Transmit antenna selection for millimeter-wave communications using multi-RIS with imperfect transceiver hardware

Abstract

This article presents a comprehensive exploration of the synergy between transmit antenna selection (TAS) and reconfigurable intelligent surfaces (RISs) in millimeter-wave (MW) communication systems, considering the impact of practical conditions. Notably, it accounts for imperfect transceiver hardware (ITH) at both the transmitter and receiver. Additionally, real-world channel models and receiver noise statistics are integrated into the analysis, providing a realistic representation of wireless systems in future networks. Mathematical formulas of outage probability (OP) and system throughput (ST) of the multi-RIS-assisted MW communications with ITH and TAS (shortened as the considered communications) are derived for analyzing the system behaviors. These formulas facilitate a comprehensive examination of system behavior. Through a series of comparative scenarios, including evaluations of OP and ST with and without TAS, with and without RISs, and with and without ITH (where the absence of ITH is denoted as perfect transceiver hardware, or PTH), the study substantiates the substantial advantages of TAS and RISs while shedding light on the significant influence of ITH. It is demonstrated that even in the presence of ITH, MW communication performance can be dramatically enhanced by optimizing the number of transmit antennas, selecting suitable carrier frequencies and RIS placements, and utilizing appropriate bandwidth. Ultimately, the derived formulas are rigorously validated through Monte-Carlo simulations, reinforcing the credibility of the findings.

1 Introduction

Indeed, it has been substantiated that transmit antenna selection (TAS) serves as an effective solution for significantly reducing signal processing complexity and markedly improving the performance of multiple-input multiple-output systems [1]. Because of its advantages, TAS was widely used in the previous generations of mobile communications (3G & 4G) and it is being an effective method to use in the future generations (5G & B5G) [1, 2]. In particular, a transmit antenna is chosen from N antennas to dramatically enhance the received power. Consequently, the effects of the negative parameters, e.g., residual self-interference and imperfect transceiver hardware (ITH) can be indirectly reduced [2]. Therefore, TAS becomes an effective manner with low cost and low computational complexity for applying in 5G & B5G networks [3].

On the other hand, 5G & B5G networks exploit the high carrier frequencies of millimeter-wave (MW) bands. Therefore, the number of antennas at wireless nodes can be considerably extended [4]. Specifically, MW communications operate within a wavelength range spanning from 1 to 10 mm, corresponding to carrier frequencies that range from 30 to 300 GHz. This extensive bandwidth allows for the deployment of numerous antennas in wireless nodes that operate within the MW bands [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. As a result, the data speeds of MW communications can be significantly increased. However, the main disadvantage of MW communications is that they are strongly affected by blockages [5]. Thus, many solutions such as TAS, beamforming, and channel estimation have been applied to reduce the effects caused by the elevated carrier frequencies inherent to MW bands. [19].

1.1 Related works

To exploit the benefits of utilizing high carrier frequencies, many new techniques such as reconfigurable intelligent surface (RIS), full-duplex, and unmanned aerial vehicle have been studied and combined in MW communications [4, 19,20,21,22]. More specifically, RIS technology has recently emerged as a powerful paradigm for applying in B5G networks [23, 24]. In particular, RIS is made from metasurfaces that can effectively reflect wireless signals without power supply, processor, converter, and amplifier [25,26,27,28]. Therefore, compared with traditional relays, the usage of advanced RIS in wireless communications can achieve many benefits, especially with larger sizes of RIS or bigger numbers of reflecting elements (REs) [29,30,31,32]. Moreover, the RIS can be deployed at the top of factories/buildings, thus, the effects caused by blockages in MW communications may be greatly reduced [33,34,35]. Besides one-RIS, multi-RIS was also utilized in MW communications [36, 37].

In recent reports, TAS has been utilized in multi-RIS-assisted MW communications [37]. In this TAS scheme, only one transmit antenna is selected from N transmit antennas based on link power to maximize the received power at the receiver/user. As a result, by utilizing TAS, the received power is much increased while the computational complexity is significantly reduced in comparison with the case that either several transmit antennas or all transmit antennas are active [2, 37,38,39,40,41,42]. Importantly, the system performance using TAS can keep as close to alternative solutions as possible [43]. In particular, by exploiting multi-antenna at base station, the outage probability (OP) and system throughput (ST) of MW communications are greatly enhanced. Thus, the effects of negative parameters in the systems can be significantly reduced. Moreover, the signal processing complexity at the mobile user is also decreased. However, a practical condition, i.e., ITH was often neglected [37, 44,45,46]. In the meanwhile, it has been demonstrated that ITH always exists in wireless communications because of hardware characteristics [47, 48]. In particular, ITH can be found in electronic components, especially in cost-effective devices. More specifically, ITH is influenced by disparities in in-phase/quadrature components, phase noise, and the nonlinearity of converters (both analog-to-digital and digital-to-analog), mixers, and amplifiers. These components encompass high-power amplifiers in transmitters and low-noise amplifiers in receivers [47, 49,50,51]. These imperfections often lead to discrepancies between intended and actual signals [49,50,51]. Several methods have been proposed to address these limitations in both analog and digital domains [47, 48, 50, 51]. Nonetheless, residual impairments stemming from inherent electronic component defects, imperfect parameter estimation of variable hardware characteristics, limitations in precision models, and basic compensation algorithms prevent complete elimination of distortion noises. Additionally, these impairments vary over time as they present new manifestations for each fresh data signal [48, 51]. Utilizing the central limit theorem and experimental observations, the cumulative impact of numerous impairments can be characterized by complex Gaussian variables [48,49,50,51,52,53]. Consequently, neglecting ITH when assessing the performance of wireless communications can result in erroneous or imprecise conclusions [49]. In addition, it was demonstrated that the ITH has a great effect on the performance of RIS-assisted wireless communications [50,51,52,53]. For improved readability, we provide a summary of existing works in Table 1. Please note that the performance metric values are evaluated at a transmit power of 30 dBm.

Table 1 Comparison with the existing works

1.2 Motivation and contributions

As the above discussions, the huge advantages of TAS and multi-RIS have been confirmed in the literature. However, only one RIS was often used to assist MW communications in the previous works [19, 57]. Moreover, exploiting both TAS and multi-RIS in MW communications is still lacking, except for the work in [37]. Unfortunately, [37] considered the case of perfect transceiver hardware (PTH). The case of ITH was not studied yet. Meanwhile, ignoring ITH can result in some inaccuracies into the system behaviors. It is because the transceiver hardware is varied and belongs to the intrinsic properties of electronic components, especially the random and time varying hardware characteristics [50, 51]. Motived from these facts, in this article, the benefits of TAS and multi-RIS are combined in the MW communications with ITH. More specifically, by utilizing TAS and RISs, the negative effects of ITH are greatly reduced. It is because the received signal-to-noise ratio (SNR) at the receiver will be dramatically increased with TAS and RISs. As a result, the received SNR with TAS and RISs is significantly higher than that without TAS and without RISs. The mathematical formulas of OP and ST of the considered communications (the multi-RIS-assisted MW communications with ITH and with TAS) are derived and then compared with those of the relevant communications (the MW communications without RISs, with PTH, and without TAS)¹. From the acquired findings and observed behaviors of the multi-RIS-assisted MW communications with ITH and with TAS, the potential for broadening the scope of the considered communications becomes apparent, encompassing comprehensive models featuring multi-RIS configurations, multiple antennas, and multiple relays within the domain of MW communications, adaptable to diverse scenarios. Notably, the considered multi-RIS-assisted MW communications with ITH and with TAS along with their extensions exhibit suitability for integration into sensor systems and internet of things (IoT) networks, particularly well-suited for operation in specialized environments such as disaster-stricken areas, mountainous terrains, and wooded areas. In these circumstances, establishing a reliable power supply for communication nodes proves arduous, leading to insufficient power reserves for extensive and prolonged communication over extended distances. As a result, employing strategies such as multi-hop relaying, multi-RIS deployment, and multi-antenna architectures emerges as effective solutions. The incorporation of RISs holds particular promise as it significantly amplifies coverage and substantially curtails energy consumption because the RISs can reflect signals without a power supply. The main contributions of this article can be shortened as follows:

• A practical scenario is considered where ITH is taken into investigation of multi-RIS-assisted MW communications. Besides ITH, the path loss models and receiver noise are also measured based on practical deployments of 5 G & B5G networks. Consequently, the multi-RIS-assisted MW communications with ITH can comprehensively reflect the behaviors of 5 G & B5G networks deployed in practice. To significantly reduce the negative effects of ITH and other parameters, TAS is utilized at the base station.

• The mathematical formulas of OP and ST of the multi-RIS-assisted MW communications with TAS are derived under the effects of ITH. We observe that mathematical derivations are challenging with ITH. Consequently, the computational complexity is significantly higher than that of previous works [37, 50, 54]. Hence, the derived equations are notably more intricate compared to those presented in prior studies [37, 50, 54, 55]. From the derived formulas, it’s easy to obtain the other formulas of relevant communications such as MW communications with and without RISs in the cases of without TAS and with ITH, with TAS and with PTH, and without TAS and with PTH. Monte-Carlo simulation results are conducted to verify the exactness of the derived formulas.

• The OP and ST of the considered MW communications (with ITH, with RISs, and with TAS) are evaluated and compared with those of MW communications in the cases with PTH, without RISs, and without TAS. Through many compared scenarios presented in numerical results, the great effects of ITH are detailedly analyzed. In this circumstance, the usage of TAS and multiple RISs are effective solutions. Specifically, besides the separate benefits of TAS and RISs, the joint benefits of them are also confirmed. Specifically, with a certain condition, the OP and ST performances with TAS are considerably higher than the OP and ST performances without TAS. Similarly, the performances with RISs are dramatically higher than those without RISs. From the communication behaviors, many solutions such as using larger number of transmit antennas, utilizing suitable carrier frequencies and bandwidths, and locating RISs in appropriate areas can be utilized for significantly reducing the OP and increasing ST of the considered communications.

The rest of this article follows this structure: Sect. 2 introduces the system model and the signal formulas relevant to the communication under consideration. Section 3 detailedly analyzes the communication performance by deriving the OP and ST formulas with ITH and with TAS. Section 4 illustrates the communication behaviors under the effects of system parameters. In conclusion, Sect. 5 provides the final remarks and conclusions for this article. For ease of reading, the mathematical notations are included in Table 2.

Table 2 Mathematical notations

2 System model

The block diagram of the multi-RIS-assisted MW communications with ITH is plotted in Fig. 1. In particular, the considered communications consist of three components, i.e., a base station (S), M RISs, and a user (U). Since the wavelengths in the MW communications are utilized, S includes N transmit antennas. Meanwhile, U is a personal device, it is equipped with only one receive antenna.²M RISs are distributed in different areas, where the $l^{th}$ RIS ($l=1,2,...,M$) has $G_l$ REs.³ In the case of PTH, the transmit signal at S and the received signal at U are, respectively, $x_\text {S}$ and $y_\text {U}$. Meanwhile, in the case of ITH, they are, respectively, $\tilde{x}_\text {S}$ and $\tilde{y}_\text {U}$. Herein, $\tilde{x}_\text {S}$ includes an ITH component of the transmitter S while $\tilde{y}_\text {U}$ consists of the ITH components of both the transmitter S and receiver U. Since the RISs are expected to deploy in B5G networks, besides the S-RIS-U links, the considered model comprises the classical S-U link that has been often exploited in 3 G/4 G/5 G networks.

Fig. 1

Illustration of the multi-RIS-assisted MW communications with ITH

At every transmission time slot, TAS is utilized at S. Thus, S selects one antenna (denoted by the $i^{th}$ antenna in the followings) from N transmit antennas for transmitting signal to U. The received signal at U with ITH can be formulated as

$$\begin{aligned} \tilde{y}_\text {U} =&\Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} a_{ilk} b_{lk} \exp (j \varphi _{lk}) + c_{isu} \Big ) (x_\text {S} + n_\text {S}^\text {t}) + n_\text {U}^\text {r} + z_\text {U} , \end{aligned}$$

(1)

where $a_{ilk}$ and $c_{isu}$ are the channels from the $i^{th}$ antenna of S to the $l^{th}/k^{th}$ and to U, respectively; $b_{lk}$ is the channel from the $l^{th}/k^{th}$ to U; $\varphi _{lk}$ is the phase shift induced by the $l^{th}/k^{th}$; $x_\text {S}$ and $n_\text {S}^\text {t}$ denote the desired signal and distortion noise induced by ITH of transmitter S, respectively; $n_\text {U}^\text {r}$ is distortion noise induced by ITH of receiver U; $z_\text {U}$ is the Gaussian noise at U, i.e., $z_\text {U} \sim C\mathcal {N}(0,\sigma _\text {U}^2)$.

It is because $a_{ilk}$, $c_{isu}$, and $b_{lk}$ are complex numbers, they can be represented as $a_{ilk} = \bar{a}_{ilk} \exp (-j \phi _{ilk})$, $c_{isu} = \bar{c}_{isu} \exp (-j \theta _{isu})$, and $b_{lk} = \bar{b}_{lk}\exp (-j \psi _{lk})$, where $\bar{a}_{ilk}$, $\bar{c}_{isu}$, and $\bar{b}_{lk}$ are the magnitudes and $\phi _{ilk}$, $\theta _{isu}$, and $\psi _{lk}$ are the phases of $a_{ilk}$, $c_{isu}$, and $b_{lk}$, respectively. On the other hand, a phase matrix is often used to express the phases of the $l^{th}$ RIS, e.g.,

$$\begin{aligned} \mathbf {\Psi }_l = \text {diag} \Big (\exp (j \varphi _{l1}),..., \exp (j \varphi _{lk}),..., \exp (j \varphi _{lG_l}) \Big ). \end{aligned}$$

(2)

Based on the magnitudes and phases of $a_{ilk}$, $c_{isu}$, and $b_{lk}$, (1) is represented as

$$\begin{aligned}&\tilde{y}_\text {U} = \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} \exp (j \varphi _{lk} - j \phi _{ilk} -j \psi _{lk}) + \bar{c}_{isu} \exp (-j \theta _{isu}) \Big ) (x_\text {S} + n_\text {S}^\text {t}) + n_\text {U}^\text {r} + z_\text {U} \nonumber \\&= \exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} \exp (j \varphi _{lk} + j \theta _{isu} - j \phi _{ilk} -j \psi _{lk}) + \bar{c}_{isu} \Big ) (x_\text {S} + n_\text {S}^\text {t}) + n_\text {U}^\text {r} + z_\text {U}. \end{aligned}$$

(3)

Then, (3) is reorganized as

$$\begin{aligned} \tilde{y}_\text {U} =&\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} \exp (j \xi _{lk}) + \bar{c}_{isu} \Big ) (x_\text {S} + n_\text {S}^\text {t}) + n_\text {U}^\text {r} + z_\text {U}, \end{aligned}$$

(4)

where $\xi _{lk} = \varphi _{lk} + \theta _{isu} - \phi _{ilk} - \psi _{lk}$.

As confirmed in the literature, one of the RISs’ advantages is that their phases can be adjusted independently [33, 51]. Specifically, due to the features of metasurfaces that are used to make RISs, the RISs’ phases are independently tuned [19]. Also, each RIS is controlled by a smart controller with communication-oriented software. In addition, the RISs’ phases are discrete because of hardware conditions [60]. As a result, with the smart controller, communication-oriented software, discrete phases, and perfect channel estimation, an expected phase can be picked up from a discrete phase set to generate $\xi _{lk}=0$ for all l and k [19, 24, 33, 51]. In other words, we have $\varphi _{lk} + \theta _{isu} - \phi _{ilk} - \psi _{lk} =0$. Consequently, the phase $\varphi _{lk}$ of the $l^{th}/k^{th}$ is chosen as

$$\begin{aligned} \varphi _{lk} = \phi _{ilk} + \psi _{lk} - \theta _{isu}. \end{aligned}$$

(5)

After picking up the ideal/optimal phase, the received signal at U from (4) becomes

$$\begin{aligned} \tilde{y}_\text {U} =&\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) (x_\text {S} + n_\text {S}^\text {t}) + n_\text {U}^\text {r} + z_\text {U}. \end{aligned}$$

(6)

It is noteworthy that the distortion noises induced by ITH can be expressed as [50, 51] $n_\text {S}^\text {t} \sim C\mathcal {N}(0,(\alpha _\text {S}^\text {t})^2 P_\text {S})$ and $n_\text {U}^\text {r} \sim C\mathcal {N}\Big (0,\Big |\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) \Big |^2 (\alpha _\text {U}^\text {r})^2 P_\text {S}\Big )$, where $P_\text {S} = \mathbb {E}\{|x_s|^2\}$; $\alpha _\text {S}^\text {t}$ and $\alpha _\text {U}^\text {r}$ denote the ITH levels at transmitter S and receiver U, respectively.

For convenience in observation, (6) is now represented as

$$\begin{aligned} \nonumber \tilde{y}_\text {U} =&\underbrace{\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) x_\text {S} + z_\text {U}}_\mathcal {I} \\&+ \underbrace{\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) n_\text {S}^\text {t} + n_\text {U}^\text {r}}_\mathcal {J} . \end{aligned}$$

(7)

It is obvious from (7) that the term $\mathcal {I}$ presents the received signal at U with PTH while the term $\mathcal {J}$ is the distortion noise induced by ITH. In other words, when the ideal hardware is considered in the RISs-assisted wireless communications [19, 33], the term $\mathcal {J}$ is set as $\mathcal {J} = 0$.

Based on the above definitions of the distortion noises, the total distortion noise power at U can be formulated as

$$\begin{aligned}&\mathbb {E} \left\{ \Big |\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) n_\text {S}^\text {t} + n_\text {U}^\text {r} \Big |^2 \right\} \nonumber \\ =&\Big |\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) \Big |^2 \Big [(\alpha _\text {S}^\text {t})^2 P_\text {S} + (\alpha _\text {U}^\text {r})^2 P_\text {S} \Big ] . \end{aligned}$$

(8)

Then, the signal-to-distortion-plus-noise ratio (SDNR) at U calculated from (7) is

$$\begin{aligned} \rho _\text {U} = \frac{\Big |\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) \Big |^2 P_\text {S}}{\mathbb {E} \left\{ \Big |\exp (-j \theta _{isu}) \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) n_\text {S}^\text {t} + n_\text {U}^\text {r} \Big |^2 \right\} + \sigma ^2_\text {U} }. \end{aligned}$$

(9)

Utilizing (8) combining with the property $|\exp (-j \theta _{isu})|^2=1$, (9) now becomes⁴

$$\begin{aligned} \rho _\text {U} = \frac{ \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) ^2 P_\text {S}}{\Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) ^2 \Big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \Big ] P_\text {S} + \sigma ^2_\text {U} }. \end{aligned}$$

(10)

On the other hand, we would like to take our considered communications practically. Thus, besides ITH, we also consider the noise power based on practical receivers [61], e.g.,

$$\begin{aligned} \sigma ^2_\text {U} = N_0 + N_F + 10\log (B_W), \end{aligned}$$

(11)

where $N_0$ is the thermal noise power density; $N_F$ is the noise figure; $B_W$ is the bandwidth. We should note that prior researches often normalized the noise power by setting $\sigma ^2_\text {U} =1$ [19, 33, 56]. Consequently, their results cannot fully reflect the impacts of $N_0$, $N_F$, and $B_W$ on the RISs-assisted wireless communication performance.

3 Performance analysis

In this section, the OP and ST formulas of the considered communications are derived over Nakagami-m channels. Because of Nakagami-m environments, the channel magnitudes are subject to Gamma distribution with the shape (denoted by m) and spread (denoted by $\Omega$) parameters. In the following, $(m_{a_l},\Omega _{a_l})$, $(m_{b_l},\Omega _{b_l})$, and $(m_{su},\Omega _{su})$ are, respectively, the channel parameters of S-$l^{th}$, $l^{th}$-U, and S-U channels. Furthermore, the channel magnitude characteristics can be expressed via their CDF and PDF using m and $\Omega$. For example, the CDF (F(.)) and PDF (f(.)) of $\bar{a}_{lk}$ are, respectively, given as [62]

$$\begin{aligned} F_{\bar{a}_{lk}}(\rho )&= \frac{1}{\Gamma (m_{a_l})} \gamma \big (m_{a_l}, \frac{m_{a_l} }{\Omega _{a_l}}\rho ^2\big ) = 1 - \frac{1}{\Gamma (m_{a_l})} \Gamma \big (m_{a_l}, \frac{m_{a_l} }{\Omega _{a_l}} \rho ^2 \big ) , \; \rho \ge 0, \end{aligned}$$

(12)

$$\begin{aligned} f_{\bar{a}_{lk}}(\rho ) = \frac{2 m_{a_l}^{m_{a_l}}}{\Gamma (m_{a_l})\Omega _{a_l}^{m_{a_l}}} \rho ^{2 m_{a_l} -1} \text {exp}\big (-\frac{m_{a_l} }{\Omega _{a_l}}\rho ^2\big ), \; \rho \ge 0. \end{aligned}$$

(13)

On the other hand, unlike the previous works where $\Omega$ is often normalized, e.g., $\Omega = 1$ [19, 33, 56], in the considered communications, $\Omega$ is computed via transmitter-receiver distance (d), carrier frequency ($f_c$), transmitter and receiver antenna gains ($G_\text {tx}$ & $G_\text {rx}$), and environmental characteristics. More specifically, $\Omega$ in the case of non-light-of-sight (NLoS) is formulated as⁵ [63]

$$\begin{aligned} \Omega = - 31.7 \log (d) - 20 \log (f_c) + G_\text {tx} + G_\text {rx} -32.4 . \end{aligned}$$

(14)

Note that the values of path losses depend on the specific distances between S and RISs, RISs and U. Consequently, when the S-RISs and RISs-U distances are different, $\Omega _{a_l}$ and $\Omega _{b_l}$ will be different.

3.1 Outage probability

The OP of the considered communications with TAS is formulated as

$$\begin{aligned} \mathcal {P}_\text {out} = \text {Pr} \{\rho _\text {U} < \rho _\text {th}\}, \end{aligned}$$

(15)

where $\rho _\text {U}$ is the SDNR of the considered communications with TAS that is given in (10); $\rho _\text {th} = 2^\mathcal {R}-1$ is defined as the SDNR threshold. Herein, $\mathcal {R}$ is the required rate of the considered communications calculated in bits per channel use (bpcu).

Substituting $\rho _\text {U}$ in (10) into (15), the OP is then formulated as

$$\begin{aligned} \mathcal {P}_\text {out} = \text {Pr} \left\{ \frac{ \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) ^2 P_\text {S}}{\Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) ^2 \Big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \Big ] P_\text {S} + \sigma ^2_\text {U} } < \rho _\text {th} \right\} . \end{aligned}$$

(16)

Then, (16) is reorganized as

$$\begin{aligned} \mathcal {P}_\text {out} = \text {Pr} \left\{ \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) ^2 \Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S} < \sigma ^2_\text {U} \rho _\text {th} \right\} . \end{aligned}$$

(17)

It is clear from (17) that when $1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \le 0$, this probability is always true. It is because the left term, $\Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) ^2 \Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}$, is either less than or equal to zero while the right term, $\sigma ^2_\text {U} \rho _\text {th}$, is greater than zero. In other words, $\mathcal {P}_\text {out} = 1$ when $1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \le 0$.

When $1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} > 0$, (17) is equivalent to

$$\begin{aligned} \mathcal {P}_\text {out} = \text {Pr} \left\{ \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \Big ) ^2 < \frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}} \right\} . \end{aligned}$$

(18)

Since S applies TAS scheme, the channel magnitudes are subject to the condition in (19), i.e.,

$$\begin{aligned} \nonumber&\sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{ilk} \bar{b}_{lk} + \bar{c}_{isu} \\&= \max \Bigg (\sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{1lk} \bar{b}_{lk} + \bar{c}_{1su}, \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{2lk} \bar{b}_{lk} + \bar{c}_{2su}, ..., \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{Nlk} \bar{b}_{lk} + \bar{c}_{Nsu} \Bigg ). \end{aligned}$$

(19)

As a result, (18) is equivalent to

$$\begin{aligned} \nonumber \mathcal {P}_\text {out} =&\; \text {Pr} \left\{ \Big (\sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{1lk} \bar{b}_{lk} + \bar{c}_{1su} \Big )^2< \frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}} \right\} \\ \nonumber&\cup \text {Pr} \left\{ \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{2lk} \bar{b}_{lk} + \bar{c}_{2su} \Big )^2< \frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}} \right\} \cup ...\\&\cup \text {Pr} \left\{ \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{Nlk} \bar{b}_{lk} + \bar{c}_{Nsu} \Big )^2 < \frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}} \right\} . \end{aligned}$$

(20)

Since

$$\begin{aligned} \nonumber \mathbb {E}\left\{ \Big (\sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{1lk} \bar{b}_{lk} + \bar{c}_{1su} \Big )^2 \right\}&= \mathbb {E}\left\{ \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{2lk} \bar{b}_{lk} + \bar{c}_{2su} \Big )^2 \right\} = ...\\&= \mathbb {E}\left\{ \Big ( \sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{Nlk} \bar{b}_{lk} + \bar{c}_{Nsu} \Big )^2 \right\} , \end{aligned}$$

(21)

$\mathcal {P}_\text {out}$ in (20) can be straightforwardly obtained when only one probability in (20) is solved. As a result, (20) is equivalent to

$$\begin{aligned} \mathcal {P}_\text {out} =&\Bigg [ \text {Pr} \Bigg \{\Big (\sum _{l=1}^{M} \sum _{k=1}^{G_l} \bar{a}_{lk} \bar{b}_{lk} + \bar{c}_{su} \Big )^2 < \frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}} \Bigg \} \Bigg ]^N , \end{aligned}$$

(22)

where $\bar{a}_{lk}$ and $\bar{c}_{su}$ are, respectively, used to denote the channel magnitudes from an arbitrary antenna of S to the $l^{th}/k^{th}$ and to U.

Let $\mathcal {X}_{lk}= \bar{a}_{lk} \bar{b}_{lk}$, $\mathcal {Y}_{l} = \sum _{k=1}^{G_l} \mathcal {X}_{lk}$, ${\mathcal {Z}}= \sum _{l=1}^{M} \mathcal {Y}_{l}$, and ${\mathcal {T}} = {\mathcal {Z}} + \bar{c}_{su}$. Then, (22) becomes

$$\begin{aligned} \mathcal {P}_\text {out} = \Bigg [ \text {Pr} \Bigg \{\mathcal {T}^2 < \frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}} \Bigg \} \Bigg ]^N . \end{aligned}$$

(23)

From (23), the OP of the considered multi-RIS-assisted MW communications with ITH and with TAS is formulated as follows.

Theorem: Under the effects of ITH, the OP of the considered multi-RIS-assisted MW communications with TAS over realistic Nakagami-m channels when $1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} > 0$ is given as

$$\begin{aligned} \nonumber&\mathcal {P}_\text {out} = \sum \limits _{n=0}^{N} {N \atopwithdelims ()n} \frac{(-1)^n}{\Gamma ^n \Big ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \Big )} \\&\times \Gamma ^n \Bigg ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2}, \frac{\Delta _\mathcal {T}(1) }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \sqrt{\frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}}} \Bigg ), \end{aligned}$$

(24)

where ${N \atopwithdelims ()n} = \frac{N!}{n!(N-n)!}$; $\Delta _\mathcal {T}(1)$ and $\Delta _\mathcal {T}(2)$ denote the first and second moments of ${\mathcal {T}}$, respectively, given in (45) and (46), respectively.⁶

Proof

Please refer to Appendix. $\square$

3.2 System throughput

Besides the OP, the ST is an essential performance metric of wireless communications. In particular, the ST of the multi-RIS-assisted MW communications with ITH and with TAS is calculated as

$$\begin{aligned} \mathcal{S}\mathcal{T} = \mathcal {R}(1 - \mathcal {P}_\text {out}). \end{aligned}$$

(25)

Substituting $\mathcal {P}_\text {out}$ in (24) into (25), the ST is then formulated as

$$\begin{aligned} \nonumber&\mathcal{S}\mathcal{T} = \mathcal {R}\Bigg [ 1 -\sum \limits _{n=0}^{N} {N \atopwithdelims ()n} \frac{(-1)^n}{\Gamma ^n \Big ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \Big )} \\&\times \Gamma ^n \Bigg ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2}, \frac{\Delta _\mathcal {T}(1) }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \sqrt{\frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}}} \Bigg ) \Bigg ]. \end{aligned}$$

(26)

4 Numerical results and discussion

In this section, the behaviors of the considered communications are provided by using the obtained formulas. Monte-Carlo simulations using MATLAB software with $10^6$ channel realizations are conducted to confirm the exactness of the obtained formulas. To clarify the great benefits of TAS, the performance of the considered communications with TAS is compared with that in the case without TAS. Moreover, the performance with RISs is also compared with that in the case without RISs. In addition, the strong impacts of ITH are demonstrated when the performances with ITH and PTH are compared. In all investigated scenarios, the number of RISs is $M=3$ and the shape parameters are $m_{su}= m_{a_l} = m_{b_l} = 2$. To derive $\sigma ^2_\text {U}$ in (11), the thermal noise power density and noise figure are, respectively, set as $N_0 = -174$ dBm/Hz and $N_F=10$ dBm while the bandwidth is changed. Similarly, to obtain $\Omega$ in (14), the antenna gains are fixed, i.e., $G_\text {tx}= G_\text {rx} = 5$ dB while $f_c$ and d are varied depending on the specific scenarios.⁷ Moreover, the certain locations of S and U are fixed, i.e., $(\hat{x}_\text {S}, \hat{y}_\text {S}) =(0,20)$ and $(\hat{x}_\text {U}, \hat{y}_\text {U}) =(100,2)$ where $(\hat{x}, \hat{y})$ is the two-dimensional coordinates. Meanwhile, the RISs’ positions are expressed via vector ${\textbf {x}}_\text {RISs}$ and ${\textbf {y}}_\text {RISs}$, e.g., ${\textbf {x}}_\text {RISs} =[20, 40, 60]$ and ${\textbf {y}}_\text {RISs} = [5, 10, 15]$ mean that $(\hat{x}_{\text {RIS}_1}, \hat{y}_{\text {RIS}_1}) =(20,5)$, $(\hat{x}_{\text {RIS}_2}, \hat{y}_{\text {RIS}_2}) =(40,10)$, and $(\hat{x}_{\text {RIS}_3}, \hat{y}_{\text {RIS}_3}) =(60,15)$.⁸ The ITH levels at S and U are aggregated as $\alpha ^2 = (\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2$. For reading conveniently, the notations in the figures are listed in Table 3 while the parameter settings are listed in Table 4.⁹

Table 3 The notations used in the figures

Table 4 Parameter settings

Fig. 2

The OP of the proposed communications versus transmit power in comparison with the OPs of relevant communications with $N=5$ antennas, $\textbf{x}_\text {RIS} = [20,40,60]$, $f_c=50$ GHz, $\mathcal {R}=3$ bpcu, $B_W =10^4$ KHz, and $\alpha ^2 =0.1$

Figure 2 depicts the OP of the considered communications (I-Wi-R & Wi-T) with $N=5$ antennas, $\textbf{x}_\text {RIS} = [20,40,60]$, $f_c=50$ GHz, $\mathcal {R}=3$ bpcu, $B_W =10^4$ KHz, and $\alpha ^2 =0.1$. To demonstrate the benefits of using RISs and TAS as well as the strong impacts of ITH, the OPs of relevant communications are also provided for clear comparisons. Specifically, the OPs with PTH, with RISs, and with TAS (P-Wi-R & Wi-T), with ITH, with RISs, and without TAS (I-Wi-R & Wo-T), with PTH, with RISs, and without TAS (P-Wi-R & Wo-T), with ITH, without RISs, and with TAS (I-Wo-R & Wi-T), with PTH, without RISs, and with TAS (P-Wo-R & Wi-T), with ITH, without RISs, and without TAS (I-Wo-R & Wo-T), with PTH, without RISs, and without TAS (P-Wo-R & Wo-T) are detailedly presented in Fig. 2. The analytical curve of the OP of the considered communications is obtained by using (24). Moreover, (24) is also used to derive the analytical curve in the case of P-Wi-R & Wi-T by setting $\alpha ^2 =0$. Firstly, the effects of ITH are significant because the OPs in the cases of I-Wi-R & Wi-T, I-Wi-R & Wo-T, I-Wo-R & Wi-T, and I-Wo-R & Wo-T are greatly higher than those in the cases of P-Wi-R & Wi-T, P-Wi-R & Wo-T, P-Wo-R & Wi-T, and P-Wo-R & Wo-T, respectively. More specifically, to reach $\mathcal {P}_\text {out} = 10^{-4}$, the considered communications have to use $P_\text {S} =47.5$ dBm while it is only 42.5 dBm in the case of P-Wi-R & Wi-T. In other words, $P_\text {S}$ of the ITH communications has to be higher than that of PTH communications. Moreover, Fig. 2 demonstrates the great benefits of TAS and RISs used in the MW communications. Particularly, with $\mathcal {P}_\text {out} = 10^{-4}$, the transmit powers are 47.5 dBm and 57 dBm corresponding to the cases I-Wi-R & Wi-T and I-Wi-R &Wo-T. In other words, the transmit power with TAS is 10 dBm lower than that without TAS. In addition, the OPs of I-Wi-R & Wi-T, P-Wi-R & Wi-T, I-Wi-R & Wo-T, and P-Wi-R & Wo-T (the MW communications with RISs) are, respectively, lower than those of I-Wo-R & Wi-T, P-Wo-R & Wi-T, I-Wo-R & Wo-T, and P-Wo-R & Wo-T (the MW communications without RISs). These results confirm the huge benefits of exploiting RISs and TAS in the proposed communication. Thus, utilizing RISs greatly improves the OP performance of the MW communications even with ITH.

Fig. 3

The impacts of $\alpha ^2$ on the OP of the considered communications with $N=5$ antennas, $P_\text {S} =40$ dBm, $\textbf{x}_\text {RIS} = [20,40,60]$, $f_c=50$ GHz, $B_W =10^4$ KHz, and $\mathcal {R}=2$ bpcu

In Fig. 3, the OPs of ITH communications are illustrated where the ITH level $\alpha ^2$ is varied from 0 to 0.3. The system parameters are $N=5$ antennas, $P_\text {S} =40$ dBm, $\textbf{x}_\text {RIS} = [20,40,60]$, $f_c=50$ GHz, $B_W =10^4$ KHz, and $\mathcal {R}=2$ bpcu. It is worth noticing that when $\alpha ^2 = 0$, the OPs of ITH communications become the OPs of PTH communications. It is clear in Fig. 3 that the impacts of $\alpha ^2$ are significant even with low values. In particular, when $\alpha ^2$ increases from 0 (PTH) to 0.06, the OPs of I-Wi-R & Wi-T, I-Wi-R & Wo-T, I-Wo-R & Wi-T, and I-Wo-R & Wo-T increase from $1.8 \times 10^{-6}$, $1.6 \times 10^{-4}$, $8.3 \times 10^{-2}$, and $1.8 \times 10^{-1}$ to $1.9 \times 10^{-5}$, $7.2 \times 10^{-4}$, $1.2 \times 10^{-1}$, and $2.3 \times 10^{-1}$, respectively. Moreover, when $\alpha ^2$ increases, the OPs with TAS (I-Wi-R & Wi-T and I-Wo-R & Wi-T) are quickly increased while the OPs without TAS (I-Wi-R & Wo-T and I-Wo-R & Wo-T) are slowly increased. As a result, the effects of $\alpha ^2$ on the OPs with TAS are higher than those without TAS. In addition, the OP of I-Wo-R & Wi-T is dramatically lower than that of I-Wi-R & Wo-T. Once more, this characteristic serves as further evidence of the significant advantages associated with the TAS scheme employed in this article. In other words, the performance of classical communications without RISs and with TAS is dramatically higher than that of advanced communications with RISs and without TAS.

Fig. 4

The benefits of TAS in the considered communications with $N=5$ antennas, $\textbf{x}_\text {RIS} = [20,40,60]$, $f_c=50$ GHz, $\mathcal {R}=3$ bpcu, $B_W =10^4$ KHz, and $\alpha ^2 =0.1$

Figure 4 visually represents the advantages of TAS in the MW communications with and without RISs, with ITH and PTH. It is obvious that an increase in N greatly reduces the OPs of MW communications. Specifically, at $P_\text {S} =46$ dBm, the OP of the considered communications (I-Wi-R & Wi-T) reduces from $8.2 \times 10^{-2}$ to $2.5 \times 10^{-2}$ and $2.1 \times 10^{-3}$ when N increases from 2 to 3 and 5. For a target of OP, i.e., $\mathcal {P}_\text {out} = 10^{-4}$, the transmit power of the considered communications are 52, 49.2, and 47.5 dBm corresponding to $N=2,3$, and 5. As a result, an increase in N can dramatically reduce the transmit power of S to obtain the specific target of OP. On the other hand, the characteristics of the OPs of P-Wi-R & Wi-T, I-Wo-R & Wi-T, and P-Wo-R & Wi-T are similar to those of the OP of the considered communications. Hence, employing a greater number of transmit antennas at S proves to be an efficient approach for enhancing the OP performance of the MW communications with ITH.

Fig. 5

The OP of the considered communications for different RISs’ positions with $N=5$ antennas, $f_c=50$ GHz, $\mathcal {R}=3$ bpcu, $B_W =10^4$ KHz, and $\alpha ^2 =0.1$

In Fig. 5, three positions of RISs are determined, i.e., ${\textbf {x}}_\text {RIS} = [90, 90, 90]$ (case 1), ${\textbf {x}}_\text {RIS} = [10, 10, 10]$ (case 2), and ${\textbf {x}}_\text {RIS} = [50, 50, 50]$ (case 3). Unlike the RISs’ positions in Figs. 2, 3,  4 where the three RISs have different positions (${\textbf {x}}_\text {RIS} = [20, 40, 60]$), the positions of the three RISs in Fig. 5 are similar in an investigated case. In other words, three RISs are grouped and located in a certain area. Among the three investigated areas, the OP in case 1 is the best while the OP in case 3 is the worst. Moreover, the OPs in cases 1 and 2 are nearly similar. It is noteworthy that the RISs are in the middle between S and U in case 3 while they are near to either S or U in cases 1 and 2. From these features, it can be concluded that the performance of the RISs-assisted MW communications can be dramatically improved by locating RISs near either the receiver or transmitter. Thus, it is better to deeply investigate the positions of RISs before deploying them in practice.

Fig. 6

The ST of the considered communications in comparison with STs of relevant communications for $N=5$ antennas, $f_c=50$ GHz, $\mathcal {R}=3$ bpcu, $B_W =10^4$ KHz, $\alpha ^2 =0.1$, and $\textbf{x}_\text {RIS} = [20,40,60]$

Figure 6 plots the ST of the considered communications in comparison with STs of relevant communications for $N=5$ antennas, $f_c=50$ GHz, $\mathcal {R}=3$ bpcu, $B_W =10^4$ KHz, $\alpha ^2 =0.1$, and $\textbf{x}_\text {RIS} = [20,40,60]$. Herein, the analytical results of I-Wi-R & Wi-T and P-Wi-R & Wi-T are plotted by using (26). It is also noted that the ST of P-Wi-R & Wi-T is obtained by setting $\alpha ^2 =0$ in (26). Similar to the OP, the STs with ITH are greatly lower than the STs with PTH. This feature demonstrates a strong effect of ITH on the ST of the considered communications. In other words, the impact of ITH is substantial and should not be disregarded. More specifically, the considered communications have to use $P_\text {S} = 46$ dBm to reach the ST target of 3 bpcu. Meanwhile, it is only 40 dBm in the case of P-Wi-R & Wi-T. On the other hand, TAS can significantly enhance the ST of the considered communications. It is because without TAS, the I-Wi-R & Wo-T communications have to use $P_\text {S} = 54$ dBm to reach the ST target of 3 bpcu. As the results, TAS dramatically reduces the OP and considerably increases the ST of the considered communications.

Fig. 7

The ST of the considered communications with different frequencies in MW bands for $N=5$ antennas, $\mathcal {R}=3$ bpcu, $B_W =10^4$ KHz, $\alpha ^2 =0.1$, and $\textbf{x}_\text {RIS} = [20,40,60]$

In Fig. 7, the carrier frequency is varied, i.e., $f_c=30,50$, and 70 GHz while the other parameters are akin or comparable to those mentioned previously. Figure 7 clarifies the great impacts of $f_c$ on the ST of the considered communications. In particular, when $f_c$ increases from 30 to 50 and 70 GHz, $P_\text {S}$ has to increase from 38 to 44 and 48 dBm to reach the ST target of 3 bpcu. In other words, when $f_c$ is 20 GHz higher, the transmit power has to increase about 4 dBm to keep the ST unchanging. Besides increasing the transmit power, an increase in N is also a good solution to indirectly take care of ITH and high carrier frequencies. As a result, it is important to use suitable frequencies in MW bands to avoid the performance reduction induced by the high carrier frequencies. When a higher carrier frequency has to be used, we should use a larger number of transmit antennas to keep similarly the ST of the considered communications.

Fig. 8

The effects of bandwidth on the ST of the considered communications with $N=5$ antennas, $\mathcal {R}=3$ bpcu, $f_c =50$ GHz, $\alpha ^2 =0.1$, and $\textbf{x}_\text {RIS} = [20,40,60]$

Figure 8 illustrates the effects of bandwidth on the ST of the considered communications with $N=5$ antennas, $\mathcal {R}=3$ bpcu, $f_c =50$ GHz, $\alpha ^2 =0.1$, and $\textbf{x}_\text {RIS} = [20,40,60]$. Five bandwidths are determined, i.e., $B_W=1, 10, 100, 1000$, and 10,000 KHz. Since higher $B_W$ leads to higher $\sigma ^2_\text {U}$ (referred to (11)), the ST of the considered communications is greatly reduced when $B_W$ increases. In particular, when $B_W$ is 10 times higher, i.e., it increases from 1 to 10 KHz, the transmit power has to be increased 10 dBm to reach a target of 3 bpcu. As a result, when a wider bandwidth is used in the considered communications, the transmit power, the number of RISs/REs, and the number of transmit antennas should be higher for dramatically enhancing the system performance. These approaches can be considered as the low-cost solutions to improve the performance of the considered communications.

5 Conclusion

In this article, the practical scenarios of wireless communications were determined. Particularly, multi-RIS-assisted MW communications with ITH and with TAS were investigated. Besides ITH, the noise power and channel model are also computed from practice. Consequently, the considered communications can deeply reflect the behaviors of wireless systems used in B5G networks. The mathematical formulas of OP and ST of the considered communications were derived for obtaining the system behaviors. Then, many comparing scenarios were provided. Specifically, the performances are compared between with and without TAS, with and without RISs, and with and without ITH. Numerical results confirmed the big advantages of TAS and RISs in the considered multi-RIS-assisted MW communications with ITH. Therefore, by utilizing TAS with a larger number of transmit antennas, the effects of ITH can be greatly decreased. Besides this method, various effective solutions such as using a larger number of REs, choosing suitable positions of RISs, and utilizing appropriate frequencies and bandwidths were deeply determined to gain more useful insights into the considered communication behaviors. Notably, optimization problems such as the positions of RISs, the numbers of REs in the RISs, and the optimal transmit power to reduce the impact of negative parameters and improve the system performance, should be continuously addressed. We leave them for future work.

Appendix

In this appendix, all mathematical derivations are detailedly presented to derive the OP formula of the considered communications with TAS and ITH in (24).

Since $\bar{c}_{su}$ is the single channel from S to U, its hth moment can be computed as [64]

$$\begin{aligned} \Delta _{\bar{c}_{su}}(h) \triangleq \mathbb {E}\{(\bar{c}_{su})^h\} = \int _{0}^{\infty } y^h f_{\bar{c}_{su}}(y) dy. \end{aligned}$$

(27)

After using (13), (27) becomes

$$\begin{aligned} \Delta _{\bar{c}_{su}}(h) = \frac{\Gamma (m_{su}+h/2)}{\Gamma (m_{su})} \Big ( \frac{m_{su}}{\Omega _{su}}\Big )^{-h/2}. \end{aligned}$$

(28)

From (28), the 1st and 2nd moments of $\bar{c}_{su}$ are, respectively, given as

$$\begin{aligned} \Delta _{\bar{c}_{su}}(1)&=\frac{\Gamma (m_{su}+1/2)}{\Gamma (m_{su})} \sqrt{\frac{\Omega _{su}}{m_{su}}}, \end{aligned}$$

(29)

$$\begin{aligned} \Delta _{\bar{c}_{su}}(2)&= \Omega _{su}. \end{aligned}$$

(30)

Since $\mathcal {X}_{lk} = \bar{a}_{lk} \bar{b}_{lk}$, its PDF is [65]

$$\begin{aligned} f_{\mathcal {X}_{lk}}(\rho ) = \int _{0}^{\infty } \frac{1}{y} f_{\bar{a}_{lk}} \Big (\frac{\rho }{y}\Big ) f_{\bar{b}_{lk}} (y) dy. \end{aligned}$$

(31)

Utilizing (13), (31) is now expressed as

$$\begin{aligned} \nonumber f_{\mathcal {X}_{lk}}(\rho ) =&\, \frac{4}{\Gamma (m_{b_l}) \Gamma (m_{a_l}) } \big (\frac{m_{b_l}}{ \Omega _{b_l}}\big )^{m_{b_l}} \big (\frac{m_{a_l}}{ \Omega _{a_l}}\big )^{m_{a_l}} \alpha ^{2 m_{a_l}-1} \\&\times \int _{0}^{\infty } y^{2 m_{b_l} - 2 m_{a_l} -1} \exp \big ( -\frac{\rho ^2 m_{a_l}}{\Omega _{a_l} y^2} - \frac{m_{b_l} y^2}{\Omega _{b_l}}\big ) dy. \end{aligned}$$

(32)

Applying [66, Eq. (3.478.4)], (32) becomes

$$\begin{aligned} \nonumber f_{\mathcal {X}_{lk}}(\rho ) =&\frac{4 }{\Gamma (m_{a_l}) \Gamma (m_{b_l}) } \Big (\frac{m_{a_l} m_{b_l} }{\Omega _{a_l} \Omega _{b_l} }\Big )^\frac{{m_{a_l} + m_{b_l}}}{2} \rho ^{ m_{a_l} + m_{b_l} -1} \\&\times \mathcal {K}_{m_{b_l} - m_{a_l}} \Bigg (2 \rho \sqrt{\frac{m_{a_l} m_{b_l} }{\Omega _{a_l} \Omega _{b_l} }} \Bigg ). \end{aligned}$$

(33)

Based on (33), the hth moment of $\mathcal {X}_{lk}$ is calculated as

$$\begin{aligned} \Delta _{\mathcal {X}_{lk}}(h) \triangleq \mathbb {E}\{\mathcal {X}_{lk}^h\} = \int _{0}^{\infty } y^h f_{\mathcal {X}_{lk}}(y) dy. \end{aligned}$$

(34)

After utilizing [66, Eq. (6.561.16)], (34) becomes

$$\begin{aligned} \Delta _{\mathcal {X}_{lk}}(h) = \Big (\frac{m_{a_l} m_{b_l} }{\Omega _{a_l} \Omega _{b_l} }\Big )^\frac{-h}{2} \times \frac{\Gamma (m_{a_l}+h/2) \Gamma (m_{b_l}+h/2)}{\Gamma (m_{a_l}) \Gamma (m_{b_l})}. \end{aligned}$$

(35)

Based on (35), the specific moments of $\Delta _{\mathcal {X}_{lk}}$ can be straightforwardly derived, i.e.,

$$\begin{aligned} \Delta _{\mathcal {X}_{lk}}(1)&= \sqrt{\frac{\Omega _{a_l} \Omega _{b_l} }{m_{a_l} m_{b_l} }} \times \frac{\Gamma (m_{a_l}+1/2) \Gamma (m_{b_l}+1/2)}{\Gamma (m_{a_l}) \Gamma (m_{b_l})}, \end{aligned}$$

(36)

$$\begin{aligned} \Delta _{\mathcal {X}_{lk}}(2)&= \frac{\Omega _{a_l} \Omega _{b_l} }{m_{a_l} m_{b_l} } \times \frac{\Gamma (m_{a_l}+1) \Gamma (m_{b_l}+1)}{\Gamma (m_{a_l}) \Gamma (m_{b_l})} = \Omega _{a_l} \Omega _{b_l}. \end{aligned}$$

(37)

From the above moments, the CDF of $\mathcal {X}_{lk}$ is obtained as [51]

$$\begin{aligned} F_{\mathcal {X}_{lk}}(\rho ) \approx \frac{1}{\Gamma \Big ( \frac{[\Delta _{\mathcal {X}_{lk}}(1)]^2 }{\Delta _{\mathcal {X}_{lk}}(2) - [\Delta _{\mathcal {X}_{lk}}(1)]^2} \Big )} \gamma \Bigg ( \frac{[\Delta _{\mathcal {X}_{lk}}(1)]^2 }{\Delta _{\mathcal {X}_{lk}}(2) - [\Delta _{\mathcal {X}_{lk}}(1)]^2}, \frac{\Delta _{\mathcal {X}_{lk}}(1) \rho }{\Delta _{\mathcal {X}_{lk}}(2) - [\Delta _{\mathcal {X}_{lk}}(1)]^2} \Bigg ) \cdot \end{aligned}$$

(38)

It is because $\mathcal {Y}_{l} = \sum _{k=1}^{G_l} \mathcal {X}_{lk}$, its CDF is formulated as [67]

$$\begin{aligned} F_{\mathcal {Y}_{l}}(\rho ) \approx \frac{1}{\Gamma \Big ( \frac{ [\Delta _{\mathcal {X}_{lk}}(1)]^2 G_l }{\Delta _{\mathcal {X}_{lk}}(2) - [\Delta _{\mathcal {X}_{lk}}(1)]^2} \Big )} \gamma \Bigg ( \frac{ [\Delta _{\mathcal {X}_{lk}}(1)]^2 G_l }{\Delta _{\mathcal {X}_{lk}}(2) - [\Delta _{\mathcal {X}_{lk}}(1)]^2}, \frac{\Delta _{\mathcal {X}_{lk}}(1) \rho }{\Delta _{\mathcal {X}_{lk}}(2) - [\Delta _{\mathcal {X}_{lk}}(1)]^2} \Bigg ) \cdot \end{aligned}$$

(39)

Now, the hth moment of $\mathcal {Y}_{l}$ is formulated [68]

$$\begin{aligned} \nonumber \Delta _{\mathcal {Y}_{l}}(h) \triangleq \mathbb {E}\{\mathcal {Y}_{l}^h\} =&\sum _{h_1=0}^{h} \sum _{h_2=0}^{h_1} \cdots \sum _{h_{G_l -1}=0}^{h_{G_l -2}} {h\atopwithdelims ()h_1} {h_1 \atopwithdelims ()h_2} \cdots {h_{G_l -2} \atopwithdelims ()h_{G_l -1}} \\&\times \Delta _{\mathcal {X}_{l1}}(t -h_1) \Delta _{\mathcal {X}_{l2}}(h_1 -h_2) \cdots \Delta _{\mathcal {X}_{lG_l}}(h_{G_l -1}). \end{aligned}$$

(40)

On the other hand, since ${\mathcal {Z}}= \sum _{l=1}^{M} \mathcal {Y}_{l}$, its hth moment is also formulated as

$$\begin{aligned} \nonumber \Delta _{{\mathcal {Z}}}(h) \triangleq \mathbb {E}\{{\mathcal {Z}}^h\} =&\sum _{h_1=0}^{h} \sum _{h_2=0}^{h_1} \cdots \sum _{h_{M -1}=0}^{h_{M -2}} {t \atopwithdelims ()h_1} {h_1 \atopwithdelims ()h_2} \cdots {h_{M -2} \atopwithdelims ()h_{M -1}} \\&\times \Delta _{\mathcal {Y}_{1}}(t -h_1) \Delta _{\mathcal {Y}_{2}}(h_1 -h_2) \cdots \Delta _{\mathcal {Y}_{M}}(h_{M -1}). \end{aligned}$$

(41)

Now, based on (35), (40), and (41), the specific moments of ${\mathcal {Z}}$ can be derived, i.e.,

$$\begin{aligned} \Delta _{\mathcal {Z}}(1)&= \sum _{l=1}^{M} \sum _{k=1}^{G_l} \Delta _{\mathcal {X}_{lk}}(1) , \end{aligned}$$

(42)

$$\begin{aligned} \nonumber \Delta _{\mathcal {Z}}(2) =&\sum _{l=1}^{M} \Big [ \sum _{k=1}^{G_l} \Delta _{\mathcal {X}_{lk}}(2) +2 \sum _{k=1}^{G_l} \Delta _{\mathcal {X}_{lk}}(1) \sum _{k'=k+1}^{G_l} \Delta _{\mathcal {X}_{lk'}}(1) \Big ] \\&+ 2 \sum _{l=1}^{M} \Big [ \sum _{k=1}^{G_l} \Delta _{\mathcal {X}_{lk}}(1) \Big ] \sum _{l'=l+1}^{M} \Big [ \sum _{k=1}^{L_{l'}} \Delta _{\mathcal {X}_{l'k}}(1) \Big ] , \end{aligned}$$

(43)

where $\Delta _{\mathcal {X}_{lk}}(1)$ and $\Delta _{\mathcal {X}_{lk}}(2)$ are, respectively, given in (36) and (37).

Then, the hth moment of ${\mathcal {T}} = {\mathcal {Z}} + \bar{c}_{su}$ can be formulated as

$$\begin{aligned} \nonumber \Delta _{{\mathcal {T}}}(h)&\triangleq \mathbb {E}\{( {\mathcal {Z}} +\bar{c}_{su})^h\} = \mathbb {E}\left\{ \sum _{i=0}^{h} {h \atopwithdelims ()i} \bar{c}_{su}^i {\mathcal {Z}}^{h-i} \right\} \\&= \sum _{i=0}^{h} {h \atopwithdelims ()i} \Delta _{\bar{c}_{su}}(i) \Delta _{\mathcal {Z}}(h-i). \end{aligned}$$

(44)

Based on (44), the specific moments of $\mathcal {T}$ can be derived, i.e.,

$$\begin{aligned} \Delta _\mathcal {T}(1)&= \Delta _{\mathcal {Z}}(1) + \Delta _{\bar{c}_{su}}(1) , \end{aligned}$$

(45)

$$\begin{aligned} \Delta _\mathcal {T}(2)&= \Delta _{\mathcal {Z}}(2) + \Delta _{\bar{c}_{su}}(2) + 2 \Delta _{\mathcal {Z}}(1) \Delta _{\bar{c}_{su}}(1) . \end{aligned}$$

(46)

Now, the CDF of ${\mathcal {T}}$ can be obtained from its moments [51], thus, we obtain

$$\begin{aligned} \nonumber F_{\mathcal {T}}(\rho )&= \frac{1}{\Gamma \Big ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \Big )} \gamma \Bigg ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2}, \frac{\Delta _\mathcal {T}(1) \rho }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \Bigg ) \\&= 1 - \frac{1}{\Gamma \Big ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \Big )} \Gamma \Bigg ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2}, \frac{\Delta _\mathcal {T}(1) \rho }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \Bigg ). \end{aligned}$$

(47)

Then, the OP from (23) is expressed as

$$\begin{aligned} \nonumber \mathcal {P}_\text {out}&= \Bigg [ \text {Pr} \Bigg \{\mathcal {T}^2< \frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}} \Bigg \} \Bigg ]^N \\ \nonumber&= \Bigg [ \text {Pr} \Bigg \{\mathcal {T} <\sqrt{\frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}}} \Bigg \} \Bigg ]^N \\&= \Bigg [F_{\mathcal {T}} \Bigg (\sqrt{\frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}}} \Bigg ) \Bigg ]^N. \end{aligned}$$

(48)

Applying (47), (48) is solved as

$$\begin{aligned} \mathcal {P}_\text {out} =&\Bigg [F_{\mathcal {T}} \Bigg (\sqrt{\frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}}} \Bigg ) \Bigg ]^N \nonumber \\ =&\Bigg [1 - \frac{1}{\Gamma \Big ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \Big )} \end{aligned}$$

(49)

$$\begin{aligned}&\times \Gamma \Bigg ( \frac{[\Delta _\mathcal {T}(1)]^2 }{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2}, \frac{\Delta _\mathcal {T}(1)}{\Delta _\mathcal {T}(2) - [\Delta _\mathcal {T}(1)]^2} \sqrt{\frac{\sigma ^2_\text {U} \rho _\text {th}}{\Big ( 1 - \big [(\alpha _\text {S}^\text {t})^2 + (\alpha _\text {U}^\text {r})^2 \big ] \rho _\text {th} \Big ) P_\text {S}}} \Bigg ) \Bigg ]^N. \end{aligned}$$

(50)

Utilizing the binomial theorem, the OP formula of the considered communications with TAS is derived as (24) in Theorem. The proof is now complete. $\square$