Capacity Performance Analysis for Terrestrial THz Channels

Abstract

The outdoor terrestrial terahertz (THz) communication links have recently attracted great research and commercial interest in response to the emerging bandwidth-hungry demands for extremely high-speed wireless data transmissions. However, their development is hindered by the random behavior of the atmospheric channel due to the molecular attenuation, adverse weather effects, and atmospheric turbulence (along with free space path loss (FSPL) and pointing errors) due to the stochastic misalignments between the transmitter and the receiver. Thus, in this work, we investigate the joint influence of these detrimental effects on both capacities, i.e., average (ergodic) and outage, of such a typical line of sight (LOS) THz communication link. Specifically, atmospheric turbulence-induced intensity fluctuations can be modeled by using either the suitable gamma or the well-known gamma–gamma distribution for weak and moderate to strong turbulence conditions, respectively. Additionally, weak to strong stochastic misalignment-induced intensity fluctuations, due to generalized pointing errors with non-zero boresight (NZB), are emulated by the appropriate Beckman distribution. Taking into additional consideration the unavoidable presence of FSPL and the different but realistic water vapor concentration values along with the influence of weather conditions, an outage performance analysis has been conducted. Considering the abovementioned significant effects, novel analytical mathematical expressions have been extracted for both average (ergodic) and outage capacity, which are critical metrics that first incorporate the total influence of all of the above significant effects on the THz links’ performance. Through the derived expressions, proper analytical results verified by simulations are presented and demonstrate the validity of our analysis. It is notable that the derived expressions can accommodate realistic parameter values involved in all the above-mentioned major effects and link characteristics. In this context, they provide encouraging quantitative results and outcomes for both capacity metrics under investigation. The latter enables the design and the establishment of modern and future high-speed THz links, which are expected to cover longer propagation distances and thus become even more vulnerable to atmospheric turbulence effect. This is modeled and incorporated in our analysis and expressions contrary to most of the previous works in the open technical literature.

1. Introduction

Over the last few years, the development of terahertz (THz) wireless communication links has been the focus of significant research and commercial efforts. This is mainly due to their potential to meet the increasing bandwidth-consuming demands for higher data rate wireless transmissions at a higher security level as well as the extremely large amount of unlicensed spectra that accommodate the THz band, which remains one of the least explored frequency bands for communication applications [1,2,3,4]. Contrary to its immediately below millimeter wave (MMW) frequency band (which has been lately utilized for fifth generation (5G) mobile communication systems), THz systems offer higher capacity in links. Indeed, they have a larger atmospheric transmission window and thus possess the potential to achieve data rates beyond several Gbps and reach capacities of 100 Gbps to 1Tbps, which is sufficient to cover the growing sixth generation (6G) future demands [5,6,7]. The latter is a vital issue, considering that the global mobile traffic volume was 7.462 EB/month in 2010, and it is predicted to be 5016 EB/month in 2030 [8]. Additionally, the THz frequency spectrum is largely unregulated in contrast with the MMW frequency spectrum. Furthermore, THz systems outperform their MMW counterparts in terms of more compact transceiver antenna equipment and inherently more directional links owing mainly to less free-space diffraction of the waves [9]. On the other hand, immediately above THz frequency band there is the infrared (IR) frequency band where optical wireless communication (OWC) links, commonly known as free space optical (FSO) links, usually operate [4,10]. Contrary to their FSO counterparts, THz have been experimentally proven to be less susceptible to signal scintillation and attenuation effects resulting from the presence of atmospheric turbulence, fog, dust, smoke, and/or clouds, respectively [11,12,13,14]. Moreover, contrary to FSO, the emitted power and, as a result, the performance and the availability of THz links is not degraded by any eye-safety limitations; plus, THz are immune to ambient and infrared light noise [15]. In view of the above distinct advantages of THz links, the latter can potentially adapt varying and vital applications such as high-speed transmission of high-definition television (HDTV) signals, health monitoring, high-speed sensor networks, secure terabit wireless communication in the military field, terabit wireless personal area networks, terabit wireless local area networks, solution for backhaul or bottleneck problem, machine-to-machine communication, and augmented reality (along with virtual reality [4,6,7,8]).

Nevertheless, regarding the establishment of outdoor THz communication systems, there are some major impairments that mainly arise from the random nature of the atmospheric channel. Still, airborne particles that emerge across the atmospheric channel during adverse weather effects that attenuate the propagating THz signals, with rain droplets being by far the most significant attenuators. Indeed, conversely to fog droplets, rain attenuates more THz radiation than IR light [16,17,18]. Additionally, molecular absorption that basically stems from oxygen and especially water vapor in the air channel plays a dominant detrimental role in the propagation of THz information-bearing waves. Specifically, from a communication perspective, molecular absorption adds a margin of noise, which limits the communication range [19]. Regarding the total attenuation that experiences the propagating THz signal through the atmospheric channel, there is also an unavoidable loss that always exists, commonly known as free space path loss (FSPL) [20].

Even in a clear sky, atmospheric turbulence originates from solar radiation absorbed by Earth’s surface that makes the air around the earth surface to be warmer than that at higher altitude. This complex stochastic phenomenon causes random refractive index fluctuations, which in turn create random intensity variations of the received THz signal, the so-called scintillation effect analogous to fading in RF systems, and results in performance degradation of the link (especially for long propagation distances [20,21,22,23,24,25,26,27]).

Another significant limitation factor for THz performance is the pointing errors (PE) effect between transceiver apertures. This effect causes random intensity fluctuations of the THz signal, which arrives at the receiver’s input [2,28,29,30]. It should be mentioned here that this effect includes boresight and jitter components. The boresight is a permanent displacement between the beam and the detector aperture, while the spatial jitter is the random offset of the beam center at the receiver [24,31].

Unlike the case with FSO communication systems [28,32,33,34,35,36,37,38,39,40], the joint impact of most of the above critical effects on the capacity performance and availability of THz links has not yet been extensively investigated. In more detail, the feasibility of establishing practical terrestrial outdoor line of sight (LOS) THz communication links under various weather and channel conditions has been recently shown in [41,42,43]. Note that during the previous years the poor development of THz devices up to this time dramatically limited the achievable outdoor THz link lengths from just 0.5 m up to 22 m operating at 300 GHz. However, owing to the impressive development of THz devices that has been witnessed just over the last few years, the practical achievable outdoor THz propagation distance has been extended in acceptable and remarkable ways for communication practical link lengths, i.e., from 186 m up to 1 km [41,42,43]. In this respect, we don’t extensively refer here to the earlier background, i.e., to what happened on the development of THz links before the publication of papers [41,42,43]. Furthermore, relying mainly on the molecular absorption loss model proposed in [44], authors in [45] suggested another THz link incorporating losses due to the propagation path as well as critical parameters concerning the transceivers and molecular absorption. According to this model, the capacity up to 0.4 THz is evaluated. In these papers it is also validated that, for such THz transmissions, attenuation caused by the water vapor is more important than the attenuation caused by the oxygen, and thus the focus on the dominant molecular attenuation factor within the frequency regime that emerging THz systems operate. Attenuation of THz radiation (which is caused by fog, clouds, or rain) has been investigated by means of laser environmental effects definition and reference (LEEDR) computation tool in [46], whereas in [7,13,14,16,17,18,42,43,47], the same effects have been studied theoretically and/or experimentally. Their findings indicate a good agreement between the expected results and the measurements and it is shown that the rain has the most significant negative influence on the THz links’ performance and availability. In this context, it has been shown that the attenuation caused by the weather effects should be considered mainly for long THz links, whereas it has been verified that, under fog conditions, THz outperforms FSO, which makes THz a viable alternative to FSO (especially in foggy channels). Moreover, atmospheric turbulence-induced scintillation effects on THz propagating signals have been initially theoretically investigated via the scintillation index metric in [48], and then experimentally in [14,49]. Their results imply that, while atmospheric turbulence is a more significant issue for IR than for THz links, it should be taken into consideration especially for the emerging near the ground THz links that cover longer propagation distances. Considering that atmospheric turbulence causes rapid fluctuations in the signal’s intensity, it should be described by using the appropriate statistical models. In this respect, authors in [20,21], used the well-known distribution models from the FSO systems for the THz links, i.e., lognormal, gamma–gamma, etc. Each one of the above-mentioned models is suitable to describe the influence of the atmospheric turbulence effect depending on its strength. More recently, authors in [50] proposed gamma distribution for weak turbulence-induced scintillations as a less complex alternative to the lognormal one to estimate the outage probability of a typical THz link. The PE effect between the transmitter and the receiver of the THz link devices was first studied as a part of the shadowing effect in [51], and next in [52,53] by deterministic models; however, these models are not able to accommodate their stochastic nature. The stochastic behavior of PE for THZ links has been recently assessed in [1,2,54] relying mainly on the Rayleigh distribution model [24], which has been used for FSO communication links. Thus, the ergodic (average) capacity has been evaluated in [2] under the presence of stochastic zero boresight pointing errors, FSPL, molecular attenuation, multipath fading along with hardware imperfections for different THz link propagation distances, relative humidity, operation frequency, signal to noise ratio (SNR), and spatial jitter values. Next, authors in [1] investigated the total impact of stochastic zero boresight PE along with FSPL, molecular attenuation, and different THz link and antenna characteristics on the THz average capacity by utilizing three different movement models. The joint impact on the average LOS THz capacity of the atmospheric turbulence effect along with zero boresight stochastic pointing errors (also including FSPL and molecular attenuation) has been first evaluated in [20]. Still, the outage capacity metric has not been reported for LOS THz links in the open technical literature as well as the joint influence of atmospheric turbulence, generalized PE with NZB, FSPL, and molecular attenuation along with different weather effects on their average capacity have not yet been investigated.

Motivated by the above, in this work we investigate, for first time, both outage and average capacity metrics for a typical LOS THz link under the combined influence of atmospheric turbulence, generalized NZB PE, FSPL, and molecular attenuation caused by the water vapor along with clear, fog or rain weather conditions. By emulating turbulence-induced scintillations by gamma or gamma–gamma distributions for weak, moderate, or strong turbulence conditions and generalized pointing errors via Beckmann distribution, an outage and average capacity analysis is performed. In this context, another key novelty of this work is that, for the first time in the open THz literature, closed form mathematical expressions are derived either for the outage capacity or for the average capacity of a typical LOS THz link, including the joint influence of all these above-mentioned major factors that affect the THz capacity performance and availability. Additionally, for realistic THz link scenarios, we present proper analytical results obtained via the extracted expressions, which are further verified by simulations. Specifically, these are Monte Carlo simulations, which have been performed with the help of well-known mathematical software. Consequently, insightful observations are provided for the influence of each major factor (much less for their total impact on the link’s capacity availability and performance), which can be used for the design of THz links.

2. Signal and Channel Model

2.1. The THz Link

The LOS THz link under investigation is assumed to employ the on-off keying (OOK) modulation format, which is well-known for industrial and commercial applications mainly due to its simplicity. Thus, the arriving THz signal at the receiver is expressed as

$$y=hx+n$$

(1)

with $h$ denoting the channel state due to turbulence, generalized pointing errors, and total signal attenuation, $x=\{0 \mathrm{o}\mathrm{r} 2P_t\}$ being the emitted OOK information signal where $P_t$ represents the average transmitted signal power, and $n$ standing for the corresponding additive Gaussian white noise with variance ${\sigma }_n^2$ [20].

The above coefficient, h, can be expressed as

$$h=h_l{h}_p{h}_a$$

(2)

where the signal’s intensity of the THz channel is obtained through the total attenuation ($h_l$), the pointing errors ($h_p$), and the turbulence effect ($h_a$) [25,28,50].

2.2. Total Attenuation

Furthermore, the factor $h_l,$ can be written as

$$h_l=h_{fl}{h}_{vl}{h}_{wl}$$

(3)

where the total attenuation of the THz channel depends on the FSPL ($h_{fl}$), the molecular attenuation ($h_{\nu l}$), and the weather effects attenuation ($h_{wl}$). Note that the $h_{wl}$ is taking various values fog or rain, while it is equal to one for clear weather conditions as has been already reported in [2,20,50], where the presence of any adverse weather effect has been neglected.

Specifically, the attenuation term due to FSPL is described by Friis equation as [54]

$$h_{fl}=\frac{c\sqrt{G_t{G}_r}}{4\pi fz}$$

(4)

with $c$ being the light’s speed in the free space, $f$ being the operation frequency of the information-bearing THz carrier wave, $z$ standing for the length of the link, and $G_t$, $G_r$ representing the transmission and reception antenna gains, respectively.

According to the Beer–Lambert law, the molecular attenuation factor due to the signal’s absorption along the propagation path is given as [11]

$$h_{vl}=\exp \left(-a_{v}z\right)$$

(5)

where $a_{\nu }$ denotes the attenuation coefficient (m⁻¹), with the link length of the THz channel, $z$, being expressed in meters. Taking into consideration that water vapor is the dominant attenuation factor for the outdoor LOS THz links at low altitudes [6,9,55], it is assumed without loss of generality that $a_{\nu }$ coincides with the attenuation coefficient due to water vapor, which at $T_0=20 °\mathrm{C}$ surface temperature and $f\le 350 \mathrm{G}\mathrm{H}\mathrm{z}$ is given as [[56], Equation (3.2)]

$$\begin{gathered} a_v\left(\mathrm{d}\mathrm{B}/\mathrm{k}\mathrm{m}\right)=\left[0.067+\frac{2.4}{{\left(f-22.3\right)}^2+6.6}+\frac{7.33}{{\left(f-183.5\right)}^2+5}\right. \\ \left.+\frac{4.4}{{\left(f-323.8\right)}^2+10}\right]f^2\rho \times 10^{-4}, \end{gathered}$$

(6)

where the frequency $f$ should be now expressed in GHz while $\rho$ represents the water vapor concentration in g/m³. It is worth noting here that it is feasible to extend Equation (6) for different surface temperature values, i.e., the lower the surface temperatures, the higher the attenuation coefficient. Specifically, for every Celsius degree, the coefficient increases by about 1% [56].

In the same context, the attenuation term, $h_{wl}$, is given as

$$h_{wl}=\exp \left(-a_{w}z\right),$$

(7)

where $a_w$ denotes the experimental coefficient of attenuation due to weather effect (in km⁻¹) and the propagation distance $z$ (in km). Furthermore, it is worth mentioning that the attenuation due to adverse weather effect can be added with the FSPL and the molecular attenuation [20]. Here, therefore, by using the above presented expressions, the total attenuation will be jointly evaluated.

2.3. Atmospheric Turbulence Models

In this work we use two different distribution models to study weak, moderate, or strong turbulence conditions. For weak THz turbulent channels we use the more compact gamma distribution, while for moderate to strong turbulence we resort to the more complex but very accurate gamma–gamma model.

The probability density function (PDF) for the gamma–gamma distribution of the random variable $h_a$ is given as [57]

$$f_{GG,h_a}\left(h_a\right)=\frac{2{\left(\alpha \beta \right)}^{\frac{\alpha +\beta }{2}}}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{h}_a^{\frac{\alpha +\beta }{2}-1}{K}_{\alpha -\beta }\left(2\sqrt{\alpha \beta h_a}\right),$$

(8)

with K_ν(.) standing for the ν-th order modified Bessel function of the second kind [58], (8.432.2), Γ(.) denoting the gamma function [58], (8.310.1), while α and β depend on the atmospheric channel and the link’s characteristics, respectively, as [59]:

$$\begin{gathered} \alpha ={\left[\exp \left(\frac{0.49{\delta }^2}{{\left(1+0.18d^2+0.56{\delta }^{12/5}\right)}^{7/6}}\right)-1\right]}^{-1} \\ \beta ={\left[\exp \left(\frac{0.51{\delta }^2{\left(1+0.69{\delta }^{12/5}\right)}^{-5/6}}{{\left(1+0.9d^2+0.62d^2{\delta }^{12/5}\right)}^{5/6}}\right)-1\right]}^{-1}, \end{gathered}$$

(9)

where $d=0.5D\sqrt{2\pi {\lambda }^{-1}{z}^{-1}}$, with $D$ being the receiver’s aperture diameter and $\lambda =c/f$ denoting the operational wavelength, while the parameter ${\delta }^2$ stands for the Rytov variance, which, for wave propagation, is given as [60,61]

$${\delta }^2=0.5C_n^2{k}^{\frac{7}{6}}{z}^{\frac{11}{6}}$$

(10)

where $k=2\pi /\lambda$ is the wavenumber and $C_n^2$ represents the refractive index structure parameter, which is proportional to atmospheric turbulence strength [14,16,60,61].

The PDF of the gamma distribution model, which focuses on weak turbulent channels, is given as [22]

$$f_{G,h_a}\left(h_a\right)=\frac{{\zeta }^{\zeta }}{\mathrm{\Gamma }\left(\zeta \right)}{h}_a^{\zeta -1}\exp \left(-\zeta h_a\right),$$

(11)

where $\zeta$ is the gamma distribution parameter, which is related, in turn, to the parameters α and β as follows [34,50]

$$\zeta ={\left[\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\alpha \beta }\right]}^{-1},$$

(12)

2.4. Generalized Pointing Errors Model with NZB

The stochastic nature of generalized PE incorporating the boresight part is accurately described by Beckmann statistical distribution model. Its PDF of the radial displacement at the receiver $\theta$ is given as [23,24,31]:

$$\begin{gathered} f_{\theta }\left(\theta \right)=\frac{\theta }{2\pi {\sigma }_x{\sigma }_y} \\ \times {\int }_0^{2\pi }\exp \left(-\frac{{\left(\theta \mathit{cos}\phi -{\mu }_x\right)}^2}{2{\sigma }_{x,n}^2}-\frac{{\left(\theta \mathit{sin}\phi -{\mu }_y\right)}^2}{2{\sigma }_{y,n}^2}\right)d\phi ,, \end{gathered}$$

(13)

with $\phi$ representing the divergence angle related to the increase of the beam radius at the receiver with distance $z$ from the transmitter’s side, while the beam width can be approximated as $w_z\approx \phi z$ for long propagation distances. Moreover, $\theta =\sqrt{{\theta }_x^2+{\theta }_y^2}$ where ${\theta }_x^2$ and ${\theta }_y^2$ are the offsets along the horizontal and elevation axes while ${\theta }_x$ and ${\theta }_y$ are considered as nonzero mean Gaussian random variables, i.e., ${\theta }_x~N\left({\mu }_x,{\sigma }_x^2\right)$ and ${\theta }_y~N\left({\mu }_y,{\sigma }_y^2\right)$, with ${\mu }_x$ and ${\mu }_y$ denoting the mean values, whereas ${\sigma }_x$ and ${\sigma }_y$ represent the standard deviations for horizontal and elevation displacements, respectively. Considering [62], the PDF of (13) can be simplified through the modified Rayleigh model:

$$f_{\theta }\left(\theta \right)=\frac{\theta }{{\sigma }_{\mathrm{m}\mathrm{o}\mathrm{d}}^2}\exp \left(-\frac{{\theta }^2}{{2\sigma }_{\mathrm{m}\mathrm{o}\mathrm{d}}^2}\right),\theta \ge 0,$$

(14)

where ${\sigma }_{\mathrm{m}\mathrm{o}\mathrm{d}}$ denotes the joint standard deviation of ${\sigma }_x$ and ${\sigma }_y$, which can be expressed as [23]

$${\sigma }_{\mathrm{m}\mathrm{o}\mathrm{d}}^2={\left(\frac{3{\mu }_x^2{\sigma }_x^4+3{\mu }_y^2{\sigma }_y^4+{\sigma }_x^6+{\sigma }_y^6}{2}\right)}^{\frac{1}{3}},$$

(15)

Hence, according to [23,62], the PDF of $h_p$ is given as

$$f_{h_p}\left(h_p\right)=\frac{{\psi }^2}{(A_{0}g{)}^{{\psi }^2}}{h}_p^{{\psi }^2-1}, 0\le h_p\le A_{0}g,$$

(16)

where $\psi =w_{z,eq}/{2\sigma }_{\mathrm{m}\mathrm{o}\mathrm{d}}$ is directly related to the total generalized pointing error at the receiver’s input. In fact, greater values of $\psi$ correspond to weaker PE effect [24]. In the same context, ${\psi }_x=w_{z,eq}/{2\sigma }_x$ and ${\psi }_y=w_{z,eq}/{2\sigma }_y$ refer to the specific generalized misalignment strengths along the horizontal and the elevation axis, respectively. It is to be mentioned that $w_{z,eq}={\left[\sqrt{\pi }\mathrm{erf}\left(v\right)w_z^2/2v\mathrm{e}\mathrm{x}\mathrm{p}(-v^2)\right]}^{1/2}$, where $A_0={\mathrm{erf}}^2\left(v\right)$ stands for the fraction of the collected power at $r=0$, with $\mathrm{e}\mathrm{r}\mathrm{f}\left(.\right)$ denoting for the error function [63] Equation (7.1.1) and $r$ representing the radius of the aperture of the receiver, while $v=\sqrt{\pi }r/\sqrt{2}{w}_z$, with $w_z$ denoting the beam waist on the receiver plane at propagating distance $z$ [20,28]. Next, the parameter $g=\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{1}{{\psi }^2}-\frac{1}{2{\psi }_x^2}-\frac{1}{2{\psi }_y^2}-\frac{{\mu }_{x,n}^2}{2{\sigma }_x^2{\psi }_x^2}-\frac{{\mu }_y^2}{2{\sigma }_y^2{\psi }_y^2}\right)$ determines the presence of boresight component. In this respect, when the boresight displacement $s=\sqrt{{\mu }_x^2+{\mu }_y^2}$ is equal to zero, i.e., ${\mu }_x={\mu }_y=0$, and ${\sigma }_x={\sigma }_y$, then $g=1$. Consequently, for $g=1$ we have only ZB pointing errors and thus the Beckmann model simplifies to a Rayleigh distribution with $s=0$. Therefore, in this case, Equation (14) reduces to [28], Equation (10) while Equation (16) also reduces to [28] and Equation (11).

2.5. Joint Influence of Effects

The PDF of the total channel state, considering the joined influence of the above presented effects, is expressed as:

$$f_h(h)=\int f_{h\left|h_a\right.}\left(h\left|h_a\right.\right)f_{h_a}\left(h_a\right)d{h}_a,$$

(17)

where, for the sake of brevity, $f_{h_a}\left(h_a\right)$ denotes either the gamma–gamma modeled PDF or the gamma modeled PDF through Equation (8) and Equation (11), respectively. In this context, $f_{h\left|h_a\right.}\left(h\left|h_a\right.\right)$ is the corresponding conditional probability of random variable $h$ given $h_a$, while it can be expressed as [28]

$$\begin{array}{l}{f}_{h\left|h_a\right.}\left(h\left|h_a\right.\right)=\frac{d{F}_{h_p}\left(h\left|h_a{h}_l\right.\right)}{dh}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\frac{{{\left(A_{0}g\right)}^{{-\psi }^2}\psi }^2}{h_a{h}_l}{\left(\frac{h}{h_a{h}_l}\right)}^{{\psi }^2-1},0\le h\le A_{0}g{h}_a{h}_l,\end{array}$$

(18)

with $F\left(.\right)$ denoting the cumulative distribution function (CDF).

By following the analysis performed in [22], after substituting Equations (8) and (18) into Equation (17) and Equations (11) and (18) into Equation (17) for gamma–gamma modeled turbulence and for gamma modeled turbulence, respectively, and by then utilizing Equation (3) we correspondingly obtain:

$$\begin{gathered} f_{GG,h}(h)=\frac{\alpha \beta {\psi }^2}{A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)} \\ \times G_{1,3}^{3,0}\left(\frac{\alpha \beta h}{A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}{\psi }^2\\ {\psi }^2-1,\alpha -1,\beta -1\end{array}\right.\right), \end{gathered}$$

(19)

and

$$\begin{gathered} f_{G,h}(h)=\frac{\zeta {\psi }^2}{A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}\mathsf{\Gamma }\left(\zeta \right)} \\ \times G_{1,3}^{3,0}\left(\frac{\zeta h}{A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}{\psi }^2\\ {\psi }^2-1,\zeta -1\end{array}\right.\right), \end{gathered}$$

(20)

where $G_{p,q}^{m,n}\left(.\right)$ denotes the Meijer G-function [64], Equation (5).

2.6. SNR Expressions

The instantaneous signal to noise ratio (SNR) for the THz link under investigation is defined as [20]

$$\gamma =\frac{2P_{\mathrm{t}}^2{h}^2}{{\sigma }_n^2}$$

(21)

Additionally, the corresponding average SNR is defined as [20],

$$\mu =\frac{2P_{\mathrm{t}}^2{\left(E\left[h\right]\right)}^2}{{\sigma }_n^2},$$

(22)

where $\mathrm{E}\left[h\right]={\int }_0^{\infty }h{f}_h\left(h\right)dh$, with $\mathrm{E}\left[.\right]$ denoting the statistical expectation. By substituting Equations (8) and (11) into the latter integral for the gamma–gamma and gamma distribution models, respectively, and by following next the analysis performed in [65], along with using Equation (3), the latter integral gives for both distribution models the following solution [25,34,35]:

$$E\left[h\right]=h_{fl}{h}_{vl}{h}_{wl}{A}_{0}g{\left(1+{\psi }^{-2}\right)}^{-1}$$

(23)

Considering Equations (21)–(23) and Equations (8) and (11) for gamma–gamma and gamma distributions respectively, along with applying standard technique of transforming random variables $f_{\gamma }\left(\gamma \right)=\frac{f_h\left(h\right)}{\left|\frac{\partial \gamma }{\partial h}\right|}\left|\begin{array}{l}\\ A_{0}g{\left(1+{\psi }^{-2}\right)}^{-1}{h}_{fl}{h}_{vl}{h}_{wl}\sqrt{\frac{\gamma }{\mu }}\end{array}\right.$ [35], we obtain their corresponding PDFs of random variable $\gamma$ after some algebraic manipulations as follows:

$$\begin{array}{l}{f}_{GG,\gamma }\left(\gamma \right)=\frac{\alpha \beta {\psi }^2}{2\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)\gamma }\\ \hspace{1em}\hspace{1em}\hspace{1em} \times G_{1,3}^{3,0}\left(\frac{\alpha \beta \sqrt{\gamma }{\sigma }_n}{\sqrt{2}{P}_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}1+{\psi }^2\\ {\psi }^2,\alpha ,\beta \end{array}\right.\right),\end{array}$$

(24)

and

$$\begin{array}{l}{f}_{G,\gamma }\left(\gamma \right)=\frac{{\psi }^2}{2\mathsf{\Gamma }\left(\zeta \right)\gamma }\\ \hspace{1em}\hspace{1em}\hspace{1em} \times G_{1,2}^{\mathrm{2,0}}\left(\frac{\zeta \sqrt{\gamma }{\sigma }_n}{\sqrt{2}{P}_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}1+{\psi }^2\\ {\psi }^2,\zeta \end{array}\right.\right).\end{array}$$

(25)

3. Average Capacity Analysis

The instantaneous channel capacity $C$, which is a very critical metric for the performance of any communication system, is defined as [34,66]

$$C=B{\mathit{log}}_2\left(1+\gamma \right),$$

(26)

where $B$ stands for the channel’s bandwidth.

Consequently, according to Equations (21) and (26), the average capacity can be expressed through the following integral [36,39]:

$$C_{av}=\frac{1}{\mathit{ln}2}{\int }_0^{\mathrm{\infty }}B\mathit{ln}\left(1+\frac{2P_t^2{h}^2}{{\sigma }_n^2}\right)f_h\left(h\right)dh,$$

(27)

By substituting Equation (8) for gamma–gamma modeled turbulence and Equation (8) for gamma modeled turbulence into Equation (27), the latter correspondingly gives

$$\begin{array}{l}{C}_{GG,av}=\frac{\alpha \beta {\psi }^{2}B}{\mathit{ln}2A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\\ \hspace{1em}\hspace{1em}\hspace{1em} \times {\int }_0^{\mathrm{\infty }}\mathit{ln}\left(1+\frac{2P_t^2{h}^2}{{\sigma }_n^2}\right)G_{1,3}^{3,0}\left(\frac{\alpha \beta h}{A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}{\psi }^2\\ {\psi }^2-1,\alpha -1,\beta -1\end{array}\right.\right)dh,\end{array}$$

(28)

and

$$\begin{array}{l}{C}_{G,av}=\frac{\zeta {\psi }^{2}B}{\mathit{ln}2A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}\mathsf{\Gamma }\left(\zeta \right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \times {\int }_0^{\mathrm{\infty }}\mathit{ln}\left(1+\frac{2P_t^2{h}^2}{{\sigma }_n^2}\right)G_{1,2}^{\mathrm{2,0}}\left(\frac{\zeta h}{A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}{\psi }^2\\ {\psi }^2-1,\zeta -1\end{array}\right.\right)dh.\end{array}$$

(29)

Bearing in mind that $\mathit{ln}\left(1+x\right)=G_{2,2}^{1,2}\left(x\left|\begin{array}{l}1,1\\ 1,0\end{array}\right.\right)$ [64], Equation (11) and Equations (28) and (29) can be written as

$$\begin{array}{l}{C}_{GG,av}=\frac{\alpha \beta {\psi }^{2}B}{\mathit{ln}2A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}\\ \hspace{1em}\hspace{1em} \times {\int }_0^{\mathrm{\infty }}{G}_{1,3}^{3,0}\left(\frac{\alpha \beta h}{A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}{\psi }^2\\ {\psi }^2-1,\alpha -1,\beta -1\end{array}\right.\right)G_{2,2}^{1,2}\left(\frac{2P_t^2{h}^2}{{\sigma }_n^2}\left|\begin{array}{l}1,1\\ 1,0\end{array}\right.\right)dh,\end{array}$$

(30)

and

$$\begin{array}{l}{C}_{G,av}=\frac{\zeta {\psi }^{2}B}{\mathit{ln}2A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}\mathsf{\Gamma }\left(\zeta \right)}\\ \times {\int }_0^{\mathrm{\infty }}{G}_{1,2}^{2,0}\left(\frac{\zeta h}{A_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}{\psi }^2\\ {\psi }^2-1,\zeta -1\end{array}\right.\right)G_{2,2}^{1,2}\left(\frac{2P_t^2{h}^2}{{\sigma }_n^2}\left|\begin{array}{l}1,1\\ 1,0\end{array}\right.\right)dh.\end{array}$$

(31)

By next applying [64], Equation (21) to solve the integrals of Equations (30) and (31), we correspondingly obtain

$$\begin{array}{l}{C}_{GG,av}=\frac{2^{\alpha +\beta -3}{\psi }^{2}B}{\pi \mathit{ln}2\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)}\\ \times G_{8,4}^{1,8}\left(\frac{32{\left(P_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}\right)}^2}{{\left(\alpha \beta {\sigma }_n\right)}^2}\left|\begin{array}{l}1,1,\frac{1-{\psi }^2}{2},\frac{2-{\psi }^2}{2},\frac{1-\alpha }{2},\frac{2-\alpha }{2},\frac{1-\beta }{2},\frac{2-\beta }{2}\\ 1,\frac{-{\psi }^2}{2},\frac{1-{\psi }^2}{2},0\end{array}\right.\right),\end{array}$$

(32)

and

$$\begin{array}{l}{C}_{G,av}=\frac{2^{\zeta -2}{\psi }^{2}B}{\sqrt{\pi }\mathit{ln}2\mathrm{\Gamma }\left(\zeta \right)}\\ \times G_{6,4}^{1,6}\left(\frac{8{\left(P_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}\right)}^2}{{\left(\zeta {\sigma }_n\right)}^2}\left|\begin{array}{l}1,1,\frac{1-{\psi }^2}{2},\frac{2-{\psi }^2}{2},\frac{1-\zeta }{2},\frac{2-\zeta }{2}\\ 1,\frac{-{\psi }^2}{2},\frac{1-{\psi }^2}{2},0\end{array}\right.\right),\end{array}$$

(33)

where we can set $\frac{C_{av}}{B}$ standing for the average capacity over channel’s bandwidth, expressed in (b/s/Hz) [32].

It therefore becomes evident that the extracted average capacity closed-from expressions (32), and (33) include the joint influence of generalized pointing errors with NZB, the FSPL, the water vapor attenuation, and the potential emergence of adverse weather attenuation, along with gamma–gamma or gamma modeled turbulence-induced scintillations, respectively, on the average THz capacity performance.

4. Outage Capacity Analysis

A very crucial outage performance metric for the availability of any communication system is its outage capacity, $C_{\mathrm{o}\mathrm{u}\mathrm{t}}$, which refers to the capacity guaranteed for a percentage rate of $\left(100-R\%\right)$ or, equivalently, $\left(1-R\right)$, of the channel realizations [39,66,67]

$$\mathrm{Pr}\left[C<C_{\mathrm{o}\mathrm{u}\mathrm{t}}\right]=R$$

(34)

with $\mathrm{P}\mathrm{r}\left[.\right]$ denoting probability.

Considering Equations (21) and (26), it is deduced that, as is the case with the instantaneous electrical SNR, the instantaneous capacity is a random variable too. Therefore, the probability defined by Equation (34) is evaluated either for gamma–gamma or for gamma modeled turbulence as [34,39]

$$R={\int }_0^{C_{\mathrm{o}\mathrm{u}\mathrm{t}}}{f}_C\left(C\right)dC$$

(35)

with $f_C\left(C\right)$ representing the PDF of the random variable $C$ either under gamma–gamma modeled turbulent or gamma turbulent channels, respectively.

Considering Equations (21)–(23) and (26) and Equations (24) and (25) for gamma–gamma and gamma distributions respectively, along with utilizing standard technique of transforming random variables $f_C\left(C\right)=\frac{f_{\gamma }\left(\gamma \right)}{\left|\frac{\partial C}{\partial \gamma }\right|}\left|\begin{array}{l}\\ \gamma =2^{\frac{C}{B}}-1\end{array}\right.$ [35], we obtain their corresponding PDFs of random variable $C$ (after some algebraic manipulations) as follows:

$$\begin{array}{l}{f}_{GG,C}(C)=\frac{{\psi }^2{2}^{\frac{C}{B}}{\left(2^{\frac{C}{B}}-1\right)}^{-1}\mathit{ln}2}{2\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)B}\\ \hspace{1em}\hspace{1em}\hspace{1em} \times G_{1,3}^{3,0}\left(\frac{\alpha \beta {\sigma }_n\sqrt{2^{\frac{C}{B}}-1}}{\sqrt{2}{P}_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}1+{\psi }^2\\ {\psi }^2,\alpha ,\beta \end{array}\right.\right)\end{array}$$

(36)

and

$$\begin{gathered} f_{G,C}\left(C\right)=\frac{{\psi }^2{2}^{\frac{C}{B}}{\left(2^{\frac{C}{B}}-1\right)}^{-1}\mathit{ln}2}{2\mathsf{\Gamma }\left(\zeta \right)B} \\ \times G_{1,2}^{\mathrm{2,0}}\left(\frac{\zeta {\sigma }_n\sqrt{2^{\frac{C}{B}}-1}}{\sqrt{2}{P}_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}1+{\psi }^2\\ {\psi }^2,\zeta \end{array}\right.\right) \end{gathered}$$

(37)

Next, by substituting Equations (36) and (37) into Equation (35), we obtain the following expressions for the latter under gamma–gamma turbulent and gamma turbulent channels, respectively:

$$\begin{array}{l}{R}_{GG}=\frac{{\psi }^2\mathit{ln}2}{2\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)B}\\ \times {\int }_0^{C_{out}}{2}^{\frac{C}{B}}{\left(2^{\frac{C}{B}}-1\right)}^{-1}{G}_{1,3}^{3,0}\left(\frac{\alpha \beta {\sigma }_n\sqrt{2^{\frac{C}{B}}-1}}{\sqrt{2}{P}_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}{\psi }^2+1\\ {\psi }^2,\alpha ,\beta \end{array}\right.\right)dC,\end{array}$$

(38)

and

$$\begin{array}{l}{R}_G=\frac{{\psi }^2\mathit{ln}2}{2\mathsf{\Gamma }\left(\zeta \right)B}\\ \times {\int }_0^{C_{out}}{2}^{\frac{C}{B}}{\left(2^{\frac{C}{B}}-1\right)}^{-1}{G}_{1,2}^{\mathrm{2,0}}\left(\frac{\zeta {\sigma }_n\sqrt{2^{\frac{C}{B}}-1}}{\sqrt{2}{P}_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}{\psi }^2+1\\ {\psi }^2,\zeta \end{array}\right.\right)dC.\end{array}$$

(39)

In order to solve the latter integrals of Equations (38) and (39), we initially set $u=\sqrt{2^{\frac{C}{B}}-1}$, and, thus, $du=\frac{2^{\frac{C}{B}}\mathit{ln}2}{2B\sqrt{2^{\frac{C}{B}}-1}}dC$. Next, by utilizing [64] and Equation (26) and after some algebraic manipulations, we correspondingly obtain

$$R_{GG}=\frac{{\psi }^2}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}{G}_{\mathrm{2,4}}^{\mathrm{3,1}}\left(\frac{\alpha \beta {\sigma }_n\sqrt{2^{\frac{C_{\mathrm{o}\mathrm{u}\mathrm{t}}}{B}}-1}}{\sqrt{2}{P}_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}1,{\psi }^2+1\\ {\psi }^2,\alpha ,\beta ,0\end{array}\right.\right)$$

(40)

and

$$R_G=\frac{{\psi }^2}{\mathsf{\Gamma }\left(\zeta \right)}{G}_{\mathrm{2,3}}^{\mathrm{2,1}}\left(\frac{\zeta {\sigma }_n\sqrt{2^{\frac{C_{\mathrm{o}\mathrm{u}\mathrm{t}}}{B}}-1}}{\sqrt{2}{P}_t{A}_{0}g{h}_{fl}{h}_{vl}{h}_{wl}}\left|\begin{array}{l}1,{\psi }^2+1\\ {\psi }^2,\zeta ,0\end{array}\right.\right)$$

(41)

where $\frac{C_{\mathrm{o}\mathrm{u}\mathrm{t}}}{B}$ stands for the outage capacity over channel’s bandwidth, expressed in (b/s/Hz), similar to what is shown above [33].

It is thus notable that the derived outage capacity closed-form expressions (40) and (41) incorporate the combined impact of generalized pointing errors with NZB, the FSPL, the water vapor attenuation, and the potential emergence of adverse weather attenuation, along with gamma–gamma or gamma modeled turbulence-induced scintillations, respectively, on the THz outage capacity availability for every value of the probability R.

5. Analytical Results

In this section, by mainly utilizing the above derived mathematical expressions (32), (33), (40), and (41), proper analytical results are presented, which are very insightful in terms of average and outage capacity performance and availability metrics for the examined THz link. Specifically, the typical LOS THz link under investigation is $z=150 \mathrm{m}$ in length, operating with a carrier frequency equal to $f=0.3 \mathrm{T}\mathrm{H}\mathrm{z}$. Regarding the transceiver antenna characteristics, both transmitter and receiver have aperture radius $r=0.15 \mathrm{m}$ along with gain ${G=G}_{\mathrm{t}}=G_{\mathrm{r}}=55 \mathrm{d}\mathrm{B}\mathrm{i}$ [20,50,68]. The C_n² parameter is assumed to be C_n² = $5\times {10}^{-10}$ m^−2/3 for weak or C_n² = $2.3\times {10}^{-9}$ m^−2/3 for strong turbulence, as is the case with the experimental results conducted in [14]. Unless otherwise stated, gamma distribution is utilized for the weak turbulent channel case by means of Equations (32) and (40), while the gamma–gamma model is utilized for the strong turbulent channel case via Equations (33) and (41) for the average capacity and the outage capacity estimation, respectively. In addition, varying practical scenarios for stochastic generalized pointing errors have been considered, i.e., $w/r=$ 9 with $\left({\mu }_x/r,{\mu }_y/r,{\sigma }_x/r,{\sigma }_y/r\right)=\left(\mathrm{0,0},\mathrm{7,7}\right)$ for strong ZB pointing errors as well as $\left({\mu }_x/r,{\mu }_y/r,{\sigma }_x/r,{\sigma }_y/r\right)=\left(\mathrm{3,1},\mathrm{6,5}\right)$ or $\left(\mathrm{3,1},\mathrm{7,7}\right)$ for weak to strong NZB PE, respectively [62]. Moreover, according to realistic channel and environmental circumstances, it is assumed that the surface temperature is $20 °\mathrm{C}$ and the air pressure is $1 \mathrm{a}\mathrm{t}\mathrm{m}$, while $\rho =7.5 \mathrm{g}/{\mathrm{m}}^3$ or $\rho =10 \mathrm{g}/{\mathrm{m}}^3$ for moderate to strong molecular attenuation due to water vapor along the THz link, respectively [2,20,56]. Regarding the impact of weather effects, apart from the clear sky scenario along the THz channel, two other weather cases are investigated. In fact, the influence of fog attenuation is $a_w=0.6 \mathrm{d}\mathrm{B}/\mathrm{k}\mathrm{m}$, whereas the influence of rain attenuation is $a_w=3 \mathrm{d}\mathrm{B}/\mathrm{k}\mathrm{m}$ at a precipitation rate of $2 \mathrm{m}\mathrm{m}/\mathrm{h}$ [7,46], (Figure 1). Considering expression (6) in each weather scenario under investigation attenuation due to vapor dominates, while attenuation due to rain droplets is expected to increase to a greater extent the total attenuation than attenuation due to fog droplets. Note also that σ_n = ${10}^{-7} \mathrm{A}/\mathrm{H}\mathrm{z}$ in all the above-mentioned link scenarios [20,50]. Finally, for the outage capacity estimation scenarios under investigation we have set $R=0.1$ which is very common for any practical wireless communication link [33,34,35,66,67,69].

Figure 1 illustrates the average capacity evolution for the investigated THz link versus the transmitted power through a strong gamma–gamma modeled turbulent channel with either clear, rain, or fog weather conditions and moderate air humidity along with the presence of FSPL and strong ZB or NZB PE. Thus, for the same transmitted power, channel attenuation and zero boresight pointing errors, the average capacity for the THz link is significantly degraded under rain compared to fog and, especially, to a clear weather scenario. Indeed, slightly decreased average capacity values for the THz link are depicted by comparing clear to fog weather scenarios, whereas significantly decreased average capacity values are illustrated by comparing fog to rain weather scenarios, much less so clear to rain weather scenarios. This average capacity performance comparison between the three different weather scenarios (clear sky, fog, and rain) highlight that rain is the most detrimental weather effect for the propagation of THz signals, which is consistent with the experiments in [11,12,14,16,17,18]. It is also notable that, by focusing on the rain weather scenario, even lower average performance values are depicted as considering NZB pointing errors (dashed line), and, therefore, the need for also considering the boresight component on the average capacity performance estimation of the THz link is revealed. Furthermore, for every illustrated channel scenario, increased transmitted power levels lead to higher average capacity values, as it was expected.

Figure 2 is devoted to the clear sky scenario and visualizes the average capacity evolution for the investigated THz link over the same transmitted power, FSPL, and the same strong amount of NZB pointing mismatch along with weak to strong turbulent channel conditions as well as moderate to strong air humidity. Note that, regarding Figure 2, weak turbulence is pictured with dashed lines and strong turbulence is pictured with solid lines, while moderate water vapor concentration is pictured with square markers and strong water vapor concentration is pictured with triangle markers. Under these circumstances, it is observed that the case of weak turbulence along with moderate water vapor concentration (dashed line with square markers) achieves the best average capacity performance. By maintaining the moderate water vapor concentration value, we observe that the increase of atmospheric turbulence from weak to strong slightly but not negligibly decreases the average capacity of the link. On the other hand, by maintaining the weak turbulence value, we observe that the increase of water vapor concentration from moderate to strong significantly decreases the average capacity of the link. In other words, it is shown that the influence of the water vapor attenuation is more significant than the detrimental impact of turbulence on the average capacity of THz links; however, the atmospheric turbulence effect should not be neglected, since it further degrades the air humidity-induced average capacity degradation. It is remarkable that the latter behavior is qualitatively in a good agreement with experimental results in [11,14] that highlight that, for THz signal transmissions, molecular attenuation is far more destructive than atmospheric turbulence-induced scintillations, which should not be omitted in their own right. Moreover, as it has been mentioned above, for weak turbulence (dashed lines) we have selected gamma distribution, whereas for strong turbulence (solid lines), we have selected the gamma–gamma distribution. In fact, gamma–gamma distribution could have very well been utilized for weak turbulence as well. A representative example of the latter is that, for the most average capacity performance effective case, when the transmitted power is equal to $5 \mathrm{d}\mathrm{B}\mathrm{m},$ it is depicted an average capacity value equal to $9.000135124 \mathrm{b}/\mathrm{s}/\mathrm{H}\mathrm{z}$ through gamma distribution, while its corresponding value would be equal to $9.003206569 \mathrm{b}/\mathrm{s}/\mathrm{H}\mathrm{z}$ through gamma–gamma distribution. Consequently, gamma distribution is accurate enough for weak turbulence, which is also consistent with the results in [50] and has been selected due to its simplicity compared to the more complex gamma–gamma distribution. Finally, once again, even in the worst depicted case (strong water vapor concentration along with strong turbulence) the average capacity increases by increasing the transmitted power.

Figure 3 presents the outage capacity of the investigated THz link versus the transmitted power through a strong gamma–gamma modeled turbulent channel with either rain or fog adverse weather effects, moderate or strong water vapor concentration, and FSPL and weak NZB PE. As is the case with the average capacity metric (where higher values lead to enhanced performance results for the THz link), higher outage capacity metric values lead to improved availability results for the THz link. It is therefore mainly shown that more important outage performance degradations are observed for rain weather conditions along with stronger water vapor concentrations, especially for lower transmission power levels. Once again (but in terms of outage capacity availability this time), it is validated that rain is a more destructive effect than fog for THz signal transmissions. Additionally, it is highlighted that, even in rain conditions, the increase in air humidity along the THz channel from moderate water vapor concentration (dashed line) to strong water vapor concentration (solid line) strong air humidity (solid lines) remains a major limiting factor in the THz performance in terms of outage capacity metric.

Figure 4 illustrates the outage capacity evolution for the investigated THz link over the same wide range of transmitted, identical rain or fog adverse weather effects, identical moderate or strong water vapor concentration along with the presence of the same FSPL, and the same amount of weak NZB pointing mismatch (but over a weak gamma modeled turbulent channel). The performance comparison between the two last figures therefore reveals the impact of atmospheric turbulence-induced scintillations on the outage capacity availability for the THz link. Indeed, the detrimental impact of both rain and strong water vapor concentration along the propagation path is once again observed, but all of the illustrated results of Figure 4 slightly but not negligibly outperform their corresponding results from Figure 3 in terms of outage capacity for the THz link. It is therefore validated that, contrary to FSO links, the outage performance for THz links (and more specifically the outage performance in terms of outage capacity metric for THz links) is not significantly degraded to as great an extent as atmospheric turbulence along the propagation path is getting stronger. The latter does not alter the fact that the influence of atmospheric turbulence should be also considered for the design of THz links. It is important to recall here that this behavior, in terms of outage capacity metric through turbulent channels, is qualitatively consistent with the experimental findings in [24], concerning both FSO and THz signal transmissions in terms of other critical performance metrics as well as with the very recently published findings in [70], where the outage performance of such THz links is assessed in terms of the average bit error rate (ABER) metric.

6. Discussion

In this work, the capacity performance and availability for a typical LOS THz wireless communication link has been evaluated in terms of average capacity and outage capacity critical metrics, including, for the first time, the joint influence of atmospheric turbulence, generalized stochastic pointing errors where the boresight has been taken into account, attenuation due the unavoidable emergence of FSPL, and molecular attenuation due to water vapor along with attenuation due to adverse weather effects such as rain or fog. In this context, a key novelty of this work is that, for the first time, novel closed form average capacity and outage capacity mathematical expressions have been derived for both gamma and gamma–gamma modeled turbulent channels along with incorporating all of the above-mentioned major effects. Their analytical results (which are further verified by simulations) demonstrate the individual influence of each of these crucial effects, let alone their joint critical impact on THz performance and availability. Additionally, the obtained results and findings have been proven to be reasonable and qualitatively consistent with previous experimental works that evaluate other significant performance and availability metrics. In short, our findings demonstrate that, although both average capacity and outage capacity metrics obtain acceptable values for the establishment of LOS THz links under realistic link parameter values, they are significantly degraded by the joint impact of all the above impairments, especially for lower transmission power levels. Specifically, they are degraded to a greater extent by molecular attenuation due to water vapor along with rain and strong stochastic pointing errors especially when including boresight component, whereas to a smaller extent they are also degraded by turbulence-induced scintillations along with fog or clear weather conditions. Consequently, our findings can be used for the design of the emerging high speed LOS THz links, taking into consideration the prime factors that affect their capacity performance.