. Due to mathematical complexity, the topic of overall system OP in TWAF FG networks was relatively less addressed upon. Recently, Ni etc.
[9]
presented exact expressions and high signaltonoise ratio (SNR) approximations of overall OP in TWAF FG relay networks with asymmetric traffic requirements under Rayleigh fading channels. As another important performance measure, channel capacity has been extensively studied as well. Thorough analysis and optimization of achievable rate for different kinds of relay networks were presented in
[10]
for AF protocol and in
[11]
for DF protocol, respectively. Generic expressions of upper and lower bounds for channel capacity of dual hop AF relay networks under different fading channels were given in
[12]
[13]
.
In practical wireless engineering, it is widely observed that different links in wireless cooperative communication systems may experience different fading characteristics, rendering the system to be asymmetric. Performance analysis of such networks has always been a hot topic in the literature. The endtoend OP and SER of dual hop AF relay networks operating over mixed Rician and Nakagami
m
fading channels was studied in
[14]
. In OW relay network operating over mixed shadowed Rician and Nakagami
m
fading channels, the authors in
[15]
derived the moment generating function (MGF) of endtoend SNR. Recently, a mixed generalized
κ

μ
and
η

μ
fading scenario is investigated in
[16]
, the endtoend OP and SER of dual hop AF relay networks with both FG and VG policies were considered.
The special case of mixed Rician and Rayleigh fading scenario is of practical interest in modeling mixed line of sight (LOS) and noneLOS (NLOS) fading channels, which occurs in various wireless applications and is recommended by popular communication standards. The endtoend OP and SER of dual hop AF relay networks under this special mixed fading scenario is investigated in
[17]
for VG policy and in
[18]
for FG policy. The authors in
[19]
presented asymptotic OP and SER study of an opportunistic TWAF VG relay network where one out of
N
candidate relays is chosen for transmission. To the best of the authors’ knowledge, the overall OP of TWAF relay networks in such a mixed fading scenario where two source nodes have different traffic requirements, along with the topic of channel capacity, under either VG or FG policy, is not seen published in the literature.
Based on the observations above, in this paper, we focus on the analysis and optimization of both the endtoend and overall OP and channel capacity in a TWAF relay network, operating over mixed Rician and Rayleigh fading channels, with different APs adopted at the relay, respectively. In particular, the main contributions of this paper are summarized as follows:
1. A uniform expression for the cumulative distribution function (CDF) of endtoend SNR that applies to different APs was presented and simplified asymptotic expressions of endtoend OP for VG and FG policies were given. Besides, the topic of optimizing the endtoend OP performance was briefly described.
2. For TWAF relay network where two source nodes have nonidentical traffic requirement, with FG policy, a general expression of the overall OP were analyzed and based on which effective power allocation (PA) schemes were obtained via bruteforce numerical exhaustive search type algorithms under different constraints, respectively. With VG policy, we were able to obtain the high SNR asymptotic expressions of the overall OP of the considered scenario and then theoretical global optimizers were presented in closedfrom under different constraints. Simulation results showed that the proposed PA schemes achieve considerable performance improvements compared to equal power allocation (EPA) scheme. To sum up, a discussion is provided about the similarities and differences between the logic of PA schemes for VG and FG policies.
3. Exact expressions of generic tight lower bounds and high SNR approximations of channel capacity of the considered scenario were presented, with FG and VG policies adopted at the relay, respectively. Global optimizers that maximize the system sum capacity under different constraints are obtained via numerical methods.
4. We compare the performance of TWAF relay network operating in the considered asymmetric scenario with than in a conventional homogenous Rayleigh/Rayleigh fading scenario and a discussion of the impact of the channel asymmetry on system performance was provided.
The rest of the paper is organized as follows, In Section 2 we briefly outline the system and channel model. In Section 3, we give elaborate analysis of endtoend OP under different APs. The overall OP and PA issue is detailed in Section 4, Section 5 deals with the topic of channel capacity. Finally, we conclude this paper in section 6.
Throughout this paper,
E
{·} and
P
{·} denote the expectation and probability, respectively. Bold italic symbols indicate vectors. [
x
]
^{+}
=
max
(0,
x
).
f_{α}
(
x
),
F_{α}
(
x
),
represents the probability density function (PDF), CDF and complementary CDF (CCDF) with respect to (w.r.t.) the random variable (RV)
α
. Γ(
a
,
x
) and
γ
(
a
,
x
) stand for the upper and lower incomplete gamma functions (
[20]
, eq. (8.350.12)), respectively.
ε
≈ 0.577 is the Euler’s constant.
I_{v}
(
z
) and
K_{v}
(
z
) denote the
v
th order modified Bessel functions of the first kind (
[20]
, eq. (8.445)) and second kind (
[20]
, eq. (8.446)), respectively.
_{p}F_{q}
(
a
_{1}
...
a_{p}
;
b
...
b_{q}
;
z
) is the generalized hypergeometric function (
[20]
, eq. (9.14.1)).
E_{n}
(
x
) is the generalized exponential integral function (
[21]
, eq. (5.1.4)).
is the firstorder Marcum
Q
function (
[22]
, eq. (4.34)).
2. System and Channel Models
The system under consideration is shown in
Fig. 1
. Two terminal nodes,
N
_{1}
and
N
_{2}
, communicate to each other with the aid of a single relay node,
R
. One round information exchange occupies two time slots. In the first time slot, both
N
_{1}
and
N
_{2}
transmit their individual data signals simultaneously to the relay. In the second time slot, the relay simply broadcasts an amplified version of the received signal to the two source terminals. The system operates in halfduplex mode and no direct link between
N
_{1}
and
N
_{2}
is available due to long distance and severe shadowing. Assume independent additive white Gaussian noise (AWGN) with zeromean and variance
N
_{0}
is present at all nodes and all channels are both reciprocal and quasistatic.
Twoway relay network model under consideration
To simplify notations, let
h
= [
h
_{1}
h
_{2}
] be the channel coefficient,
α_{i}
= 
h_{i}

^{2}
, let Ω
_{i}
=
E
{
h_{i}

^{2}
},
i
∈ {1,2} denote the instantaneous and statistical channel power, respectively.
E_{s}
indicates the total power constraint during two time slots and
ρ
=
E_{s}
/
N
_{0}
represents the system SNR.
q
= [
q
_{1}
q
_{2}
q
_{3}
],
q
_{1}
+
q
_{2}
+
q
_{3}
= 1,
q_{i}
˃ 0,
i
∈ {1,2,3} stands for the PA vector with elements corresponding to
N
_{1}
,
N
_{2}
, and
R
, respectively. Particularly, the logical
N
_{1}
→
R
→
N
_{2}
transmission is termed the forward link and
N
_{1}
←
R
←
N
_{2}
transmission the reverse link, respectively.
In order to incorporate the effect of relay geometry into our analysis, we adopt a simple linear model that
R
is located somewhere on the line between
N
_{1}
and
N
_{2}
. The distance between
N
_{1}
and
N
_{2}
is normalized to unity, while the distances between
N
_{1}
and
R
,
N
_{1}
and
R
are denoted as
d
and 1 
d
, respectively. Let
u
be the path loss exponent and Ω
_{1}
=
d
^{–u}
, Ω
_{2}
= (1 
d
)
^{–u}
.
Without loss of generality, we define that
h
_{1}
is subject to Rician fading with parameters
K
_{1}
and Ω
_{1}
where
k
_{1}
is the Rician factor, while
h
_{2}
is subject to Rayleigh fading with parameter Ω
_{2}
. The PDF and CDF of
α
_{1}
are given in (1) and (2), respectively.
For Rayleigh fading,
α
_{2}
is an exponentially distributed RV, the PDF and CDF of
α
_{2}
are written as
f
_{α2}
(
x
) =
ω
_{2}
exp
(–
ω
_{2}
x
),
F
_{α2}
(
x
) = 1 –
exp
(–
ω
_{2}
x
), where
ω
_{2}
= 1/Ω
_{2}
.
3. Endtoend Outage Probability Analysis
We consider four different APs at the relay. First, VG policy where
R
has full knowledge of
h
_{1}
and
h
_{2}
. Scond, FG policy where
R
only knows Ω
_{1}
and Ω
_{2}
. Third, a mixed amplification (MA1) policy where
h
_{1}
and Ω
_{2}
is available at
R
. Fourth, another mixed amplification (MA2) policy where
h
_{2}
and Ω
_{1}
is available at
R
.
The endtoend received SNR at
N_{i}
,
γ_{Ni}
, can be expressed as
where
i
∈ {1,2},
k
= 3 
i
and
is the amplification coefficient. The choice of
under different APs is listed in
Table 1
.
Amplification coefficients of different APs
Amplification coefficients of different APs
Hereinafter, for ease of notations, we define the following instantaneous and average received SNRs during the transmission,
γ
_{1}
=
ρq
_{1}
α
_{1}
,
γ
_{2}
=
ρq
_{2}
α
_{2}
,
γ
_{3}
=
ρq
_{3}
α
_{1}
,
γ
_{4}
=
ρq
_{3}
α
_{2}
,
, l ∈ {1,2,3,4} is the expectation. In addition,
is determined by power allocation policies. It is observed that, by choosing proper parameters, the exact expressions of endtoend received SNR under different APs can be formulated uniformly as
where
γ_{eq}
denotes the equivalent endtoend SNR,
μ_{a}
,
μ_{b}
are constants and
γ_{a}
,
γ_{b}
are instantaneous received SNRs during the transmission. Without loss of generality, let
γ_{a}
denote the Rician SNR as in (1) and (2) with parameters
K_{a}
and
,
γ_{b}
represents an Rayleigh SNR with
. The exact expression and uniform parameterization of endtoend received SNR under different APs are presented in
Table 2
.
Exact expression and uniform parameterization of endtoend SNR
Exact expression and uniform parameterization of endtoend SNR
Furthermore, a uniform expression for the CDF of endtoend SNRs under different APs is presented in the following proposition.
Proposition 1:
In TWAF relay networks operating over mixed Rician and Rayleigh fading channels, the exact CDF of
γ_{eq}
,
F_{eq}
(
γ_{th}
), is written as
where
Λ =
λ_{a}μ_{b}
+
λ_{b}μ_{a}
,
is the binomial coefficient.
Proof:
According to probability,
F_{eq}
(
γ
) can be expressed as
Substitute (1) into this equation, expand
I
_{0}
(
z
) into series (
[20]
, eq. (8.445)), use a change of variables
y
= (
x
–
μ_{b}γ_{th}
) /
γ_{th}
and expand the power term into binomial form, we get
The integral can be resolved in closed form (
[18]
, eq. (3.471.9)). After rearrangement we get the desired result. ■
Concerning the convergence of the infinite series in (5), substitute an asymptotic expression for
K_{v}
(
z
) (
[15]
, eq. (15)), the residual of
M
terms truncation,
R_{M}
, is written as
It can be observed that Γ(
m
+ 1,
μ_{b}λ_{a}γ_{th}
) ˂ Γ(
m
+ 1) = m! and lim
_{x→∞}
K_{a}^{x}
/
x
! = 0. As will be seen later, (5) converges quickly for a finite number of
M
terms thus truncation does not sacrifice numerical accuracy.
For certain special cases (5) could be simplified. For instance, when
μ_{b}
= 0,
F_{eq}
(
γ_{th}
) is rewritten as
In communications system, the outage event is identified as the received SNR falls below a predetermined threshold,
γ_{th}
, i.e.,
P_{out}
=
P
{
γ_{eq}
≤
γ_{th}
} =
F_{eq}
{
γ_{th}
} which is the value of the CDF function at
γ_{th}
. In order to gain more insight we further investigate into the asymptotic behavior of the endtoend OP in high SNR regime and present the following proposition.
Proposition 2:
With VG and FG polices, the asymptotic OP of the endtoend transmission in high SNR regime of the considered scenario is given is
Table 3
.
upper bound of SNR and asymptotic endtoend OP in high SNR regime
upper bound of SNR and asymptotic endtoend OP in high SNR regime
Proof:
We apply an upper bound for the exact individual endtoend SNR based on an inequality that for
a, b
˃ 0,
ab
/ (
a
+
b
) ≤
min
(
a, b
) always holds true. This method is widely adopted in the literature and calculating CDF of the obtained upper bound the finals results directly arise. For detailed derivation please refer to
[11]
[13]
[14]
and the reference therein. ■
In
Fig. 2
, we plot the OP of the endtoend transmissions in the considered scenario as functions of
d
and
ρ
, respectively. Particularly, equal power allocation (EPA) is adopted, i.e.,
q
_{1}
=
q
_{2}
=
q
_{3}
= 1/3, besides,
M
=20 terms truncation in (5) is adopted. In
Fig. 3
, we plot the exact and asymptotic endtoend OP as functions of system SNR
ρ
.
exact endtoend outage probability with γ_{th} = 1, u = 3, K_{1} = 3. (a):ρ = 33dB. (b):d = 0.5.
exact and asymptotic endtoend outage probability with γ_{th} = 1, u = 3, K_{1} = 3, d = 0.5 with different APs (a): the reverse link. (b): the forward link.
As seen in
Fig. 2
and
Fig. 3
, the MonteCarlo simulation results well matched the analytical results which corroborate the correctness of the analysis. In general, we see that VG provides the most desirable OP performance, FG suffers from obvious performance loss compared with VG, and they together may be viewed as upper and lower bounds for the performance of mixed APs. It can be observed that when the relay is located very close to one source node, the performance gain by acquiring the CSI of the other link at the relay is negligible, in such conditions
the CSI of the stronger link itself is sufficient to obtain similar performance with VG
. With
K
_{1}
= 3, the vertical black dotted line at
d
_{3}
≈ 0.63 represents a boundary for relay location, where
d
˂
d
_{3}
guarantees a better performance of the forward link compared with the reverse link. For TWAF with homogenous Rayleigh/Rayleigh fading channels (
K
_{1}
= 0), we have
d
_{0}
= 0.5. In this sense,
the Rician channel is able to ‘push’ the relay more far away while maintaining a better forward link performance
. In fact, the value of
d
_{k}
is associated with
K
_{1}
, generally,
d
_{K}
becomes larger with the increment of
K
_{1}
and the reason of this phenomenon shall be discussed in Section 4.3. Besides, we see that the asymptotic behavior with VG policy approaches the exact performance more quickly than FG policy, and there are more obvious vibrations in low SNR regime for the forward link than the reverse link. This phenomenon is caused by the channel asymmetry and different convergence speed for the seriesform PDFs of the distribution for Rican and Rayleigh SNRs.
Though not our primary concern, we very briefly look into the topic of optimizing the endtoend OP performance. Take the forward link with VG policy for example, from
Table 3
, substitute
therein it can be shown that in high SNR regime, the OP of the forward link is asymptotically approximated as
The optimization problem under total power constraint can be written as
Based on partial derivative of
q
_{2}
, it can be seen that the global optimizer lies on the boundary of –
q
_{2}
˂ 0. So we relax the constraint to –
q
_{2}
≤ 0, and the optimal solution can be written as
With
q
_{2}
= 0 the TWAF relay network reduces to a conventional oneway relay network and no information can be acquired at
N
_{1}
, this conflicts with the bidirectional transmission nature of twoway relay networks. However, we see that
in high SNR regime the OP performance of individual endtoend link in TWAF relay network is upbounded by that in OW relay network
.
4. Overall Outage Probability and Power Allocation
As aforementioned the OP performance of VG and FG policies may be viewed as bounds for MA1 and MA2, in this section we focus on the overall OP analysis and PA schemes for VG and FG policies. Since TWAF relay network carries two data streams of opposite directions concurrently, the system is in outage if either
N
_{1}
or
N
_{2}
is in outage. Besides,
N
_{1}
and
N
_{2}
may have different traffic requirements. Thus, the overall OP is defined as
where
γ
_{th}
_{1}
and
γ
_{th}
_{2}
denote the prescribed SNR threshold at
N
_{1}
and
N
_{2}
, respectively, and define
η
=
γ
_{th}
_{1}
/
γ
_{th}
_{2}
as traffic pattern indicator representing the level of traffic requirement asymmetry.
 4.1. Overall OP and PA Schemes with VG Policy
With VG policy, in order to simplify the analysis, we give an asymptotic lower bound for the overall OP in the following proposition.
Proposition 3:
When two source nodes have nonidentical traffic requirements, the overall OP of a TWAF relay network with VG policy can be lower bounded by
where
λ
_{α1}
= (1 +
K
_{1}
) /
Ω
_{1}
,
ω
_{2}
= 1 /
Ω
_{2}
,
ζ
_{1}
= 1 / (1 +
η
),
ζ
_{2}
=
η
/ (1 +
η
), and Case1, Case2, and Case3 are characterized by
q
_{1}
≥
ζ
_{1}
,
q
_{2}
≥
ζ
_{2}
, and (
q
_{1}
˂
ζ
_{1}
,
q
_{2}
˂
ζ
_{2}
) , respectively.
And in high SNR regime an asymptotic expression of (14) is given by
where
ω
_{1}
=
e
^{–K1}
(1 +
K
_{1}
) /
Ω
_{1}
.
Proof:
Apply the upper bound for endtoend SNR from
Table 3
, wet get a lower bound of the OP for the reverse link as
Note that
α
_{1}
and
α
_{2}
are mutually independent. For the forward link we have
Note that if
q
_{1}
˂
ζ
_{1}
,
holds and if
q
_{2}
˂
ζ
_{2}
,
holds. Along with the total power constraint of
q
_{1}
+
q
_{2}
+
q
_{3}
= 1, (13) could be rewritten as
Substitute into (18) the CDF expressions for
α
_{1}
and
α
_{2}
, we directly arrive at (14). It is interesting to see that if
q
_{1}
≥
ζ
_{1}
, the overall OP is determined by the reverse link itself. If
q
_{2}
≥
ζ
_{2}
, the forward link dominates. Only when
q
_{1}
˂
ζ
_{1}
,
q
_{2}
˂
ζ
_{2}
holds simultaneously, the overall OP is codetermined by both links.
Substitute the wellknown small value approximate for
I
_{0}
(
z
) (
[21]
, eq. (9.6.7)) into (
[22]
, eq. (4.41)), after some algebraic manipulations it can be shown that with
α
fixed and
β
approaches 0, we have
In high SNR regime, i.e., when 1 /
ρ
→ 0, substituting (19) and
e
^{–}
^{x}
≈ 1 –
x
into (14) and omit the higher order infinitesimals directly yields (15). ■
Next, we present the optimal power allocation (OPA) scheme to minimize the asymptotic overall OP in (15). Obviously, under the total power constraint the following three cases should be considered.
Case1 (
q
_{1}
≥
ζ
_{1}
):
Case2 (
q
_{2}
≥
ζ
_{2}
):
Case3 (
q
_{1}
˂
ζ
_{1}
,
q
_{2}
˂
ζ
_{2}
)
It is interesting to see that, though via different methods, (18)(20) bears basically similar mathematical forms with (
[5]
, eq. (18, 22, 33)) which deals with TWAF relay network with VG policy under homogenous Rayleigh/Rayleigh fading channels, of course in our analysis the Rician channel introduces a new definition for
ω
_{1}
. Following standard analysis and applying Lagrange dual method with KarushKhunTucker (KKT) conditions to solve (20)(22), we arrive at the following lemma.
Lemma 1:
the OPA that minimize the asymptotic overall OP (15) of TWAF relay network with VG policy operating over mixed Rician and Rayleigh fading channels, under the total power constraint of
q
_{1}
+
q
_{2}
+
q
_{3}
= 1,
q_{i}
˃ 0,
i
= 1,2,3, is given by
where
Note that if
ω
_{1}
=
ω
_{2}
,
q
^{*}
_{A}
and
q
^{*}
_{B}
reduces to
q
^{*}
_{A}
= [
ζ
_{1}
;
ζ
_{2}
/ 2 ;
ζ
_{2}
/ 2 ;],
q
^{*}
_{B}
= [
ζ
_{1}
/2 ;
ζ
_{2}
;
ζ
_{1}
/ 2 ;] ■
As a special case, in practical applications where equal source power (ESP) is mandated upon two source nodes, i.e., with an extra
q
_{1}
=
q
_{2}
constraint, the optimal power allocation scheme with such constraint, OESP, is given by the following lemma.
Lemma 2:
The OESP that minimize the asymptotic system overall OP (15) of TWAF relay network with VG policy operating over mixed Rician and Rayleigh fading channels, where two source nodes are loaded with equal transmission power, is given by
where
Note that when the relay is located too close to the source and thus rendering
Δ_{d}
/ (1 + 2
Δ_{d}
) ˂
ζ
_{2}
,
q
^{*}
_{D}
reduces to
q
^{*}
_{D}
= [
ζ
_{2}
;
ζ
_{2}
; 1  2
ζ
_{2}
; ]. Similarly, when
Δ_{f}
/ (1 + 2
Δ_{f}
) ˂
ζ
_{1}
,
q
^{*}
_{F}
reduces to
q
^{*}
_{F}
= [
ζ
_{1}
;
ζ
_{1}
; 1  2
ζ
_{1}
; ]. ■
 4.2 Overall OP and PA Scheme with FG Policy
In this subsection we redirect our attention to FG policy. According to probability, the overall outage could be written as
[9]
First we look into
I_{a}
. From
Table 2
,
I_{a}
could be expressed as
Similarly,
I_{b}
could be expressed as
The integral boundaries in (27), (28) are defined as follows
It is difficult, if not impossible, for
I_{a}
and
I_{b}
to be resolved in closed form. In fact, after expanding the Marcum
Q
function (
[22]
, eq. (4.35)) and Bessel function (
[20]
, eq. (8.447.1)) into infinite series, and with the aid of Taylor series,
I_{a}
and
I_{b}
can be expressed in multifold infinite series, which is not quite efficient for mathematical computation. Alternatively, we resort to numerical methods (
[21]
, eq. (25.4.38)) to calculate
I_{a}
and
I_{b}
.
Take the second term of
I_{a}
for example, after simple manipulations we get
Let
w_{n}
=
π
/
N_{t}
,
z_{n}
=
cos
(2
n
 1)
π
/2
N_{t}
,
c_{n}
=
sin
(2
n
 1)
π
/2
N_{t}
θ_{n}
= (
z_{n}
+ 1)
π
/4,
x_{n}
=
tan
θ_{n}
+
δ
_{2}
, (28) could be evaluated as
For space limitations we omit the overall expression for (25). In simulation experiments, the choice of
N_{t}
always guarantees the stability of 7th decimal place and it is noted that
N_{t}
is affected by the average channel power and pathloss exponent. In general, when the relay is close to source nodes, larger value of
N_{t}
should be utilized. The OPA and OESP schemes that minimizes the OP performance with FG policy can be obtained by numerically solving (25), via exhaustive search type algorithms.
 4.3 Simulation Results of Overall OP with VG and FG polices
In this subsection, we carry out Monte Carlo simulation experiments to examine the analytical results presents in the previous two subsections.
In
Fig. 4
we plot the overall OP performance of different PA schemes in TWAF relay network with FG and VG policies when source nodes have nonidentical traffic requirements. We can see that with both VG and FG policies, OPA brings the most desirable OP performance compared with OESP and EPA in medium and high SNR, irrespective of relay locations. And it is a surprise that OESP works just slightly better than EPA with VG policy, but with FG policy OESP obviously outperforms EPA.
Overall OP of TWAF relay network with nonidentical traffic requirements, γ_{th}_{1} = 0.5, γ_{th}_{2} = 1, η = 0.5, u = 3, K_{1} = 3. (a): ρ = 33dB. (b): d = 0.75
In
Fig. 5
we plot the overall OP performance of VG and FG policies when
h
_{1}
is characterized by different Rician factors. In addition, we compare the system performance under the considered mixed Rican and Rayleigh fading scenario, with that of a homogeneous Rayleigh fading scenario (
K
_{1}
= 0). It can be observed that when two source nodes have identical traffic requirement, the performance of the homogeneous Rayleigh scenario is strictly ‘symmetric’ w.r.t.
d
_{0}
= 0.5, with both VG and FG policies. Besides, we also see that with the increment of the Rician factor, both FG and VG policies achieve better performance and with large
K
_{1}
value, the OP of VG can be times better than FG. The vertical black dotted line
d_{k}
indicates the relay location where OESP best approaches OPA, i.e., at
d_{k}
we have
q^{oesp}
=
q^{opa}
and it is an interesting turning point in the OP curve. At last, we note that when the relay is located very close to
N
_{1}
, the performance of VG policy with different
K
_{1}
is not as clearly distinguished from each other as in FG policy. For the VG policy with large
K
_{1}
and the proposed OPA in (23), we do not see a ‘best’ relay location that provides the best overall OP performance. The overall OP performance is more like a monotonic decreasing function of relay location
d
.
Overall OP of TWAF relay network with identical traffic requirements and different Rician factors, γ_{th}_{1} = γ_{th}_{2} = 1, η = 1, u = 3, ρ = 33dB. (a): With VG policy, K_{1} = 0,3,5,7. (b): With FG policy, K_{1} = 0,3,7.
In
Fig. 6
we plot the PA schemes presented in closedform in (23) (24) for VG policy, and those acquired from exhaustive search type algorithms for FG policy, when two source nodes have equal traffic requirement. From (23) and (24), it can be proved that for a given
η
,
d_{k}
is the solution to
thus we have
d_{k}
= 1 / (1 +
θ
) where
And from (12), we see that with EPA, a change of the worse directional transmission link occurs if (2 
η
)
ω
_{1}
= (2
η
1)
ω
_{2}
holds true, with
η
= 1, this point is also at
d_{k}
, that is why
Fig. 2
(a) and
Fig. 5
shares the same
d
_{3}
. In this sense
d_{k}
represents a change of power allocation philosophy, to give more power to source node of the ‘new’ weaker link. Though theoretical expressions for PA schemes with FG is still missing, we see
d_{k}
applies to FG policy as well, at least roughly. Besides, we see that in high SNR regime FG policy always allocates more than half of the total power to the relay node, and both FG and VG policies share the same logic of allocating more power to the source node of weaker link.
PA schemes of TWAF relay network, γ_{th}_{1} = γ_{th}_{2} = 1, η = 1, u = 3, ρ = 33dB. K_{1} = 3. (a): closedfrom solutions from (23) (24). (b): numerical solutions via exhaustive search.
5. Channel Capacity Analysis
In this section, we study the channel capacity of TWAF relay networks, with FG and VG policies, respectively. First, for each AP, we give tight lower bounds for the endtoend links. Next, high SNR approximations were presented. At last, based on the high SNR approximations, we resort to numerical methods to obtain PA schemes that maximize the overall channel capacity under different constraints. We present the following lemma about the integral in the form
E
{
ln
(
v
_{1}
α
_{1}
+
v
_{2}
α
_{2}
)}
v
_{1}
˃ 0,
v
_{2}
˃ 0 which is needed in the following subsections.
Lemma 3:
If
α
_{1}
is a Rician Power RV with
K
_{1}
and
Ω
_{1}
,
α
_{2}
is a Rayleigh Power RV with
Ω
_{2}
, the integral
E
{
ln
(
v
_{1}
α
_{1}
+
v
_{2}
α
_{2}
)}
v
_{1}
˃ 0,
v
_{2}
˃ 0 can be resolved by
where
θ
=
v
_{1}
ω
_{2}
/
v
_{2}
λ_{α}
_{1}
,
ω
_{2}
= 1 /
Ω
_{2}
,
λ_{α}
_{1}
= (1 +
K
_{1}
) / Ω
_{1}
and
J
(
a
,
x
) is given by
Proof :
First, perform the integration w.r.t.
α
_{2}
. With the aid of (
[20]
, eq. (4.337.1)), we have
Substitute (1) herein and expand the Bessel function into series, (32) can be rewritten as
The integral in (33) can be solved with the aid of (
[23]
, eq. (4.1.8), (4.2.17), (4.2.20)) according to different values of
θ
=
v
_{1}
ω
_{2}
/
v
_{2}
λ_{α}
_{1}
. Concerning the convergence of the infinite series, it will be shown later via simulations that a finite number of
M
terms truncation does not sacrifice numerical accuracy. ■
 5.1 Lower Bound and High SNR Approximation of Channel Capacity
In this subsection, we apply the method proposed by Zhong, etc. (
[12]
, eq. (4)), which is a clever manipulation of Jensen’s inequality, to present the lower bounds, and the method proposed by Rodríguez (
[10]
, eq. (24)(25)), to present the high SNR approximations. Though the lower bounds were originally proposed for FG policy, it can be extended to VG policy as well. We shall see that though via different methods, the results are closely related. In general, we have the following proposition.
Proposition 4:
The lower bounds and high SNR approximations for the channel capacity of the endtoend links in TWAF relay network operating over mixed Rician and Rayleigh fading channels, with both FG and VG polices, are given in
Table 4
with
F
_{1}
(
x
) = ln(1 +
e^{x}
) / 2
ln
2,
F
_{2}
(
x
) =
x
/ 2
ln
2,
c
_{1}
= 1 +
ρq
_{1}
Ω
_{1}
+
ρq
_{2}
Ω
_{2}
,
Lower bounds and high SNR approximations of channel capacity
Lower bounds and high SNR approximations of channel capacity
P
_{3}
(
a
),
P
_{4}
(
a
),
P
_{5}
and
P
_{6}
are defined as
where
Δ
(
a, M, K_{1}
) =
E
_{1}
(
a
λ
_{3}
)
e
^{k1}
[
γ
(
M
 1,
K
_{1}
)(
M
 1)
K
_{1}
+
γ
(
M
,
K
_{1}
)] / Г(
M
) ,
and
J
(
a, x
) is given in (31).
Proof:
For simplicity, take the reverse link for example. For lower bounds, applying Zhong’s method (
[12]
, eq. (4)), the lower bound of the reverse link with FG policy can be written as
and for VG policy, let
c
_{2}
= 0 which is efficient in medium and high SNRs, the lower bound can be written as
As for high SNR approximations, according to (
[10]
, eq. (24)(25)), the high SNRs approximations of channel capacity can be written as
To evaluate (36)(38), note that for Rayleigh fading, the integral
E
{
lnα
_{2}
} and
E
{
ln
(
a
+
α
_{2}
)},
a
˃ 0 were given in (
[10]
, eq. (9)(10)). And for Rician fading,
E
{
lnα
_{1}
} and
E
{
ln
(
a
+
α
_{1}
)},
a
˃ 0 were given in (
[12]
, eq. (43), (47)). Therefore, the key task is to evaluate the integral in (30). Substitute (31) into (34)(36) yields the desired result. ■
 5.2 Simulation Results of Channel Capacity
In this subsection, we resort to simulation experiments to validate the analytical results. The same as before, we use exhaustive search type algorithms to find the global optimizers that maximizes the sum capacity
under the total power constraint (OPA) and equal source power constraint (OESP), respectively, and compare them to EPA. Note that
M
≤ 20 terms truncation is applied when calculating the functions defined in
Table 4
.
In
Fig. 7
, we plot the system sum capacity of TWAF relay network against SNR and relay location, respectively. In
Fig. 8
, we plot the OPA and OESP schemes obtained via exhaustive search type algorithms that maximize the sum capacity. As shown in
Fig. 7
(a), the lower bounds and high SNR approximations of sum capacity are in fine agreement with the simulation results, especially in medium and high SNRs. In addition, we see that the sum capacity with VG policy is better than FG, and with the increment of
K
_{1}
, the sum capacity slowly grows. However, in general the capacity performance is less sensitive to the increment of Rician factor compared with outage probability. In
Fig. 7
(b), we compare the sum capacity of different PA schemes. It is noteworthy that the performance of OPA obtained via exhaustive search is, to a large extent, slightly better than and generally very similar to that of the suboptimal scheme (subOPA) proposed in
[10]
. Though subOPA was originally proposed in a homogenous Rayleigh/Rayleigh fading scenario, it is solely determined by statistical channel powers and therefore to some extent it is “fading irrelevant”. Furthermore, the OESP with both FG and VG policies obviously outperforms the EPA scheme, and still suffers from considerable performance loss compared with the OPA scheme. As seen in
Fig. 8
, the OPA and OESP schemes for FG and VG polices share the same trend as
d
varies from 0 to 1, and the subOPA scheme are very similar with the OPA scheme.
Sum capacity of TWAF relay network with VG and FG polices. u = 3. (a): d = 0.5. (b): k_{1} = 3, ρ = 33dB.
Power allocation schemes via exhaustive search to maximize sum capacity with VG and FG polices. u = 3, k_{1} = 3, ρ = 33dB. (a): OPA schemes. (b): OESP schemes.
6. Conclusions
In this paper, we focus on analyzing and optimizing the OP and channel capacity of TWAF relay network operating over mixed Rician and Rayleigh fading channels, with different APs adopted at the relay, respectively. First, a uniform expression for the exact OP of endtoend links which applies to different APs and simplified asymptotic expressions of for VG and FG policies were presented. Next, we investigated into the system overall OP with two source nodes having nonidentical traffic requirements, under FG and VG polices, respectively. We were able to derive theoretical global optimizers for the overall OP with VG policy, while for FG policy we resort to numerical exhaustive search methods. Concerning channel capacity with both FG and VG polies, we presented explicit expressions for tight lower bounds and high SNR approximations of endtoend links and then utilized them to search for the global optimizers that maximize the sum capacity. Extensive simulation experiments were carried out to validate the analytical results and demonstrated that the proposed OPA and OESP schemes offer substantial performance improvements as compared to EPA scheme.
BIO
Qi Yanyan was born in 1985. He received the Master Degree in Enginnering in 2010 from North China Electric Power University. He is currently a Ph.D. candidate at Beijing University of Posts and Telecommunications, Beijing, China. His research interests include cooperative relay communications, multimedia communications and signal processing.
Wang Xiaoxiang was born in 1969. Prof. Wang is doctor supervisor in school of information and telecommunication Engineering. She received her PH.D. degree from BIT in 1998. She was a visit scholar in Austria University of Technology in Vienna from 2001 to 2002. Her research interests include cooperative communications, MIMO & OFDM technique, and MBMS systems.
Rankov B.
,
Wittneben A.
2007
“Spectral efficient protocols for halfduplex fading relay channels,”
IEEE J. Sel. Areas Commun.
25
(2)
379 
389
DOI : 10.1109/JSAC.2007.070213
Rodríguez L. J.
,
Tran N.
,
LeNgoc T.
“Achievable rates and power allocation for twoway AF relaying over Rayleigh fading channels,”
in Proc. of IEEE Int. Conf. Communications (ICC)
June 913, 2013
5914 
5918
Li D.
,
Xiong K.
2012
“SER analysis of physical layer network coding over Rayleigh fading channels with QPSK modulation,”
InformationAn International Interdisciplinary Journal
15
(11(A))
4573 
4578
Ji X.
,
Zheng B.
,
Cai Y.
,
Zou L.
2012
“On the study of halfduplex asymmetric twoway relay transmission using an amplifyandforward relay,”
IEEE Trans. Veh. Technol.
61
(4)
1649 
1664
DOI : 10.1109/TVT.2012.2188108
Zhang C.
,
Ge J.
,
Li J.
,
Hu Y.
2013
“Performance evaluation for asymmetric twoway AF relaying in Rician fading,”
IEEE Wireless Commun. Lett.
2
(3)
DOI : 10.1109/WCL.2013.030613.130050
Yang J.
,
Fan P.
,
Duong Trung Q.
,
Lei X.
2011
“Exact performance of twoway AF relaying in Nakagamim fading environment,”
IEEE Trans. Wireless Commun.
10
(3)
980 
987
DOI : 10.1109/TWC.2011.011111.101141
Xu F.
,
Lau Francis C. M.
,
Yue D.
2010
“Diversity order for amplifyandforward dual hop systems with fixedgain relay under Nakagami fading channels,”
IEEE Trans. Wireless Commun.
9
(1)
92 
98
DOI : 10.1109/TWC.2010.01.090510
Ni Z.
,
Zhang X.
,
Yang D.
2014
“Outage performance of twoway fixed gain amplifyandforward relaying systems with asymmetric traffic requirements,”
IEEE Commun. Lett.
18
(1)
78 
81
DOI : 10.1109/LCOMM.2013.112513.132232
Rodríguez L. J.
,
Tran N. H.
,
LeNgoc T.
2014
“Achievable rate and power allocation for singlerelay AF systems over Rayleigh fading channels at high and low SNRs,”
IEEE Trans. Veh. Technol.
63
(4)
1726 
1739
DOI : 10.1109/TVT.2013.2287997
Xiong K.
,
Shi Q.
,
Fan P.
,
Letaief K. B.
“Resource allocation for twoway relay networks with symmetric data rates: an information theoretic approach,”
in Proc. of IEEE Int. Conf. Communications (ICC)
June 913, 2013
6060 
6064
Zhong C.
,
Matthaiou M.
,
Karagiannidis G. K.
,
Ratnarajah T.
2011
“Generic ergodic capacity bounds for fixedgain AF dualhop relaying systems,”
IEEE Trans. Veh. Technol.
60
(8)
3814 
3824
DOI : 10.1109/TVT.2011.2167362
Waqar O.
,
Ghogho M.
,
McLernon D.
2011
“Tight bounds for ergodic capacity of dualhop fixedgain relay networks under Rayleigh fading,”
IEEE Commun. Lett.
15
(4)
413 
415
DOI : 10.1109/LCOMM.2011.022411.102027
Soliman S. S.
,
Beaulieu N. C.
2014
“The bottleneck effect of Rician fading in dissimilar dualhop AF relaying systems,”
IEEE Trans. Veh. Technol.
63
(4)
1957 
1965
DOI : 10.1109/TVT.2013.2288038
Bhatnagar M. R.
,
Arti M. K.
2013
“Performance analysis of AF based hybrid satelliteterrestrial cooperative network over generalized fading channels,”
IEEE Commun. Lett.
17
(10)
1912 
1915
DOI : 10.1109/LCOMM.2013.090313.131079
Peppas K. P.
,
Alexandropoulos G. C.
,
Mathiopoulos P. T.
2013
“Performance analysis of dualhop AF relaying systems over mixedη−μandκ−μfading channels,”
IEEE Trans. Veh. Technol.
62
(7)
3149 
3163
DOI : 10.1109/TVT.2013.2251026
Suraweera H. A.
,
Louie H. Y.
,
Li Y.
,
Karagiannidis G. K.
,
Vucetic B.
2009
“Two hop amplifyandforward transmission in mixed Rayleigh and Rician fading channels,”
IEEE Commun. Lett.
13
(4)
227 
229
DOI : 10.1109/LCOMM.2009.081943
Suraweera H. A.
,
Karagiannidis G. K.
,
Smith P. J.
2009
“Performance analysis of the dualhop asymmetric fading channel,”
IEEE Trans. Wireless Commun.
8
(6)
2783 
2788
DOI : 10.1109/TWC.2009.080420
Fan Z.
,
Guo D.
,
Zhang B.
,
Zeng L.
2012
“Performance analysis and optimization for AF twoway relaying with relay selection over mixed Rician and Rayleigh fading,”
KSII Transactions on Internet and Information Systems
6
(12)
3275 
3295
Gradshteyn I. S.
,
Ryzhik I. M.
2007
Table of Integrals, Series, and Products
7th Edition
Academic Press
San Diego, CA
Abramowitz M.
,
Stegun I. A.
1972
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
10th Edition
Dover
New York
Simon M. K.
,
Alouini M. S.
2005
Digital Communication over Fading Channels
2nd Edition
Wiley & Sons
New Jersey
Gradshteyn I. S.
,
Ryzhik I. M.
2007
Table of Integrals, Series, and Products
7th Edition
Academic Press
San Diego, CA
Murray Geller
,
Edward W. Ng
1969
“A table of integrals of the exponential integral,”
Journal of Research of the National Bureau of StandardsB: Mathematics and Mathematical Science
73B
(3)
191 
210