Performance Analysis in DF Based Cooperative SWIPT Networks with Direct Link
Yingting Liu Yinghui Ye Haiyang Ding Jianmei Shen Hongwu Yang

 




Performance Analysis in DF Based Cooperative SWIPT Networks with Direct Link
    Yingting Liu, Yinghui Ye, Haiyang Ding, Jianmei Shen, and Hongwu Yang
This work was supported in part by the National Natural Science Foundation of China (Grant No. 61861041, 61871387 and 11664036), and in part by the National Key Research and Development Plan of China (Grant No. 2018YFE0100500).
Yingting Liu, Jianmei Shen and Hongwu Yang are with the College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou, 730070, China.

Yinghui Ye and Haiyang Ding are with the Integrated Service Networks Lab of Xidian University, Xi'an, China. Haiyang Ding is also with School of Information and Communications, National University of Defense Technology, Xi'an, China. The corresponding author is Yingting Liu (e-mail: liuyt2018@163.com)

    December 30, 2023
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This letter proposes a dynamic power splitting scheme (DPSS) for decode-and-forward (DF) based cooperative simultaneous wireless information and power transfer (SWIPT) networks with direct link. The relay node adopts an optimal dynamic power splitting factor determined by instantaneous channel state information (CSI) to harvest energy and process information. The expressions for the optimal dynamic power splitting factor, outage probability and ergodic capacity of the proposed network are derived. Numerical results show that the proposed scheme is better than or the same as the existing PS schemes in terms of outage probability, while it achieves higher ergodic capacity compared to the existing PS schemes.









§ INTRODUCTION


Compared to the non-cooperative network, cooperative communication can obviously improve system performance. Radio frequency (RF) signals have the ability of carrying energy as well as carrying information at the same time. Using the ability of RF signals, simultaneous wireless information and power transfer (SWIPT) based cooperative communication network has attracted much attention in the literatures, such as [1-6] and references therein.

The authors studied outage probability, ergodic capacity and symbol error rate in one-way [2-3] or two-way [4-6] network. In above literatures, the relay node can use power splitting (PS) scheme and time switching (TS) scheme to harvest energy, and adopt amplify-and-forward (AF) or decode-and-forward (DF) mode to forward signals. In a PS scheme, the received power is divided into two portions in proportion for harvesting energy and processing information simultaneously. This results in a better performance for the PS scheme by comparing to the TS scheme, and thus we focus on the design of PS in this letter

Until now, the design of PS scheme is based on the statistic or instantaneous channel state information (CSI).  We refer the statistic CSI  based PS scheme as a fixed power splitting scheme (FPSS), where the PS factor is constant over all transmission blocks if the statistic CSI remains unchanged [2-5]. On the contrary, we terms the instantaneous CSI based PS as the dynamic power splitting scheme (DPSS), in which the PS factor is constant over a transmission block and may be changed in the next block. The authors in [6, 10] proposed a DPSS, in which the power splitter adjusts its parameter in each transmission based on the instantaneous CSI and the target data rate. As shown in [7], in the cooperative communication, the direct link (DL) between the source node and the destination node can usually supply the additional diversity gain that can enhance the performance of the network. In [2-6], the authors neglected the DL from the source to the destination. The authors in [8] analyzed the average symbol error rate for TS based one-way DF networks with or without DL in Nakagami-m fading channels. The authors in [9] and [10] studied the outage probability of one-way PS based SWIPT network with DL. In [9], the relay adopts FPSS and AF protocol, and in [10], the relay adopts DPSS and DF protocol, respectively. The authors in [11] presented a novel DPSS, in which the relay adjusts the PS factor aiming to maximizing the end-to-end signal to noise ratio (SNR). As described in [11], the existing DPSS for DF based SWIPT network without DL [6, 10] is able to minimize the outage probability but  fails to maximize the ergodic capacity. However, the authors do not consider the effect of the DL. Accordingly, an proper DPSS scheme is required for DF based cooperative SWIPT networks with DL and this motivates this work.

Our contributions of this letter are summarized as follows.


  * We propose a novel DPSS for DF based SWIPT network with DL. We derive the closed-form expression of the optimal dynamic PS factor.

  * The approximate analytical expressions of the outage probability and ergodic capacity are derived. The simulations verify our derived results and show that the proposed scheme can minimize the outage probability and maximize the ergodic capacity at the same time compared to the existing schemes.





§ SYSTEM MODEL







In the proposed DF based cooperative SWIPT network, the source node (denoted by S) transmits its information to the destination node (denoted by D) via the help of an energy constrained relay node (denoted by R). Each transmission T can be divided into two equal time duration slots T/2. The channel coefficients of the corresponding links S→ D, S→ R and R→ D are denoted by h_0, h_1 and h_2, respectively. The channels are modeled as quasi-static, which means the channels keep constant over each transmission but may vary in different transmissions. The channel fading coefficients are assumed to follow Rayleigh distributions and the means of the channel gains |h_0|^2, |h_1|^2 and |h_2|^2 are denoted by 1/λ_0, 1/λ_1 and 1/λ_2. It is assumed that all node are equipped with single antenna [2-6] and the instantaneous CSI can be obtained by channel estimation [6, 10-11].

In the first time slot T/2, S sends the signal to R and D, and the received signals for processing information at R and D can be written as

    y_r(t)=√((1-ρ)P_s)h_1s(t)+n_r(t),


    y_sd(t)=√(P_s)h_0s(t)+n_d(t),

where s(t) denotes the transmitted signal by S and has unit energy. P_s denotes the transmitted power and 0 ≤ρ≤ 1 is the dynamic PS factor determined by instantaneous CSI, which will be detailed in the following. n_r(t) and n_d(t) denote the received additive white Gaussian noises (AWGNs) with zero means and the variances σ_r^2 and σ_d^2 at R and D, respectively. n_r(t) and n_d(t) both consist of the AWGN introduced by the receiving antenna and the AWGN produced by the RF band to baseband conversion. For simplicity, it is assumed that σ^2=σ_r^2=σ_d^2, as in [6, 9-11].

Based on (1) and (2), the SNR in the first slot at R and D can be derived as γ_r=(1-ρ)γ_in |h_1|^2 and γ_sd=γ_in |h_0|^2, respectively, where γ_in=P_s/σ^2 denotes the SNR transmitted by S.
The harvested energy at R is written as E_r=ηρ P_s|h_1|^2T/2. The transmitted power by R in the second time slot can be derived as P_r=ηρ P_s|h_1|^2, where 0<η<1 is the energy conversion efficiency.
The received signal in the second slot at D can be derived as

    y_rd(t)=√(ηρ P_s|h_1|^2)h_2s(t)+n_d(t).

And the corresponding received SNR can be expressed as γ_rd=ηργ_in |h_1|^2|h_2|^2.



§ PS SCHEME DESIGN AND PERFORMANCE ANALYSIS

Adopting the maximal ratio combining (MRC), the achievable data rate at D is given as [7, eq. (15)]

    R=1/2min{log_2(1+γ_r),log_2(1+γ _d)},

where γ _d = γ _sd + γ _rd.
The problem to solve the maximal achievable rate at D is equivalent to following problem, given by

    γ_op = maxmin__0 ≤ρ≤ 1( γ _r,γ _d).



Observing the expressions of γ_r, γ_sd and γ_rd, it is found that γ_r and  γ_d are the decreasing and increasing functions with respect to the dynamic PS factor ρ, respectively.








Lemma 1: The optimal ρ^∗ leading to maximal achievable rate can be derived as

    ρ ^ *  = {[                               0,           |h_1|^2 < |h_0|^2; |h_1|^2-|h_0|^2/|h_1|^2+η|h_1|^2|h_2|^2,     |h_1|^2≥|h_0|^2 ].




Proof: When |h_1|^2<|h_0|^2, the curve of γ_r has no crossing point with respect to that of γ_d, and the optimal received SNR at D is achievable when ρ^∗=0 and equal to γ_op^1=γ_in|h_0|^2. When |h_1|^2≥|h_0|^2, the optimal received SNR at D is achievable by equating γ_r to γ_d. In this case, 0≤ρ^∗=|h_1|^2-|h_0|^2/|h_1|^2+η|h_1|^2|h_2|^2<1. Substituting the value of ρ^∗ into (5), the corresponding maximal SNR can be obtained as γ_op^2=γ_in|h_0|^2+η|h_1|^2|h_2|^2/1+η|h_2|^2.

Remark: As exhibited in [7, eq. (15)] or (4),  the achievable rate is determined by the minimum of γ_r and γ _sd + γ _rd. This  is valid in conventional cooperative communication networks, since all nodes are powered by a battery and all the SNRs, i.e., γ_r, γ_sd and γ_rd, are larger than zero.
However, eq. (4) may be invaild in our considered  DF based cooperative SWIPT network. This is because the PS based relay is powered by the RF signals from the source and the transmit power of such relay may equal zero. For example, when |h_1|^2<|h_0|^2, we have ρ ^ * =0 and γ_rd=ηρ ^ *γ_in |h_1|^2|h_2|^2=0. In such case,  the achievable rate by (4) is  1/2log _2( 1 + γ _in| h_1|^2), which is smaller than that achieved by the DL link 1/2log _2( 1 + γ _in| h_0|^2).
In this case, the proposed model is equivalent to the non-cooperative  transmission and we can obtain γ_op^1=γ_in|h_0|^2.






 §.§ Outage Probability Analysis

The outage probability is defined as the probability that the achievable data rate is below the target data rate R_th. The outage probability of the proposed network can be written as

    P_out=P( γ _op < γ _th) = P_1 + P_2,

where γ_th=2^2R_th-1, P_1 = P( γ _op^1 < γ _th,ρ ^* = 0) and P_2 = P( γ _op^2 < γ _th,ρ ^* = |h_1|^2 - |h_0|^2/|h_1|^2 + η |h_1|^2|h_2|^2).

Let X=|h_0|^2, Y=|h_1|^2 and Z=|h_2|^2, so the corresponding probability destiny functions (PDFs) of the variables X, Y and Z can be written as f_X( x ) = λ _0e^ - λ _0x, f_Y( y ) = λ _1e^ - λ _1y and f_Z( z ) = λ _2e^ - λ _2z.

Accordingly, P_1 can be calculated as

    P_1   = P( γ _in|h_0|^2 < γ _th,|h_1|^2 < |h_0|^2)
       = P( x < a,x > y) = ∫_0^a ∫_y^a λ _0e^ - λ _0xλ _1e^ - λ _1ydxdy
       = ∫_0^a ( e^ - λ _0y - e^ - λ _0a)λ _1e^ - λ _1ydy
       = λ _1/λ _0 + λ _1( 1 - e^ - ( λ _0 + λ _1)a) - e^ - λ _0a( 1 - e^ - λ _1a),

where a=γ_th/γ_in.

On the other hand, P_2 can be calculated as

    P_2   = P( γ _in|h_0|^2 + η |h_1|^2|h_2|^2/1 + η |h_2|^2 < γ _th,|h_0|^2≤ |h_1|^2)
        = P( ( η y - η a)z < a - x,x ≤ y).

Considering the fact X>0, Y>0, Z>0 and the condition X≤ Y, and omitting the intermediate manipulation operations, P_2 can be further derived as

    P_2   = ∫_0^a ∫_0^y λ _0e^ - λ _0xλ _1e^ - λ _1ydxdy_P_21( 0 < x < y < a)
       + ∫_a^∞∫_0^a ( 1 - e^ - λ _2( a - x/η y - η a))λ _0e^ - λ _0xλ _1e^ - λ _1ydxdy_P_22( x < a,y > a).

In (10), the first term P_21 can be calculated as

    P_21 = λ _0/λ _0 + λ _1 + λ _1/λ _0 + λ _1e^ - ( λ _0 + λ _1)a - e^ - λ _1a.

The second term P_22 can be calculated as


    P_22 = P_221 - λ _0λ _1∫_a^∞∫_0^a ( e^ - λ _2( a - x/η y - η a))e^ - λ _0xe^ - λ _1ydxdy_P_222,

where P_221=( 1 - e^ - λ _0a)e^ - λ _1a.

Invoking the change of variables u=y-a, P_222 can be derived as


    P_222 = e^ - λ _1a∫_0^a e^ - λ _0x∫_0^∞e^ - λ _1u - λ _2a - x/η ududx
        = e^ - λ _1a/λ _1∫_0^a e^ - λ _0x√(4( a - x)λ _1λ _2/η)K_1( √(4( a - x)λ _1λ _2/η))dx
       ≈e^ - λ _1a/λ _1∫_0^a e^ - λ _0x( 1 + ( a - x)λ _1λ _2/ηln( ( a - x)λ _1λ _2/η))dx
        = e^ - λ _1a/λ _0λ _1( 1 - e^ - λ _0a) + Φ( ae^λ _0a - 1/λ _0( e^λ _0a - 1))
       + Λ( aln ae^λ _0a - 1/λ _0( e^λ _0a - 1) - ∫_0^a e^λ _0xln xdx).

In (13), we use the integration ∫_0^∞e^ - β/4x - γ xdx = √(β/γ)K_1( √(βγ)) [12, 3.324.1] and in (14), we utilize the fact that a→ 0 in the high SNR regime and the equivalent infinitesimal replacement θK_1( θ) ≈ 1 + θ ^2/2ln( θ/2) when θ→0 [6, 10]. K_1( ·) is the the first order modified Bessel function of the second kind. In (15), Φ  = λ _2e^ - ( λ _0 + λ _1)aln( λ _1λ _2/η)/ηλ _0 and Λ  = λ _2e^ - ( λ _0 + λ _1)a/ηλ _0.

To the best of our knowledge, the integration ∫_0^a e^λ _0xln xdx  shown in the last term of (15) has no closed-form result. For obtaining the closed-form result, we can use the Gaussian-Chebyshev quadrature [10] to approximate the the integration as

    ∫_0^a e^λ _0xln xdx≈a/2ω∑_i = 1^N √(1 - f_i^2)e^λ _0a/2c_iln( a/2c_i),

where ω=π/N, f_i=cos((2i-1)π/2N) and c_i = f_i + 1. N is the parameter determining the tradeoff between the complexity and the accuracy.

Substituting (8)-(16) into (7), the analytical expression of the proposed model for the outage probability can be obtained. For the limit of the page, it is neglected in the letter.

The diversity order of the proposed model can derived as D =  - lim_γ _in→∞log( log( γ _in)/γ _in^2)/log( γ _in) = 2.











 §.§ Ergodic Capacity Analysis

The optimal ergodic capacity of the proposed model can be written as

    C = C_1 + C_2,

where C_1 and C_2 are the ergodic capacities corresponding to the cases ρ ^ * =0 and ρ ^ *=|h_1|^2 - |h_0|^2/η |h_1|^2|h_2|^2 + |h_1|^2, respectively.

C_1 can be derived as (18)-(21), which are exhibited at the top of next page.
mytempeqncnt

Using the integration by parts with respect to the variable x, we can obtain (19). After using the integrations ∫_0^∞e^ - μ xln( 1 + β x)dx =  - 1/μe^μ/βE_i(  - μ/β) [12, 4.337.2] and ∫_u^∞e^ - μ x/x + βdx =  - e^μβE_i(  - μ u - μβ) [12, 3.352.2], (20) can be obtained. First using the change of variables y=tanθ and then using the Gaussian-Chebyshev quadrature versus the second term of (21), we can obtain (21). In (21), ζ =π/M, f_j = cos( ( 2j - 1)π/2M), θ_j = π/4(f_j + 1), and  M is the parameter determining the tradeoff between the calculating complexity and the accuracy.

Substituting γ_op^2 into C_2, C_2 can be derived as (22)-(25), which are detailed at the top of next page. Step (b_1) utilizes the Gaussian-Chebyshev quadrature with respect to the  variable x, and step (b_2) and (b_3) use the Gaussian-Chebyshev quadratures with respect to the variables y and z, after changing the variables of y=tanθ and z=tanθ, respectively. In (23)-(25), ω _1=π/N_1, ω _2=π/N_2, ω _1=π/N_3, f_i_1=cos((2i_1-1)π/2N_1), c_i_1 = f_i_1 + 1, f_i_2=cos((2i_2-1)π/2N_2), θ _i_2 = π/4(f_i_2 + 1),
f_i_3=cos((2i_3-1)π/2N_3), θ _i_3 = π/4(f_i_3 + 1), Φ  = λ _0λ _1πω _1/16 and Φ _i_1i_2i_3 = λ _0λ _1λ _2π ^2/64ω _1ω _2ω _3∑_i_1 = 1^N_1√(1 - f_i_1^2)∑_i_2 = 1^N_2√(1 - f_i_2^2)∑_i_3 = 1^N_3√(1 - f_i_3^2). N_1, N_2 and N_3 are the parameters determining the tradeoff between the calculating complexity and the accuracy.
mytempeqncnt1






§ NUMERICAL RESULTS AND DISCUSSION

This section presents the numerical results to validate above analysis. The parameters are set to as follows in the simulations [9]: η=0.5, 1/λ_0=1, 1/λ_1=5 and  1/λ_2=5.


As shown in Fig.1, both the approximate and accurate results both match well with the simulation results over the entire SNR regime for the proposed scheme. The outage probability of the proposed DPSS is smaller than that of the existing DPSS without DL and the random scheme with DL, and compared to the non-cooperative scheme, the DL indeed improve the performance in the proposed DF based cooperative SWIPT network.

In the existing DPSS in [6, 10], the dynamic PS factor is selected based on the target data rate of the system and the instantaneous CSI and can be written as ρ ^* = max( 1 - γ _th/γ _in| h_1|^2,0) [10, eq.16]. If the selected dynamic PS factor in [6, 10] cannot support the target rate, no other dynamic PS factor can accomplish the transmission at the target rate in the proposed DPSS. So, two scheme have the same performance in terms of outage probability which is verified in Fig.1.


As shown in Fig.2, the proposed DPSS can maximize the received SNR at the destination and utilize the DL, so the proposed scheme has the highest ergodic capacity compared to other existing schemes over the entire SNR regime.



§ CONCLUSIONS

In this letter, a DPSS for the DF based cooperative SWIPT
network with direct link has been proposed. The proposed DPSS can minimize the outage probability and maximize the ergodic capacity simultaneously. Numerical results showed that the proposed scheme has the best performance compared to other existing schemes.

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