# Transmit Power Optimization in Multihop Amplify-and-Forward Relay Systems with Simultaneous Wireless Information and Power Transfer

Kunju Kim, Derek Kwaku Pobi Asiedu, Prince Anokye, Eunkyung Kim, Kyoung-Jae Lee

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Kunju Kim, Derek Kwaku Pobi Asiedu, Prince Anokye, Eunkyung Kim, Kyoung-Jae Lee. Transmit Power Optimization in Multihop Amplify-and-Forward Relay Systems with Simultaneous Wireless Information and Power Transfer. Electronics, 2024, 13 (21), pp.4232. 10.3390/electronics13214232. hal-04910052

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Article

# Transmit Power Optimization in Multi-hop AF Relay Systems with Simultaneous Wireless Information and Power Transfer

Kunju Kim ¹D, Derek Kwaku Pobi Asiedu ²D, Prince Anokye ²D, Eunkyung Kim ³D, and Kyoung-Jae Lee <fnref n="1" />,*

**Abstract:** In this paper, we study a multi-hop amplify-and-forward (AF) simultaneous wireless information and power transmission (SWIPT) relay system. Each relay node harvests power using a power split (PS) method from a portion of the received signal, amplifies the remaining received signal, and passes it to the next relay. Based on this system model and signal flow, we derived and solved the convex power minimization problem with the optimal PS ratio. In this case, it was found that using the optimal PS ratio consumed lower power than when using a fixed PS ratio (0.5). We then investigated the impact of processing cost on the AF-SWIPT system using decoding and forwarding SWIPT as benchmarks, and found that AF-SWIPT was superior.

**Keywords:** multi-hop amplify-and-forward (AF) relays; power splitting (PS) ratio; simultaneous wireless information and power transfer (SWIPT)

## 1. Introduction

The Internet-of-Things (IoT) connects network-enabled devices communicating with each other over the Internet. Hence, the objective of IoT is to integrate the physical world and the virtual world to attain a self-sustaining system [1]. Currently, research into the application of IoT for the creation of smart-cities, smart-homes, smart-energy, intelligent-transportation system, and many more is growing. An IoT network consists of the use of routing schemes, a gateway supporting communication between the different nodes and a central system [1]. These routing schemes facilitate either direct or relaying communication between a gateway and a particular node. As demand for data service provision rapidly increases worldwide, cooperative communication through relays has recently been recommended as an effective solution. The use of cooperative relay systems significantly enhances the reliability of data transmission [4], [5]. Additionally, relay technology is applied to effectively minimize path loss, expanding the system's coverage area and capacity [6]. The relaying nodes can be opportunistically implemented by non-active user devices [1]. This results in the selected user nodes consuming their power in facilitating communication between the two user pairs [2], [3]. This implies that, the selected users sacrifice their resources to facilitate relaying procedures [2], [12]. The strain on the relay nodes can be mitigated by employing wireless power transfer (WPT) [1]-[3].

Recently, research into alternative wireless sources of energy (e.g., radio frequency (RF) and light energy harvesting) to power IoT devices is also increasing [1]-[11]. The WPT can be accomplished by using two different techniques, namely, simultaneous wireless information and power transfer (SWIPT), and wireless powered communication network (WPCN) [1]-[3], [8].

SWIPT is a technology proposed to extend the life of power-constrained wireless networks and provide an alternative solution for battery-operated devices with limited

**Citation:** Kim, K.; Asiedu, D.K.P.; Anokye, P.; Kim, E.; Lee, K.-J. Transmit Energy Optimization in Multi-hop AF Relay Systems with Simultaneous Wireless Information and Power Transfer. *Electronics* **2024**, *1*, 0. https://doi.org/

Received:

Revised:

Accepted:

Published:

**Copyright:** © 2024 by the authors. Submitted to *Electronics* for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

<footnote n="1">¹ Department of Electronic Engineering, Hanbat National University, Daejeon, South Korea; golho2006@naver.com (K.K.)</footnote>

<footnote n="2">² IMT Atlantique, Brest 29200, France; derek.asiedu@imt-atlantique.fr (D.K.P.A.); prince.anokye@imt-atlantique.fr (P.A.)</footnote>

<footnote n="3">³ Department of Artificial Intelligence Software, Hanbat National University, Daejeon, South Korea; ekim@hanbat.ac.kr (E.K.)</footnote>

<footnote>* Correspondence: kyoungjae@hanbat.ac.kr; Tel.: +82-42-821-1730</footnote>
lifespan [13]. SWIPT involves the transmission of wireless information signal and wire-
less power signal concurrently [1]-[3], [8]. SWIPT can benefit significantly from power
consumption, spectrum efficiency, interference management, and transmission latency by
enabling simultaneous transmission of power and information [14], [19]. Thus, SWIPT
is an essential solution for beyond fifth-generation (B5G), providing the energy needed
to wirelessly charge energy-limited devices and transmit and receive information [15].
Furthermore, SWIPT can be effectively integrated with B5G small cell systems, includ-
ing Millimeter-Wave (mmWave) technology and massive multiple-input multiple-output
(MIMO) systems, to enhance both throughput and energy efficiency [9], [10]. To implement
the SWIPT system, time switching (TS) and power splitting (PS) are the two main tech-
niques [1]-[3]. Zhou et al. [16] and Zhang and Ho [17] investigated both PS and TS methods
and compared the performance of these two methods.

The successive transmission of wireless information signal and wireless power signal is used to accomplish WPCN [1]-[3]. WPCN can be deployed efficiently to power multiple communication devices with different physical conditions and service requirements. Moreover, RF-enabled wireless energy transfer also allows information to be transmitted along with energy using the same waveform [18]. Compared with existing battery-powered networks, WPCN does not require manual battery replacement/charging, which can effectively reduce operating costs and improve communication performance [22].

Research on RF-based WPT has evolved from point-to-point systems, dual-hop co-
operative systems to multi-hop cooperative systems. A few literature on both dual-hop
and multi-hop cooperative systems can be found in [1]-[3]. Asiedu et al. [3] considered
a single-antenna amplify-and forward (AF) dual-hop configuration with multiple relay
nodes. Both the power control elements and the PS ratio of each relay node have been
optimized with the goal of maximizing the achievable speed. In [1], a single-antenna
multi-hop decode-and-forward (DF) SWIPT system was optimized. Both transmission
power and data rate were optimized based on PS ratio.

In this paper, we extend the work in [1] which focused on a DF-SWIPT multi-hop
system model to an AF-SWIPT multi-hop system model. We consider single-antenna relay
nodes, SWIPT PS mode at each node, unlike [1] each relay node operated as AF mode.
And we derive the transmit power optimization problem based on PS scheme. Finally, the
closed-form solution are obtained for our optimization problem. In the simulation results,
we compare AF-SWIPT multi-hop systems to the DF-SWIPT in [1].

The rest of the paper is organized as follows, Sections 2, 3, and 4 contains the the step-
wise process in formulating the AF-SWIPT multi-hop optimization problem, the solutions
for our optimization problem, and simulation results and discussion, respectively. Finally,
concluding remarks are provided in Section 5.

## 2. System Model and Problem Formulation

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Multi-hop AF relay systems with SWIPT architecture.</figcaption></figure>
Consider the system model shown in Figure 1, which consists of a single base station (BS), a single destination and K relay nodes. All nodes have single antennas. The BS node, i.e., communicates with the destination node through multi-hop relays. Each relay node uses the AF protocol to forward its signals to the next node. We assume that each relay does not have a dedicated power but utilizes energy harvest to forward its signals during coherence time. The relay nodes operate according to the SWIPT technique, where the received signal is split into information and energy harvesting (EH) parts. In this work, the PS ration technique is adopted for the SWIPT EH process at the multi-hop nodes. The relays store the harvested energy in their batteries for signal retransmission.

<figure><img src="image_3.png" /><figcaption>Figure 2. Multi-hop AF relay node SWIPT architecture.</figcaption></figure>

The structure of our AF-SWIPT relay is shown in Figure 2. Each relay node has a battery that can store energy harvested from the received RF signal and use it for only retransmit the relaying data after amplification. The signal transmitted to the relay node is divided into an information part and an EH part by the PS ratio $\rho_k$, ($0 \le \rho_k \le 1$).

We assume that the source node knows the channel state information (CSI) for all communication nodes. However, each relay and the destination node only know the CSI of their own communication channel. It is assumed that there is no direct communication link between the current node and the next second or more nodes. For example, there is no direct link between the first node and the third node.

The received RF signal at node ($k, \forall k = 1, \dots, K+1$) from the previous node is given as

$$y_k = h_k x_{k-1} + n_k, \quad (1)$$

where $h_k$ is the channel coefficient between the current node and the previous node, and $n_k \sim CN(0, \delta_k^2)$ represents the antenna noise at the current node. The channel $h_k$ is defined as $h_k = \sqrt{\zeta_k \tilde{h}_k}$, where $\zeta_k = G_k(d_k/d_0)^{-\alpha_k}$ is the large-scale fading coefficient, $G_k$ is the attenuation constant at a reference distance $d_0$, $\alpha_k$ is the pathloss exponent, $d_k$ is distance between the transmit and receive nodes, $\tilde{h}_k \sim CN(0,1)$ is the Rayleigh fading component. The received signal at the kth relay node is split into EH and information signal which are written, respectively, as

$$y_k^{EH} = \sqrt{\rho_k} y_k, \quad (2)$$

$$y_k^{ID} = \sqrt{1 - \rho_k} y_k, \quad (3)$$

And the harvested energy at the kth node and the destination node are given by, respectively,

$$E_k = \beta_k \rho_k |h_k|^2 E_{k-1}, \quad (4)$$

$$E_{K+1} = \beta_{K+1} \rho_{K+1} |h_{K+1}|^2 E_K - P_c, \quad (5)$$
where $P_c$ is the processing power for information decoding at destination node, $E_0$ is the transmit energy at source node, $0 \le \beta_k \le 1$ is the energy conversion efficiency for kth node, and signal for information transmission at the kth relay is represented by

$$y_k^{TR} = \sqrt{1 - \rho_k} y_k + z_k, \qquad (6)$$

where $z_k \sim CN(0, \sigma_k^2)$ is the additional noise introduced by the information decoding (ID) circuitry. The signal-to-noise ratio (SNR) is expressed as

$$\gamma_k = \frac{\prod_{j=1}^{k-1} \beta_j \rho_j \prod_{j=1}^{k} (1 - \rho_j) |h_j|^2 E_0}{\sum_{i=1}^{k} [((1 - \rho_i) \delta_i^2 + \sigma_i^2) \prod_{j=1}^{k-1} \beta_j \rho_j \prod_{j=1, j \neq i}^{k} (1 - \rho_j) \prod_{j=i+1}^{k} |h_j|^2]}, \qquad (7)$$

The signal to be transmitted to the kth relay is represented by

$$x_k = g_k y_k^{TR}, \qquad (8)$$

where $g_k = \sqrt{\frac{E_k}{(1-\rho_k)E_{k-1}|h_j|^2}} \approx \sqrt{\frac{\beta_k \rho_k}{1-\rho_k}}$ is amplification factor [31], and $x_0$ is the information signal at source node.

We now consider the problem of optimizing the PS ratio $\{\rho_k\}_{k=1}^K$ at the relay nodes and energy at the source node $E_0$. We aim to minimize the source transmit energy subject to the destination node SNR and PS ratio constraints. Consequently, we formulate the optimization problem as

$$\begin{array}{ll}
\min_{E_0, \{\rho_k\}_{k=1}^K} & E_0 \\
\text{subject to} & \gamma_{K+1} \ge \tilde{\gamma}_{K+1} \\
& 0 \le \rho_k \le 1, \quad k = 1, \dots, K+1 \\
& E_0 \ge 0,
\end{array}
\qquad (9)$$

The first constraint means that the SNR threshold value at the destination is greater or equal than SNR value at the destination node.

### 3. Problem Solution

In this section, we provide the solution for the source transmit power minimization problem. The optimal transmit energy at the source node is determined to be

$$E_0 = \tilde{\gamma}_{K+1} \frac{\sum_{i=1}^{K+1} [((1-\rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \rho_j \prod_{j=1, j \neq i}^{K+1} (1-\rho_j) \prod_{j=i+1}^{K+1} |h_j|^2]}{\prod_{j=1}^{K} \beta_j \rho_j \prod_{j=1}^{K+1} (1-\rho_j) |h_j|^2}, \qquad (10)$$

where $\tilde{\gamma}_{K+1}$ is the SNR threshold constraint of the destination node. The optimal PS ratio at the kth node is written as

$$\rho_k = \frac{C_k + \sum_{i=k+1}^{K} D_i - \sqrt{F_k \sum_{i=k+1}^{K+1} D_i}}{C_k + \sum_{i=k+1}^{K} D_i - F_k}, \quad k = 1, \dots, K \qquad (11)$$

where $C_k$, $D_i$, and $F_k$ are defined as

$$C_k = ((1 - \rho_{K+1}) \delta_{K+1}^2 + \sigma_{K+1}^2) \prod_{j=k}^{K} (1 - \rho_{j+1}), \qquad (12)$$
$$D_i = ((1 - \rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \prod_{j=k+1, j \neq i}^{K+1} (1 - \rho_j) \prod_{j=i}^{K} \rho_j |h_{j+1}|^2, \quad (13)$$

$$F_k = \prod_{j=k}^{K} \beta_j \prod_{j=k+1}^{K+1} (1 - \rho_j) |h_j|^2 \prod_{j=k+1}^{K} \rho_j \sigma_k^2, \quad (14)$$

Proof: The proof in acquiring the optimal values is provided in the appendix A.

## 4. Simulation Results

This section contains the evaluation of the performance of the AF-SWIPT system compared with the DF-SWIPT system as benchmark [1]. In the simulation setup, for the large-scale fading component, the attenuation constant $G = G_0 = \dots = G_K = -10$ dB, and the pathloss exponent $\alpha = 3$ are assumed. The distance between each node is $d = d_1 = \dots = d_K = 2$ m, the antenna noise variance $\sigma^2 = \sigma_1^2 = \dots = \sigma_{K+1}^2 = -80$ dBm, and the energy conversion efficiency $\beta = \beta_1 = \dots = \beta_K = 0.7$. We assume that $P_c = 0$ W at destination node, because the destination node possesses enough power for decoding the received signal. The PS ratio used for the suboptimal scheme is $\rho_k = 0.5$. The simulation results are obtained over $10^4$ random channel realizations.

<figure><img src="image_4.png" /><figcaption>Figure 3. The influence of increasing each relay $P_c$ ($K = 3, \bar{\gamma} = -5$ dB).</figcaption></figure>

Figure 3 shows the effect of each relay circuit power $P_c$ on $E_0$. When the PS ratio is fixed, the performance is the same when the relay circuit power $P_c$ is very low, but from -80 dBm, the transmission power $E_0$ of AF-SWIPT is lower than that of DF-SWIPT. In the case of optimal PS ratio, DF-SWIPT has lower transmission power than AF-SWIPT for very low $P_c$. However, above -70 dBm, the trend reverses. Through this, it can be seen that above a certain $P_c$, the performance of AF-SWIPT is better than DF-SWIPT in terms of transmission power.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Average $E_0$ against the number of relay nodes at $\tilde{\gamma} = -5$ dB, $P_c = -50$ dBm.</figcaption></figure>

Figure 4 shows a plot of the average power $E_0$ against the number of relays $K$ for different circuit powers $P_c$. When $P_c$ is 0, there is little difference in performance between both DF-SWIPT and AF-SWIPT, but when $P_c$ is -50 dBm, the transmission power of DF-SWIPT is actually higher than that of AF-SWIPT. This shows the same trend for all number of relays. This is because there is no signal decoding (i.e., processing power) at the relays for AF-SWIPT, unlike in the DF-SWIPT. In the AF-SWIPT signal decoding occurs only at the destination. Therefore, processing power is used only at the destination for the AF-SWIPT, while processing power is used at all nodes in the DF-SWIPT.

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Average $E_0$ relative to $\tilde{\gamma}$ at $K = 3, P_c = -40$ dBm.</figcaption></figure>

Figure 5 shows a plot of the average $E_0$ versus the SNR threshold $\tilde{\gamma}$ (in dB). Similar to the previous results, when $P_c$ is 0, the transmission power of DF-SWIPT and AF-SWIPT are similar, but when $P_c$ is -40dBm, DF-SWIPT consumes more power than AF-SWIPT. We can
also see that the lower the SNR threshold, the larger the transmit power difference between DF-SWIPT and AF-SWIPT.

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Average $E_0$ relative to $d_k$ at $K = 3, P_c = -50$ dBm.</figcaption></figure>

Figure 6 shows the $E_0$ against the internode distance, $d_k$. As $d_k$ increases, more transmission power is required. Additionally, when $P_c$ is -50 dBm, it can be seen that AF-SWIPT consumes less power than DF-SWIPT.

## 5. Conclusion

This paper investigated an AF-SWIPT multi-hop cooperative relaying in IoT networks. The AF-SWIPT source energy minimization optimization problem is presented, studied and evaluated in this paper. The reason behind the energy minimization is to promote energy efficiency during signal transmission and reduce energy cost and lengthen network lifespan in the netowrk operation. When considering computational overhead and operational cost, the AF-SWIPT is shown to be more efficient compared to the DF-SWIPT benchmark. Simulation results showed the AF-SWIPT has better energy efficiency in terms of decoding cost compared to the DF-SWIPT. Possible extensions of this research are in the area of implementing the time-switching SWIPT protocol, MIMO multi-hop systems, WPCN multi-hop networks and development of routing algorithms for the multi-hop wireless powered networks.

**Author Contributions:** Conceptualization, D.K.P.A. and K.K.; methodology, K.K.; software, K.K.; validation, D.K.P.A. and K.-J.L.; writing—original draft preparation, K.K. and D.K.P.A.; writing—review and editing, P.A., E.K., and K.-J. L.; visualization, K.K.; supervision, K.-J.L.; All authors have read and agreed to the published version of the manuscript.

**Conflicts of Interest:** The authors declare no conflicts of interest.

## Appendix A.

Problem 9 is nonconvex with respect to $E_0$ and $\{\rho_k\}_{k=1}^K$. We rewrite problem 9 into an equivalent convex problem given as
$$
\begin{array}{ll}
\displaystyle \min_{Q, \{\rho_k\}_{k=1}^K} & 1/Q \\
\text{subject to} & \gamma_{K+1} \ge \tilde{\gamma}_{K+1} \\
& 0 \le \rho_k \le 1, \quad k=1, \dots, K+1 \\
& Q \ge 0
\end{array}
\qquad \text{(A1)}
$$

where $Q = (1/E_0)$. The Lagrangian of Problem A1 is defined with its KKT conditions given as

182

$$
L\{Q, \lambda\} = \frac{1}{Q} + \lambda \left( \tilde{\gamma}_{K+1} - \frac{\prod_{j=1}^{K} \beta_j \rho_j \prod_{j=1}^{K+1} (1-\rho_j) |h_j|^2}{Q \sum_{i=1}^{K+1} [((1-\rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \rho_j \prod_{j=1, j \neq i}^{K+1} (1-\rho_j) \prod_{j=i+1}^{K+1} |h_j|^2]} \right), \quad \text{(A2)}
$$

$$
\begin{equation}
\begin{split}
\frac{\partial L}{\partial Q} = -\frac{1}{Q^2} &+ \lambda \left( \tilde{\gamma}_{K+1} + \frac{\prod_{j=1}^{K} \beta_j \rho_j \prod_{j=1}^{K+1} (1-\rho_j) |h_j|^2}{Q^2 \sum_{i=1}^{K+1} [((1-\rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \rho_j \prod_{j=1, j \neq i}^{K+1} (1-\rho_j) \prod_{j=i+1}^{K+1} |h_j|^2]} \right) = 0,
\end{split}
\tag{A3}
\end{equation}
$$

$$
\lambda \left( \tilde{\gamma}_{K+1} - \frac{\prod_{j=1}^{K} \beta_j \rho_j \prod_{j=1}^{K+1} (1-\rho_j) |h_j|^2}{Q \sum_{i=1}^{K+1} [((1-\rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \rho_j \prod_{j=1, j \neq i}^{K+1} (1-\rho_j) \prod_{j=i+1}^{K+1} |h_j|^2]} \right) = 0, \quad (A4)
$$

From A3, we can calculate $\lambda$ as

185

$$
\lambda = \frac{\sum_{i=1}^{K+1} [((1-\rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \rho_j \prod_{j=1, j \neq i}^{K+1} (1-\rho_j) \prod_{j=i+1}^{K+1} |h_j|^2]}{\prod_{j=1}^{K} \beta_j \rho_j \prod_{j=1}^{K+1} (1-\rho_j) |h_j|^2}, \quad (A5)
$$

and substituting A5 into A3, results in the equation,

186

$$
\frac{1}{Q} = \tilde{\gamma}_{K+1} \frac{\sum_{i=1}^{K+1} [((1-\rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \rho_j \prod_{j=1, j \neq i}^{K+1} (1-\rho_j) \prod_{j=i+1}^{K+1} |h_j|^2]}{\prod_{j=1}^{K} \beta_j \rho_j \prod_{j=1}^{K+1} (1-\rho_j) |h_j|^2}, \quad (A6)
$$

Next, Problem A1 is changed into a convex problem with respect to $\rho_k$, and introduce new variables $x, y$. The new problem is defined as

187

188

$$
\begin{array}{ll}
\displaystyle \min_{x,y, \{\rho_k\}_{k=1}^K} & \frac{x^2}{y} \\
\text{subject to} & \tilde{\gamma}_{K+1} \sum_{i=1}^{K+1} [((1-\rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \rho_j \prod_{j=1, j \neq i}^{K+1} (1-\rho_j) \prod_{j=i+1}^{K+1} |h_j|^2] \ge x^2 \\
& \prod_{j=1}^{K} \beta_j \rho_j \prod_{j=1}^{K+1} (1-\rho_j) |h_j|^2 \ge y,
\end{array}
$$
(A7)

and we formulate the Lagrangian of changed problem A7 as

$$
\begin{equation}
\begin{aligned}
L\{x, y, \{\rho_k\}_{k=1}^K, \lambda_1, \lambda_2\} &= \frac{x^2}{y} + \\
&\quad \lambda_1 \left( x^2 - \bar{\gamma}_{K+1} \sum_{i=1}^{K+1} \left[ ((1-\rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^K \beta_j \rho_j \prod_{j=1, j \neq i}^{K+1} (1-\rho_j) \prod_{j=i+1}^{K+1} |h_j|^2 \right] \right) \\
&\quad + \lambda_2 \left( y - \prod_{j=1}^K \beta_j \rho_j \prod_{j=1}^{K+1} (1-\rho_j) |h_j|^2 \right),
\end{aligned}
\tag{A8}
\end{equation}
$$

The KKT conditions are represented as

$$
\frac{\partial L}{\partial \rho_k} = \bar{\gamma}_{K+1} \left( -\delta_k^2 A_k - \sum_{i=k+1}^{K+1} B_i \frac{1}{\rho_k} + B_k \frac{1}{1-\rho_k} \right) = 0, \quad k=1, \dots, K \quad (A9)
$$

where $A_k$ and $B_k$ are defined as follows

$$
A_k = \frac{1}{\prod_{j=1}^{k-1} \beta_j \rho_j (1 - \rho_j) \prod_{j=1}^{k} |h_j|^2}, \quad (A10)
$$

$$
B_k = ((1 - \rho_k)\delta_k^2 + \sigma_k^2)A_k, \quad (A11)
$$

From A9, we can find the each optimal PS ratios as

$$
\rho_k = \frac{C_k + \sum_{i=k+1}^{K} D_i - \sqrt{F_k \sum_{i=k+1}^{K+1} D_i}}{C_k + \sum_{i=k+1}^{K} D_i - F_k}, \quad k = 1, \dots, K \quad (A12)
$$

$$
\rho_{K+1} = 0, \tag{A13}
$$

where the constants $C_k$, $D_i$, and $F_k$ are defined as

$$
C_k = ((1 - \rho_{K+1})\delta_{K+1}^2 + \sigma_{K+1}^2) \prod_{j=k}^{K} (1 - \rho_{j+1}), \quad (A14)
$$

$$
D_i = ((1 - \rho_i)\delta_i^2 + \sigma_i^2) \prod_{j=i}^{K} \beta_j \prod_{j=k+1, j \neq i}^{K+1} (1 - \rho_j) \prod_{j=i}^{K} \rho_j |h_{j+1}|^2, \quad (A15)
$$

$$
F_k = \prod_{j=k}^{K} \beta_j \prod_{j=k+1}^{K+1} (1 - \rho_j) |h_j|^2 \prod_{j=k+1}^{K} \rho_j \rho_k^2. \quad (A16)
$$

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