Performance Optimization of Two-Way AF Relaying in Asymmetric Fading Channels

Abstract

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 two-way amplify-and-forward (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 end-to-end and overall performance metrics were considered. In detail, both uniform exact expressions and simplified asymptotic expressions for the end-to-end outage probability (OP) were presented, based on which the system overall OP was studied under the condition of the two source nodes having non-identical 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

Recently, 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 two-way (TW) relay network, compared with conventional one-way (OW) relay network, it brings considerable spectral efficiency improvement as completing one round of bidirectional information exchange in two time slots [1] between two half-duplex terminals. Different transmission protocols employing different signal processing technique can be utilized in cooperative relay networks such as amplify-and-forward (AF), decode-and-forward (DF), compress-and-forward (CF) and so on, among which the AF protocol is a popular concern because it is low-cost 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 two-way amplify-and-forward (TWAF) relay network with VG policy under Rayleigh fading channels, when two source terminals have non-identical 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, end-to-end 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 signal-to-noise 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 end-to-end 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 end-to-end SNR. Recently, a mixed generalized κ - μ and η - μ fading scenario is investigated in [16], the end-to-end 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 none-LOS (NLOS) fading channels, which occurs in various wireless applications and is recommended by popular communication standards. The end-to-end 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 end-to-end 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 end-to-end SNR that applies to different APs was presented and simplified asymptotic expressions of end-to-end OP for VG and FG policies were given. Besides, the topic of optimizing the end-to-end OP performance was briefly described.

2. For TWAF relay network where two source nodes have non-identical 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 brute-force 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 closed-from 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 end-to-end 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). fα(x), Fα(x), represents the probability density function (PDF), CDF and complementary CDF (C-CDF) 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. (8.350.1-2)), respectively. ε ≈ 0.577 is the Euler’s constant. Iv(z) and Kv(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. pFq(a1 ... ap; b ... bq; z) is the generalized hypergeometric function ([20], eq. (9.14.1)). En(x) is the generalized exponential integral function ([21], eq. (5.1.4)). is the first-order Marcum-Q function ([22], eq. (4.34)).

 

2. System and Channel Models

The system under consideration is shown in Fig. 1. Two terminal nodes, N1 and N2, 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 N1 and N2 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 half-duplex mode and no direct link between N1 and N2 is available due to long distance and severe shadowing. Assume independent additive white Gaussian noise (AWGN) with zero-mean and variance N0 is present at all nodes and all channels are both reciprocal and quasi-static.

Fig. 1.Two-way relay network model under consideration

To simplify notations, let h = [h1 h2] be the channel coefficient, αi = |hi|2, let Ωi = E{|hi|2}, i ∈ {1,2} denote the instantaneous and statistical channel power, respectively. Es indicates the total power constraint during two time slots and ρ = Es / N0 represents the system SNR. q = [q1 q2 q3], q1 + q2 + q3 = 1, qi ˃ 0, i ∈ {1,2,3} stands for the PA vector with elements corresponding to N1, N2, and R, respectively. Particularly, the logical N1 → R → N2 transmission is termed the forward link and N1 ← R ← N2 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 N1 and N2. The distance between N1 and N2 is normalized to unity, while the distances between N1 and R, N1 and R are denoted as d and 1 - d, respectively. Let u be the path loss exponent and Ω1 = d–u, Ω2 = (1 - d)–u.

Without loss of generality, we define that h1 is subject to Rician fading with parameters K1 and Ω1 where k1 is the Rician factor, while h2 is subject to Rayleigh fading with parameter Ω2. The PDF and CDF of α1 are given in (1) and (2), respectively.

For Rayleigh fading, α2 is an exponentially distributed RV, the PDF and CDF of α2 are written as fα2(x) = ω2exp(–ω2x), Fα2(x) = 1 – exp(–ω2x), where ω2 = 1/Ω2.

 

3. End-to-end Outage Probability Analysis

We consider four different APs at the relay. First, VG policy where R has full knowledge of h1 and h2. Scond, FG policy where R only knows Ω1 and Ω2. Third, a mixed amplification (MA1) policy where h1 and Ω2 is available at R. Fourth, another mixed amplification (MA2) policy where h2 and Ω1 is available at R.

The end-to-end received SNR at Ni, γNi, 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.

Table 1.Amplification coefficients of different APs

Hereinafter, for ease of notations, we define the following instantaneous and average received SNRs during the transmission, γ1 = ρq1α1, γ2 = ρq2α2, γ3 = ρq3α1, γ4 = ρq3α2, , 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 end-to-end received SNR under different APs can be formulated uniformly as

where γeq denotes the equivalent end-to-end SNR, μa, μb are constants and γa, γb are instantaneous received SNRs during the transmission. Without loss of generality, let γa denote the Rician SNR as in (1) and (2) with parameters Ka and , γb represents an Rayleigh SNR with . The exact expression and uniform parameterization of end-to-end received SNR under different APs are presented in Table 2.

Table 2.Exact expression and uniform parameterization of end-to-end SNR

Furthermore, a uniform expression for the CDF of end-to-end 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 γeq, Feq(γth), is written as

where Λ = λaμb + λbμa, is the binomial coefficient.

Proof: According to probability, Feq(γ) can be expressed as

Substitute (1) into this equation, expand I0(z) into series ([20], eq. (8.445)), use a change of variables y = (x – μbγth) / γth 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 Kv(z) ([15], eq. (15)), the residual of M terms truncation, RM, is written as

It can be observed that Γ(m + 1, μbλaγth) ˂ Γ(m + 1) = m! and limx→∞ Kax / 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 μb = 0, Feq(γth) is rewritten as

In communications system, the outage event is identified as the received SNR falls below a predetermined threshold, γth, i.e., Pout = P{γeq ≤ γth} = Feq{γth} which is the value of the CDF function at γth. In order to gain more insight we further investigate into the asymptotic behavior of the end-to-end OP in high SNR regime and present the following proposition.

Proposition 2: With VG and FG polices, the asymptotic OP of the end-to-end transmission in high SNR regime of the considered scenario is given is Table 3.

Table 3.upper bound of SNR and asymptotic end-to-end OP in high SNR regime

Proof: We apply an upper bound for the exact individual end-to-end 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 end-to-end transmissions in the considered scenario as functions of d and ρ, respectively. Particularly, equal power allocation (EPA) is adopted, i.e., q1 = q2 = q3 = 1/3, besides, M=20 terms truncation in (5) is adopted. In Fig. 3, we plot the exact and asymptotic end-to-end OP as functions of system SNR ρ.

Fig. 2.exact end-to-end outage probability with γth = 1, u = 3, K1 = 3. (a):ρ = 33dB. (b):d = 0.5.

Fig. 3.exact and asymptotic end-to-end outage probability with γth = 1, u = 3, K1 = 3, d = 0.5 with different APs (a): the reverse link. (b): the forward link.

As seen in Fig. 2 and Fig. 3, the Monte-Carlo 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 K1 = 3, the vertical black dotted line at d3 ≈ 0.63 represents a boundary for relay location, where d ˂ d3 guarantees a better performance of the forward link compared with the reverse link. For TWAF with homogenous Rayleigh/Rayleigh fading channels (K1 = 0), we have d0 = 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 dk is associated with K1, generally, dK becomes larger with the increment of K1 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 series-form PDFs of the distribution for Rican and Rayleigh SNRs.

Though not our primary concern, we very briefly look into the topic of optimizing the end-to-end 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 q2, it can be seen that the global optimizer lies on the boundary of –q2 ˂ 0. So we relax the constraint to –q2 ≤ 0, and the optimal solution can be written as

With q2 = 0 the TWAF relay network reduces to a conventional one-way relay network and no information can be acquired at N1, this conflicts with the bidirectional transmission nature of two-way relay networks. However, we see that in high SNR regime the OP performance of individual end-to-end link in TWAF relay network is up-bounded 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 N1 or N2 is in outage. Besides, N1 and N2 may have different traffic requirements. Thus, the overall OP is defined as

where γth1 and γth2 denote the prescribed SNR threshold at N1 and N2, respectively, and define η = γth1 / γth2 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 non-identical traffic requirements, the overall OP of a TWAF relay network with VG policy can be lower bounded by

where λα1 = (1 + K1) / Ω1, ω2 = 1 / Ω2, ζ1 = 1 / (1 + η), ζ2 = η / (1 + η), and Case1, Case2, and Case3 are characterized by q1 ≥ ζ1, q2 ≥ ζ2, and (q1 ˂ ζ1, q2 ˂ ζ2) , respectively.

And in high SNR regime an asymptotic expression of (14) is given by

where ω1 = e–K1(1 + K1) / Ω1.

Proof: Apply the upper bound for end-to-end SNR from Table 3, wet get a lower bound of the OP for the reverse link as

Note that α1 and α2 are mutually independent. For the forward link we have

Note that if q1 ˂ ζ1 , holds and if q2 ˂ ζ2 , holds. Along with the total power constraint of q1 + q2 + q3 = 1, (13) could be rewritten as

Substitute into (18) the CDF expressions for α1 and α2, we directly arrive at (14). It is interesting to see that if q1 ≥ ζ1, the overall OP is determined by the reverse link itself. If q2 ≥ ζ2, the forward link dominates. Only when q1 ˂ ζ1, q2 ˂ ζ2 holds simultaneously, the overall OP is codetermined by both links.

Substitute the well-known small value approximate for I0(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 (q1 ≥ ζ1):

Case2 (q2 ≥ ζ2):

Case3 (q1 ˂ ζ1, q2 ˂ ζ2)

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 ω1. Following standard analysis and applying Lagrange dual method with Karush-Khun-Tucker (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 q1 + q2 + q3 = 1, qi ˃ 0, i = 1,2,3, is given by

where Note that if ω1 = ω2, q*A and q*B reduces to q*A = [ζ1; ζ2 / 2 ; ζ2 / 2 ;], q*B = [ζ1 /2 ; ζ2 ; ζ1 / 2 ;] ■

As a special case, in practical applications where equal source power (ESP) is mandated upon two source nodes, i.e., with an extra q1 = q2 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 Δd / (1 + 2Δd) ˂ ζ2, q*D reduces to q*D = [ζ2; ζ2; 1 - 2ζ2; ]. Similarly, when Δf / (1 + 2Δf) ˂ ζ1, q*F reduces to q*F = [ζ1; ζ1; 1 - 2ζ1; ]. ■

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 Ia. From Table 2, Ia could be expressed as

Similarly, Ib could be expressed as

The integral boundaries in (27), (28) are defined as follows

It is difficult, if not impossible, for Ia and Ib 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, Ia and Ib can be expressed in multi-fold infinite series, which is not quite efficient for mathematical computation. Alternatively, we resort to numerical methods ([21], eq. (25.4.38)) to calculate Ia and Ib.

Take the second term of Ia for example, after simple manipulations we get

Let wn = π / Nt , zn = cos (2n - 1)π/2Nt , cn = sin (2n - 1)π/2Nt   θn = (zn + 1)π/4, xn = tanθn + δ2, (28) could be evaluated as

For space limitations we omit the overall expression for (25). In simulation experiments, the choice of Nt always guarantees the stability of 7th decimal place and it is noted that Nt is affected by the average channel power and path-loss exponent. In general, when the relay is close to source nodes, larger value of Nt 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 non-identical 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.

Fig. 4.Overall OP of TWAF relay network with non-identical traffic requirements, γth1 = 0.5, γth2 = 1, η = 0.5, u = 3, K1 = 3. (a): ρ = 33dB. (b): d = 0.75

In Fig. 5 we plot the overall OP performance of VG and FG policies when h1 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 (K1 = 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. d0 = 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 K1 value, the OP of VG can be times better than FG. The vertical black dotted line dk indicates the relay location where OESP best approaches OPA, i.e., at dk we have qoesp = qopa and it is an interesting turning point in the OP curve. At last, we note that when the relay is located very close to N1, the performance of VG policy with different K1 is not as clearly distinguished from each other as in FG policy. For the VG policy with large K1 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.

Fig. 5.Overall OP of TWAF relay network with identical traffic requirements and different Rician factors, γth1 = γth2 = 1, η = 1, u = 3, ρ = 33dB. (a): With VG policy, K1 = 0,3,5,7. (b): With FG policy, K1 = 0,3,7.

In Fig. 6 we plot the PA schemes presented in closed-form 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, dk is the solution to thus we have dk = 1 / (1 + θ) where And from (12), we see that with EPA, a change of the worse directional transmission link occurs if (2 - η)ω1 = (2η -1)ω2 holds true, with η = 1, this point is also at dk, that is why Fig. 2(a) and Fig. 5 shares the same d3. In this sense dk 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 dk 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.

Fig. 6.PA schemes of TWAF relay network, γth1 = γth2 = 1, η = 1, u = 3, ρ = 33dB. K1 = 3. (a): closed-from 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 end-to-end 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(v1α1 + v2α2)} v1 ˃ 0, v2 ˃ 0 which is needed in the following subsections.

Lemma 3: If α1 is a Rician Power RV with K1 and Ω1, α2 is a Rayleigh Power RV with Ω2, the integral E{ln(v1α1 + v2α2)} v1 ˃ 0, v2 ˃ 0 can be resolved by

where θ = v1ω2 / v2λα1, ω2 = 1 / Ω2, λα1 = (1 + K1) / Ω1 and J(a, x) is given by

Proof : First, perform the integration w.r.t. α2. 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 = v1ω2 / v2λα1. 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 end-to-end 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 F1(x) = ln(1 + ex) / 2ln2, F2(x) = x / 2ln2, c1 = 1 + ρq1Ω1 + ρq2Ω2,

Table 4.Lower bounds and high SNR approximations of channel capacity

P3(a), P4(a), P5 and P6 are defined as

where Δ(a, M, K1) = E1(aλ3)ek1 [γ(M - 1, K1)(M - 1)K1 + γ(M, K1)] / 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 c2 = 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{lnα2} and E{ln(a + α2)}, a ˃ 0 were given in ([10], eq. (9)-(10)). And for Rician fading, E{lnα1} and E{ln(a + α1)}, 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 K1, 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 sub-optimal scheme (sub-OPA) proposed in [10]. Though sub-OPA 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 sub-OPA scheme are very similar with the OPA scheme.

Fig. 7.Sum capacity of TWAF relay network with VG and FG polices. u = 3. (a): d = 0.5. (b): k1 = 3, ρ = 33dB.

Fig. 8.Power allocation schemes via exhaustive search to maximize sum capacity with VG and FG polices. u = 3, k1 = 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 end-to-end 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 non-identical 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 end-to-end 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.