Dual-hop Relaying Networks for Short-Packet URLLCs: Performance Analysis and Optimization

I. INTRODUCTION

IN the dramatical development of emerging communication systems, the technologies that support the fifth-generation and beyond networks are positive factors in realizing the era of the Internet of things and hyperlinked society by enabling enhanced mobile broadband, massive machine-type communications, and ultra-reliable low-latency Communications (uRLLCs) [1]–[3]. Known as one of the key services, uRLLC requires the reliability more than 99.9999%, which refers to packet error rate lower than [TeX:] $$10^{-5}$$ [4], [5], and latency around 1 − 10 ms [6]. Inspired by these requirements, shortpacket communications (SPCs) with finite-blocklength have become a potential solution, where some pioneering frameworks on SPCs have been considered in the literature. For example, Polyanskiy et al. in [7] explored the approximation of the maximum achievable rates as a function of error probability and blocklength. Meanwhile, Yang et al. in [8] determined the appropriate blocklength for the best ergodic capacity. Motivated by the pioneering works, the outstanding performance and capability of SPCs to support the aforementioned uRLLC requirements has been well confirmed with the integrations of various technologies and network models, such as the nonorthogonal multiple access [9], energy harvesting [6], and the multiple-input multiple-output (MIMO) systems [10], etc.

Besides that, the relaying system is known as an effective solution to extend coverage, limit transmit power support, and improve the efficiency of data transmission by using intermediate relaying nodes to support information transmission and reception over short distances [11]. Recently, many scientists have paid more attention to work on SPCs in relay networks to enhance system performance and fulfill service requirements for 5G and beyond networks. Specifically, Yifan Gu et al. in [12] compared the performance of full-duplex relaying (FDR) and half-duplex relaying (HDR) dual-hop networks under SPCs in terms of the block error rate (BLER). The results of [12] have revealed that the FDR scheme achieves better performance than that of HDR with less stringent BLER requirements and low transmit power constraints. in [13], a dual-hop hybrid automatic repeat request IoT network was investigated with SPCs, where the users utilized either selection combining or maximum ratio combining (MRC) to detect the signals. Analyzing results in [13] have confirmed that the MRC scheme achieves superior performance compared to the selection combining scheme with big-enough transmit power and a long-enough blocklength. Subsequently, in [14], Makki et al. considered SPCs in multi-relay networks, where relays applied either amplitude-and-forward (AF) or decodeand- forward (DF) schemes to transmit and process the information. It is shown that the end-to-end (e2e) throughput of the DF scheme achieves better performance than that of the AF scheme when increasing the number of hops in a wider range of decoding delays.

A joint optimization investigation of power allocation (PA) and relay location (RL) configurations for multihop MIMO relay networks with SPCs was provided in [15]. Meanwhile, the work [16] considered the partial relay-selection (PRS) technique, in which the best relay was selected to forward the signal to the destination. The results in [16] have yielded a tradeoff that increasing SNR and blocklength improve system performance while increasing latency as the longer packets.

In this paper, to leverage the above aforementioned benefits of the relay networks, we first study a DF multi-relay scheme in a dual-hop network with SPCs to support not only ultrareliability but also low-latency, where (i) the transmit antenna selection (TAS) scheme is conducted for the signal processing of multiple transmit antennas of the source, (ii) the PRS technique is utilized to determine the best relay having the highest SNR among multiple relays, and (iii) the destination uses the MRC technique for multiple installed antennas to achieve the highest diversity gain. Our main contributions are summarized as follows:

We derive the approximated and asymptotic closed-form expressions of the e2e BLERs for both the selective DF (SDF) and fixed DF (FDF) schemes in the finiteblocklength regime and quasi-static Rayleigh fading channels. The result indicates that the BLERs of the two schemes are identical in the average and high-SNR regions but distinguished in the low-SNR regime.

We also investigate the performance in terms of e2e latency and system throughput to provide a comprehensive overview of not only ultra-reliability but also low-latency standpoints.

We propose two asymptotic approaches for the e2e BLER, in which the second asymptotic approach is a way to validate the correctness of the first approach. This reveals that both of the asymptotic approaches for the e2e BLER perfectly match with each other.

We analyze two important practical optimization problems, including PA and RL, which are obtained in the closed-form expressions for the designed configurations.

The remaining of this paper is organized as follows. In Section II, we describe the proposed system model and derive the preliminary essential cumulative distribution function (CDF) forms. The BLER analysis of the proposed system model with both approximated and asymptotic expressions is presented in Section III. Subsequently, the optimization problems of PA and RL are analyzed in Section IV. Numerical and simulation results are discussed in Section V. Eventually, the conclusion is provided in Section VI.

II. SYSTEM MODEL

As illustrated in Fig. 1, we consider a dual-hop network consisting of a based station performed as a source [TeX:] $$(\mathcal{S})$$ equipped with antennas [TeX:] $$\mathcal{S}_{t}(t=1,2, \cdots, T), N$$ single-antenna relays [TeX:] $$\mathcal{R}_{n}(n=1,2, \cdots, N)$$ and a destination [TeX:] $$(\mathcal{D})$$ installed with antennas [TeX:] $$\mathcal{D}_{m}(m=1,2, \cdots, M)$$¹ We assume that there is no direct link between the source and destination. The system operates in a time-division multiple-access method with two consecutive time slots via the help of the relays.

¹ The value of [TeX:] $$t=1,2, \cdots, T, n=1,2, \cdots, N, \text { and } m=1,2, \cdots, M$$ will be used throughout this paper if we do not specifically mention.

In the first time slot, the source transmits its signals to all relays, where the TAS and PRS techniques are simultaneously utilized. Based on the TAS principle, the system selects only one optimal transmit antenna, which maximizes the received SNR, out of T antennas of the source to perform the transmission. We note that the TAS scheme on the transmit side can achieve a significant reduction in hardware cost and power consumption because only a single radio-frequency chain is used. Subsequently, the PRS scheme is used with the principle that only the relay having the highest SNR is selected as a forwarder for the next hop. Denoting [TeX:] $$\mathcal{R}_{\mathrm{b}}$$ as the selected relay after utilizing the TAS and PRS schemes [17], [18], we have can achieve a significant reduction in hardware cost and power consumption because only a single radio-frequency chain is used. Subsequently, the PRS scheme is used with the principle that only the relay having the highest SNR is selected as a forwarder for the next hop. Denoting [TeX:] $$\mathcal{R}_{\mathrm{b}}$$ as the selected relay after utilizing the TAS and PRS schemes [17], [18], we have

where [TeX:] $$\gamma_{S \mathcal{R}_{\mathrm{b}}}$$ denotes the output SNR in the first hop and [TeX:] $$\gamma_{\mathcal{S} \mathcal{R}_{t, n}}$$ is the SNR of the link from [TeX:] $$\mathcal{S}_{t} \rightarrow \mathcal{R}_{n}.$$ Here, [TeX:] $$\gamma_{\mathcal{S} \mathcal{R}_{t, n}}$$ follows the exponential distribution with the average SNR being calculated as

where [TeX:] $$P_{\mathcal{S}}$$ is the transmit power of [TeX:] $$\mathcal{S}, h_{t, n} \text { and } \Omega_{\mathcal{S} \mathcal{R}_{t, n}}$$ are the channel coefficient and average channel gain of the link from [TeX:] $$\mathcal{S}_{t} \rightarrow \mathcal{R}_{n},$$ respectively, and [TeX:] $$\sigma^{2}$$ denotes the additive white Gaussian noise variance. The CDF of [TeX:] $$\gamma_{\mathcal{S} \mathcal{R}_{\mathrm{b}}}$$ is given as

We assume that the channel coefficients in each hop are quasi-static block-fading channels [8], where channel fading coefficients remain the constant value during each transmission block and change independently in the another transmission. The independent and identically distributed (i.i.d.) channels are also assumed with the links of [TeX:] $$\mathcal{S}_{t} \rightarrow \mathcal{R}_{n} \text { and } \mathcal{R}_{\mathrm{b}} \rightarrow \mathcal{D}_{m}.$$ In such a scenario, the average SNRs for all of the branches [TeX:] $$\bar{\gamma}_{\mathcal{S} \mathcal{R}_{t, n}}$$ are the same, i.e., [TeX:] $$\bar{\gamma}_{S \mathcal{R}_{t, n}}=\bar{\gamma}_{S \mathcal{R}}.$$ As a result, the CDF of the output SNR in the first hop [TeX:] $$\gamma_{\mathcal{S} \mathcal{R}_{\mathrm{b}}}$$ is given by [18, Eq. (3)]

In the second time slot, the best relay node uses DF technique to decode successfully the received signal and transfer this signal to the destination [TeX:] $$\mathcal{D}.$$ At the destination [TeX:] $$\mathcal{D},$$ the spatial diversity MRC protocol is investigated, i.e., it combines all channels [TeX:] $$\mathcal{R}_{\mathrm{b}} \rightarrow \mathcal{D}_{m}$$ at the destination. Therefore, the output SNR at the destination will increase linearly with the number of diversity branches. As a result, the SNR at the output of the MRC selector is determined by

where [TeX:] $$\gamma_{\mathcal{R}_{\mathrm{b}} \mathcal{D}_{m}}$$ is the received SNR at the antenna [TeX:] $$\mathcal{D}_{m}.$$ Under i.i.d. Rayleigh fading assumption for the link of [TeX:] $$\mathcal{R}_{b} \rightarrow \mathcal{D}_{m},$$ we also have [TeX:] $$\bar{\gamma}_{\mathcal{R}_{\mathrm{b}} \mathcal{D}_{m}}=\bar{\gamma}_{\mathcal{R}_{\mathrm{b}} \mathcal{D}}=P_{\mathcal{R}_{\mathrm{b}}} \Omega_{\mathcal{R}_{\mathrm{b}} \mathcal{D}} / \sigma^{2} \text {, where } P_{\mathcal{R}_{\mathrm{b}}}$$ is the transmit power at [TeX:] $$\mathcal{R}_{\mathrm{b}} \text { and } \Omega_{\mathcal{R}_{\mathrm{b}} \mathcal{D}}$$ is the average channel gain of the link from [TeX:] $$\mathcal{R}_{\mathrm{b}} \rightarrow \mathcal{D}.$$ We note that [TeX:] $$\left|g_{m}\right|^{2}$$ follows the exponential distribution so that [TeX:] $$\gamma_{\mathcal{R}_{\mathrm{b}}} \mathcal{D}_{m}$$ follows the chisquared distribution with a mean of [TeX:] $$M \bar{\gamma} \mathcal{R}_{\mathrm{b}} \mathcal{D}$$ and variance 2M [TeX:] $$\bar{\gamma} \mathcal{R}_{\mathrm{b}} \mathcal{D}$$ [19]. Accordingly, the CDF of [TeX:] $$\gamma_{\mathcal{R}_{\mathrm{b}} \mathcal{D}}$$ is given by

III. PERFORMANCE ANALYSIS

In this section, we derive the approximated closed-form expression for the e2e BLER under the finite-blocklength regime. The asymptotic analysis are further studied to gain more insights into the performance behavior in the high-SNR regime. Furthermore, the performance in terms of e2e latency and throughput is also considered.

A. Average BLER

We assume that each transmission blocklength is k channel uses (CUs). When [TeX:] $$\mathcal{S}$$ transmits [TeX:] $$\beta$$ information bits to [TeX:] $$\mathcal{D}$$ over k during each packet transmission, the coding rate of the proposed system is [TeX:] $$r=2 \beta / k.$$ For SPCs, the minimum blocklength should be larger than 100 CUs [20], the average BLER can be tightly approximated as [15, Eq. (9)]

where [TeX:] $$\mathcal{X} \in\left\{\mathcal{S} \mathcal{R}_{\mathrm{b}}, \mathcal{R}_{\mathrm{b}} \mathcal{D}\right\}, C(\gamma \mathcal{X})=\log _{2}(1+\gamma \mathcal{X})$$ is the Shannon channel capacity, [TeX:] $$V\left(\gamma_{\mathcal{X}}\right)= \left(1-1 /(1+\gamma \mathcal{X})^{2}\right)\left(\log _{2} e\right)^{2}$$ is the channel dispersion measuring the random variation of a channel compared to a defined channel for the same capacity, [TeX:] $$\mathbb{E}\{.\}$$ is the expectation operator, and [TeX:] $$Q(.)$$ is the Gaussian Q-function [21]. From (7), we observe that the form of Gaussian Q-function is too complicated, which yields that it is intractable to solve (7) directly. We now envoke the approximated linear form of [TeX:] $$Q\left(\left(C\left(\gamma_{\mathcal{X}}\right)-r\right) / \sqrt{V\left(\gamma_{\mathcal{X}}\right) / k}\right) \approx \Phi\left(\gamma_{\mathcal{X}}\right)$$ [20, Eq. (14)] with

where [TeX:] $$v=1 / \sqrt{2 \pi\left(2^{2 r}-1\right)}, \theta \triangleq 2^{r}-1, \rho_{H}=\theta+1 /(2 v \sqrt{k}),$$ and [TeX:] $$\rho_{L}=\theta-1 /(2 v \sqrt{k}).$$

By inserting (8) into (7), the average BLER for the scheme S can be tightly approximated as

where [TeX:] $$f_{\gamma_{X}}(\gamma)$$ denotes the probability density function of [TeX:] $$\gamma_{\mathcal{X}}.$$ Subsequently, (9) is calculated by utilizing the partial integration method, which yields

By substituting (4) and (6) into (10), the average BLERs at the first and second hop are obtained, respectively, as

where [TeX:] $$\Upsilon(\cdot, \cdot)$$ is the lower incomplete Gamma function with [TeX:] $$\Upsilon(m, x)=\int_{0}^{x} t^{m-1} \exp (-t) d t$$ [22].

Proposition 1. Since the SDF scheme is considered at relays [23], where the relay node supports only when it is able to decode successfully the source information. As a result, the e2e BLER can be calculated as

Otherwise, since FDF is utilized at relays [24], where the selected relay always decodes and forwards the source information without regarding the ability of prosperous decoding. In such a scenario, the e2e BLER is given by

B. Asymptotic BLER

In this subsection, to observe the qualitative conclusions on the system performance, we present the asymptotic forms of the e2e BLERs in the high-SNR regime, where the first asymptotic approach is utilized to highlight the illustration of asymptotic characteristics and the second approach is to validate the first one.

1) The first approach: In the high-SNR regime, by applying the equivalent infinitesimal series of [TeX:] $$1-\exp (-x) \stackrel{x \rightarrow 0}{\sim} x$$ to (10), the asymptotic BLER of the first hop is calculated as

The asymptotic BLER of the second hop is solved by envoking the helps of [22, Eq.(8.352.6)] and [22, Eq.(8.354.1)] for (10), as given respectively by

As a result, we get

We note the value of BLER will be very small in the high- SNR range, [TeX:] $$\text { i.e., } \bar{\varepsilon}_{\mathcal{S} \mathcal{R}_{\mathrm{b}}} \ll 1 \text { and } \bar{\varepsilon}_{\mathcal{R}_{\mathrm{b}} \mathcal{D}} \ll 1.$$ Accordingly, Proposition 1 can be simplified, yielding the e2e asymptotic BLER for the first approach as

2) Riemann integral approximation: By revisiting (10), we find that the value of [TeX:] $$\rho_{H}-\rho_{L}=\sqrt{2 \pi\left(2^{2 r}-1\right) / k}$$ is small since r is small as [TeX:] $$k \geq 100$$ [20]. In such a scenario, the first-order Riemann integral approximation [TeX:] $$\int_{a}^{b} f(x) d x= (b-a) f\left(\frac{a+b}{2}\right)$$ is valid for (10). As a result, the asymptotic BLERs for the first hop and second hop are given, respectively, by

Similar to (19), the e2e asymptotic BLER in for the second approach is obtained as

Remark 1. We observe that the average BLER of each hop in (10) is only dependent on the CDF function of [TeX:] $$\gamma_{\mathcal{X}}.$$ When [TeX:] $$F_{\gamma_{\mathcal{X}}}(\gamma)$$ becomes too complicated, the first asymptotic approach can be intractable to obtain the closed-form expression for the average BLER. Instead of using the first asymptotic method, we can utilize the Riemann integral approximation to simplify the calculation. In this paper, we do not only utilize the second context to reduce the calculating complexity but also validate the correctness of the first approach.

Proposition 2 (Diversity order). By denoting the average total transmit SNR as [TeX:] $$\bar{\gamma}=P_{0} / \sigma^{2},$$ the transmit power constraints with [TeX:] $$P_{\mathcal{S}} / \sigma^{2}=\phi \bar{\gamma} \text { and } P_{\mathcal{R}_{\mathrm{b}}} / \sigma^{2}=(1-\phi) \bar{\gamma}$$ are satified, where [TeX:] $$\phi \in(0,1)$$ denotes the PA coefficient for the source S. The diversity order of the proposed system is determined by [25]

Proof. From (19) and (22), we express the diversity order of the considered system, respectively, in the forms of [25]

where

The proof is concluded.

C. e2e Latency and Throughput

The e2e latency is characterized by the average delay during transmission and decoding. Let [TeX:] $$\mathcal{L}(k)$$ denote the delay for decoding a packet with the blocklength k. Based on [14, Eq. (6)], the e2e packet transmission latency (measured in the number of CUs) is given by

where [TeX:] $$\mathcal{L}(k)=\alpha k \text { with } \alpha$$ being a constant that represents the decoding delay factor [26, Lemma 6]. We note that in (30), the total latency is taken into account only when the best relay successfully decodes the signal and then the message arrives at the destination, regardless of whether this message is successfully decoded at the destination or not, which is precise to the analysis in [14, Eq. (6)].

Besides that, the e2e throughput is defined as the ratio of the number of information bits successfully received at the destination and the e2e packet transmission latency, which is measured in bits per CU (BPCU). As a result, the e2e throughput for the considered system is expressed as

IV. OPTIMIZATION ISSUES

This section investigates the optimization problems for PA and RL to minimize the e2e BLER under the system configuration and uRLLC constraints.

A. PA Optimization

In this subsection, we will answer the important question of the relay system under the fixed coordinations of the network nodes, how is transmit power distributed to the source and relay nodes for the best system performance? The optimal PA optimization problem is given by

where P is a set of transmit power variables, i.e., P = [TeX:] $$\left(P_{\mathcal{S}}, P_{\mathcal{R}_{\mathrm{b}}}\right), \text { and } P_{0}$$ denotes the total transmit power budget of the system.

It is noted that the optimization problem (32) is non-convex because the objective function [TeX:] $$\bar{\varepsilon}_{e 2 e}(\mathbf{P})$$ is a non-convex function. Therefore, we attempt to transform (32) into a convex optimization problem. For uRLLCs, the e2e BLER must be lower than [TeX:] $$10^{-5}$$ [5], which requires a high SNR. As a result, we can invoke (22)²for the objective function in (32). Specifically, the problem (32) can be approximated as

²We note that since the results of the first and second asymptotic approaches are identical, the second strategy, however, provides lower calculating complexity. Therefore, we consider (22) as the objective function for simplification.

By inserting (20) and (21) into (33), the optimization problem (33) becomes

where [TeX:] $$d_{1}$$ is the distance between the source and multiple relays and [TeX:] $$d_{2}$$ is the distance between multiple relays and the destination. We assume the simplified pathloss model [19] with the average channel power gains [TeX:] $$\Omega_{\mathcal{S R}_{n}}=d_{1}^{-\eta} \text { and } \Omega_{\mathcal{R}_{\mathrm{b}} \mathcal{D}}=d_{2}^{-\eta},$$ where [TeX:] $$\eta$$ denotes the pathloss exponent.

Proposition 3. The optimization problem (34) is convex. Therefore, the Lagrangian multiplier method is beneficial to find a single global optimal solution P∗.

Proof. By considering the Hessian matrix [27, Theorem. (A.2)]

where

Because all diagonal elements of H(P) are positive, H(P) is positive definite. As a result, [TeX:] $$\tilde{\varepsilon}_{\mathrm{e} 2 \mathrm{e}}^{(2)}(\mathrm{P})$$ in (34) is a strictly convex function. Since the constraint is affine, the optimization problem (34) is convex, which completes the proof.

Based on Proposition 3, we use the Lagrangian multiplier method to determine P∗, where the Lagrange function of (34) is given by

where [TeX:] $$\psi_{1}$$ is a Lagrange multiplier constant. By considering the partial derivatives of [TeX:] $$\Delta_{1}\left(\mathrm{P}, \psi_{1}\right)$$ with respect to [TeX:] $$P_{\mathcal{S}}, P_{\mathcal{R}_{\mathrm{b}}},$$ and [TeX:] $$P_{0}$$ equal to zero, we get

By considering [TeX:] $$\left(\mathcal{F}_{1}\right)-\left(\mathcal{F}_{2}\right),$$ we obtain

By substituting (41) into [TeX:] $$\left(\mathcal{F}_{3}\right),$$ we get

where [TeX:] $$\mathcal{Z}_{1}=\left(\frac{\left(\theta \sigma^{2} d_{2}^{\eta}\right)^{M} M}{\left(\theta \sigma^{2} d_{1}^{\eta}\right)^{N T} M ! N T}\right)^{\frac{1}{M+1}}.$$ We note that, when [TeX:] $$P_{S},\mathcal{Z}_{1}>0, \text { and } M, N, T \in \mathbb{N}^{*}, g\left(P_{S}\right)$$ is an monotonically increasing function because of [TeX:] $$\frac{\partial g\left(P_{S}\right)}{\partial P_{S}}=1+\mathcal{Z}_{1} \frac{N T+1}{M+1} P_{S}^{\frac{N T-M}{M+1}}> 0.$$ Furthermore, the right hand side of (42) is a constant function. When [TeX:] $$g(0)=0, g\left(P_{S}\right)>0, \text { and } P_{0}>0,$$ (42) only has a unique solution. To solve (42), we consider the following proposition.

Proposition 4 (Newton–Raphson iterative root-finding algorithm). We consider f (x), from a well-chosen value [TeX:] $$x_{0},$$ the iterations for a unique root of f (x) can be determined as

Eventually, [TeX:] $$x_{n+1}$$ converges to a unique root.

Proof. Based on the definition of the derivative, we have [TeX:] $$\frac{\partial f(\hat{x})}{\partial \hat{x}}=\lim _{x \rightarrow \hat{x}} \frac{f(x)-f(\hat{x})}{x-\hat{x}}.$$ This also means

where the notation [TeX:] $$\rightarrow$$ represents the convergence. Assuming that [TeX:] $$\hat{\boldsymbol{x}}$$ is a solution, we have [TeX:] $$f(\hat{x})=0.$$ From (44), we get [TeX:] $$\frac{\partial f(\hat{x})}{\partial \hat{x}} \rightarrow \frac{f(x)}{x-\hat{x}},$$ which yields

where [TeX:] $$\hat{x}$$ is a converged unique solution. The proof is concluded.

Remark 2. We note that the non-linear function (42) with its constraints has sufficient conditions to satisfy the guaranteed convergence requirement for both the Newton–Raphon and other search methods (e.g., Bisection, Secant, Bairstow), i.e., a single variable function with a unique real root and an easily determinable derivative. In this context, since a fair comparison between the Newton-Raphson and other search methods is guaranteed, the Newton-Raphson method offers the quickest convergence rate [28]. Although the Bisection search method is the simplest method to understand, it has the linear rate of convergence, which is the search method with the lowest convergence rate. Meanwhile, the Secant and Bairstow search methods have aureal number and quaradtic convergence rates, respectively, which are higher ones than that of the Bisection search method but lower than that of the Newton-Raphson method. As a result, the used Newton-Raphson search method is the most efficient selection in our scope.

By utilizing Proposition 4, Algorithm 1 is utilized to find the solution [TeX:] $$P_{\mathcal{S}}^{*}$$ for (42). Based on the optimal solution [TeX:] $$P_{\mathcal{S}}^{*}$$ attained from Algorithm 1 and [TeX:] $$\left(\mathcal{F}_{3}\right),$$ the optimal [TeX:] $$P_{\mathcal{R}_{\mathfrak{b}}}^{*}$$ is given by

Remark 3. We note that, in a special case with NT = M, (42) can be easily solved and yields a unique solution as

B. RL Optimization

In this subsection, the optimal RL configuration is designed to minimize the e2e BLER with the constraint of the normalized transmission distance and the given PA configuration. Since the original RL optimization is non-convex, we formulate the optimal RL problem as a convex optimization by utilizing the same hypotheses as those in the PA optimization problem. Specifically,

where [TeX:] $$\mathbf{d}=\left(d_{1}, d_{2}\right) \text { and } D$$ represents the normalized transmit distance. By substituting (20) and (21) into (48), we get

By following the same footsteps as Proposition 3, it is the fact that (49) is a convex optimization problem. Therefore, the Lagrangian multiplier method is beneficial to find the global optimal solution d∗. We define the Lagrangian cost function for (49) as

where [TeX:] $$\psi_{2}$$ is a Langrange multiplier constant. Similarly, by performing the partial derivatives of [TeX:] $$\Delta_{2}\left(\mathbf{d}, \psi_{2}\right)$$ with respect to [TeX:] $$d_{1}, d_{2}, \text { and } \psi_{2}$$ equal to zero, we obtain

By considering [TeX:] $$\left(\mathcal{G}_{1}\right)-\left(\mathcal{G}_{2}\right),$$ we get

Subsequently, by inserting (52) into [TeX:] $$\left(\mathcal{G}_{3}\right),$$ we obtain

where [TeX:] $$\mathcal{Z}_{2}=\left(\frac{\left(\theta \sigma^{2}\right)^{N T} N T P_{\mathcal{R}_{\mathrm{b}}}^{M} M !}{\left(\theta \sigma^{2}\right)^{M} M P_{\mathcal{S}}^{N T}}\right)^{\frac{1}{M \eta-1}}.$$ Similar to (42), (53) also has a unique solution. By utilizing Proposition 4, we design Algorithm 2 to determine the optimal solution [TeX:] $$d_{1}^{*}$$ for (53). Eventually, we replace the obtained [TeX:] $$d_{1}^{*}$$ from Algorithm 2 into [TeX:] $$\left(\mathcal{G}_{3}\right),$$ which yields

Remark 4. For a special case with NT = M, the optimal RL configuration is given by

Remark 5. It is worth noting that, from the derived optimal distances given in this part, we can easily indicate the optimal relay coordination. Specifically, the optimal distances can be expressed as

where [TeX:] $$\left(x_{\mathcal{S}}^{*}, y_{\mathcal{S}}^{*}\right),\left(x_{\mathcal{R}}^{*}, y_{\mathcal{R}}^{*}\right), \text { and }\left(x_{\mathcal{D}}^{*}, y_{\mathcal{D}}^{*}\right)$$ denote the coordinations of the source, relays, and destination, respectively. Because of the linear topology assumption, [TeX:] $$\left|y_{\mathcal{R}}^{*}-y_{\mathcal{S}}^{*}\right|= \left|y_{\mathcal{D}}^{*}-y_{\mathcal{R}}^{*}\right|=0.$$ Accordingly, [TeX:] $$d_{1}^{*} \text { and } d_{2}^{*}$$ are simplified to [TeX:] $$d_{1}^{*}=\left|x_{\mathcal{R}}^{*}-x_{\mathcal{S}}^{*}\right| \text { and } d_{2}^{*}=\left|x_{\mathcal{D}}^{*}-x_{\mathcal{R}}^{*}\right|,$$ respectively. In such a scenario, a system of 3 linear equations is represented in a matrix multiplication form as Ax = b, where

Because all row vectors of A are linearly independent, the determinant of A becomes zero, which implies that the inverse of A, denoted by [TeX:] $$\mathrm{A}^{-1},$$ exists. As a result, the optimal relay coordination is determined as [TeX:] $$\mathrm{x}=\mathrm{A}^{-1} \mathrm{~b}.$$

V. SIMULATION RESULTS AND DISCUSSION

In this section, we perform Monte Carlo simulations to verify our theoretical analysis presented in the previous sections.

A. Performance Evaluation

For parameter settings, we assume the information with [TeX:] $$\beta=256 \text { bits },$$ blocklength k = 256 CUs, and pathloss exponent [TeX:] $$\eta=3.$$ We consider the asymmetric system with [TeX:] $$d_{1}=0.7 \cdot D,d_{2}=0.3 \cdot D, P_{\mathcal{S}}=0.7 \cdot P_{0}, \text { and } P_{\mathcal{R}_{\mathrm{b}}}=0.3 \cdot P_{0}.$$ The normalized transmission distance D = 10 is assumed.

First, we compare the e2e BLER between the FDF and SDF schemes, as shown in Fig. 2. Here, N = 2 and M = 3 are assumed for the parameter settings. In this figure, we observe that although the e2e BLERs of the SDF and FDF schemes are identical in the medium and high-SNR regimes, their performance, however, is different in the low-SNR range. In particular, the performance of FDF is better than that of the SDF scheme in the low-SNR regime. Consequently, we will only consider the FDF scheme for the later figures. In addition, the influence of the number of source antennas T on the e2e BLER is also depicted with some typical values of T = 1, 2, and 3. It is clear that the performance improvement is significant only when T goes from 1 to 2, whereas even in the low-SNR regime, there is a negligible gap in the case of T = 3 compared to that of T = 2. When [TeX:] $$\bar{\gamma}$$ gets sufficiently high, the e2e BLERs with T = 2 and T = 3 are entirely identical, which yields that the value of T should be conservatively chosen to balance the tradeoff between the cost implementation and performance. To see this behavior in the wider range of T, Fig. 3 is dominated. Importantly, the simulation results completely coincide with the theoretical analysis results, which confirm the correctness of our analysis.

Fig. 3 is dedicated to investigating the effect of T on the system performance, whereN = 2,M = 3, and [TeX:] $$\bar{\gamma}=10,15,20 \mathrm{~dB}$$ are assumed. As observed in Fig. 3, the e2e BLER becomes saturated when [TeX:] $$T \geq 2.$$ As a result, T = 2 is an efficient choice to balance the tradeoff between the performance and cost of implementation. One way to improve the system performance as expected is by increasing the average transmit SNR [TeX:] $$\bar{\gamma},$$ as shown in Fig. 3. However, the transmit power is usually limited in practice; and even when the transmit power can be increased as much as possible, it can cause interference with the other propagation channels. Meanwhile, increasing the number of relays N and destination antennas M can be considered as an alternative way to significantly improve the system performance, which is illustrated in Fig. 4.

Fig. 4 investigates the effect of the number of relays and destination antennas on the system BLER, which is presented in three-dimensional space. Each plane represents the BLER analysis result with a specific average transmit SNR, where we consider three levels of [TeX:] $$\bar{\gamma}$$ as 5 dB, 10 dB, and 15 dB. Here, T = 2 is assumed. The simulation results, which are presented by dots, are math well with the analysis plane. As seen in Fig. 4, the e2e BLERs dramatically drop when [TeX:] $$\bar{\gamma}, N \text {, and } M$$ increase, as expected. We note that, as stated in Proposition 2, the diversity gain of the proposed system is dependent on the design of the number of relays, source antennas, and destination antennas, which yields the important role of the designed numbers of relays and antennas for the network.

Subsequently, Fig. 5 shows the effect of blocklength on the system performance in three typical cases of average SNRs as 10 dB, 15 dB, and 20 dB, where the parameters of T = 2, N = 4, and M = 3 are set. In Fig. 5, we recognize that the system performance is improved by increasing the average SNR and blocklength. However, we note that the longer blocklength also causes the higher e2e latency, as depicted in Fig. 8. This tradeoff on the blocklength value should be conservatively considered. For example, for a given average SNR [TeX:] $$\bar{\gamma}=20 \mathrm{~dB},$$ to ensure the reliability requirement in the order of e2e BLER of [TeX:] $$10^{-5},$$ the blocklength of 450 CUs can be chosen.

Fig. 6 depicts the suitability between the approximated and asymptotic BLERs. Here, the results are derived for the scenario of T = 2, N = 4, M = 3, and k = 256, 512, 1024 CUs. As seen in Fig. 6, the e2e BLER is decreased when increasing [TeX:] $$\bar{\gamma}$$ the second asymptotic approach is considered as a validated strategy of the first asymptotic one, where they are in a good agreement with each other.

Eventually, in Fig. 7, we can observe the system performance in terms of the e2e latency and throughput that are the crucial evaluation parameters in the uRLLC requirements, where T = 2, N = 4, M = 3, [TeX:] $$\bar{\gamma}=20 \mathrm{~dB}, \text { and } \alpha=0.5,1.5,2.5$$ are assumed. While increasing blocklength can improve the reliability, they suffer from the higher latency and throughput. Besides that, Fig. 7 also yields the impact of the decoding delay factor. Specifically, with the higher value of [TeX:] $$\alpha,$$ the system performance is negatively affected in terms of the e2e latency and throughput. We note that, for a CU duration of [TeX:] $$3 \mu \mathrm{s}$$ [6], [TeX:] $$\alpha=1.5,$$ and the parameter settings given in Fig. 7, if we choose the blocklength of k = 450 CUs, the system will suffer from the latency of around 2250 CUs corresponding to 6.75 ms [TeX:] $$(<10 \mathrm{~ms}),$$ which satisfies the low-latency constraint of uRLLCs to serve factory-act applications [6].

B. Performance Evaluation of Optimization Strategies

In this subsection, we compare the system performance of the optimal and non-optimal configurations for PA and RL. For a fair comparison, [TeX:] $$\beta=256 \text { bits },$$ blocklength k = 256 CUs, normalized transmit distance D = 1, and pathloss exponent [TeX:] $$\eta=3$$ are set in this subsection. The general case for T, N, and M settings is considered with T = 2, N = 3, and M = 4. Furthermore, we also show the optimization performance in a special case with T = 1 and N = M = 2, where the optimal configurations of PA and RL are dominated in (47) and (55).

Fig. 8 shows the performance comparison between the optimal and non-optimal PA configurations in both the general and special cases of N and M. In this scheme, the distance with [TeX:] $$d_{1}=0.7 \cdot D \text { and } d_{2}=0.3 \cdot D$$ are fixed, as assumed in Section V-A. To highlight the outstanding performance of the optimal PA configuration designed in Section IV-A, we consider the non-optimal scheme with the equal PA, i.e., [TeX:] $$P_{\mathcal{S}}= P_{\mathcal{R}_{\mathrm{b}}}=0.5 \cdot P_{0}.$$ Fig. 8 reveals that the optimal PA performance is significantly higher than that of the non-optimization in the entire range of the average transmit SNR for both the general and special cases. For e2e BLER of [TeX:] $$10^{-5},$$ the array gains of 2.5 dB are achieved in both the (T = 1,N = M = 2) and (T = 2,N = 3,M = 4) cases.

In the performance comparison for RL configurations, we assume that the power allocation is fixed with [TeX:] $$P_{S}=0.7 \cdot P_{0}$$ and [TeX:] $$P_{\mathcal{R}_{\mathrm{b}}}=0.3 \cdot P_{0},$$ as assumed in Section V-A. In Fig. 9, the RL optimization designed in Section IV-B is compared to the performance of the non-optimal RL scheme with [TeX:] $$d_{1}=d_{2}=0.5 \cdot D.$$ First, we recognize that the performance behavior shown in Fig. 9 is the same as Fig. 8, which confirms the superior performance of the designed optimal RL configuration. However, the array gains achieved in Fig. 9 are much higher than that of Fig. 8. Specifically, for e2e BLER of [TeX:] $$10^{-5},$$ the array gains of 6.25 dB and 9.5 dB are achieved with the (T = 1,N = M = 2) and (T = 2,N = 3,M = 4) cases, respectively.

VI. CONCLUSIONS

In this paper, we have investigated SPCs for the dual-hop DF relaying system. The proposed relay network is beneficial in the context of limited transmit power and co-channel noise degradation, whereas the SPC framework is considered to achieve uRLLCs. Besides that, the TAS, PRS, and MRC techniques are utilized at the corresponding terminals to exploit the diversity gain of the proposed system. In the quasi-static Rayleigh fading channels and finite-blocklength regime, the approximated e2e BLERs for both the SDF and FDF schemes are derived from the closed-form expression, from which the e2e latency and throughput are also investigated to provide a performance overview of both ultra-reliability and low-latency. Subsequently, the asymptotic e2e BLER is considered to get more insights into the performance behavior in the high-SNR regime. Furthermore, two optimization strategies of PA and RL are analyzed to minimize the e2e BLER objective function under the system constraints. The results have shown that the BLERs of the SDF and FDF schemes are identical in the average and high-SNR regimes but different in the low-SNR range. Besides that, two asymptotic approaches for the e2e BLER are well matched with each other. The simulation and analysis results are also in good agreement and converge to the asymptotic curve in the high-SNR regime, which validates the correctness of our analysis. Most importantly, the performance of the designed PA and RL optimization schemes is significantly better than the non-optimal configuration.