{'fraction_non_alphanumeric': 0.039627610706192196, 'fraction_numerical': 0.014577080908923868, 'mean_word_length': 4.693165969316597, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Terahertz Wireless Transmissions with Maximal Ratio Combining over Fluctuating Two-Ray Fading

Mitigating channel fading and transceiver impairments are desirable for high-speed terahertz (THz) wireless links. This paper analyzes the performance of a multi-antenna THz wireless system by considering the combined effect of pointing errors and fluctuating two-ray (FTR) fading model. We provide a statistical characterization of the maximal ratio combining (MRC) receiver over independent and nonidentical (i.ni.d.) channel conditions in terms of multi-variate Fox's H by deriving density and distribution functions of the signal-to-noise ratio (SNR) of a single-link THz link using incomplete Gamma function. We develop exact analytical expressions of outage probability, average bit-error-rate (BER), and ergodic capacity for both single-antenna and MRC receivers. We also present the diversity order of the system by deriving asymptotic expressions for outage probability and average BER at high SNR to obtain insights into the system performance. We validate our derived analytical expressions with Monte-Carlo simulations and demonstrate the effect of various system and channel parameters on the performance of single and multi-antenna THz wireless communications.


I. INTRODUCTION
Terahertz (THz) wireless is an upcoming technology to provide new spectrum resources for future communication systems. The availability of contiguous high bandwidth transmissions in the THz spectrum can be potential for wireless backhaul/fronthaul technology [1]- [3]. The THz spectrum is mostly unlicensed and can support secured terabits per second (Tbps) data transmissions with low latency for various highend applications. The line-of-sight (LOS) THz technology requires high directional antennas with higher gain to compensate for severe path loss due to the molecular absorption of transmitted signals. Nevertheless, the THz link is susceptible to the random pointing errors caused by the misalignment between transmitter and receiver antenna beams and may incur transceiver distortion at higher frequencies in addition to the stochastic multi-path fading [4]- [7]. Alleviating the adverse effects of signal attenuation and fading is desirable for highspeed THz links.
Recently, dual-hop and multi-hop relaying at THz frequencies have been investigated [8]- [16]. More specifically, the authors in [8] formulated an optimal relaying distance for THzband communication to maximize the network throughput. In [9], a relay selection approach was suggested to mitigate the impact of antenna misalignment and shadowing due to the human blockage in a multi-relay setup. A reconfigurable intelligent surface (RIS) assisted multi-hop THz system over Rician fading was considered in [11] to mitigate the signal attenuation using deep reinforcement learning (DRL) based beam-forming technique. Considering the generalized α-µ fading combined with stochastic pointing errors, the decodeand-forward (DF) protocol was employed to link THz and radio frequency (RF) technologies [12], [13]. Using multiantenna transceivers, the DF relaying was studied for a dualhop THz-THz link [10].
There has been an increased research interest to model the short-term fading for THz communications [17]- [20]. A Gamma mixture channel model for THz transmissions for a short (< 1 m) link is proposed in [18]. The authors in [19] find α-µ fading model suitable for the THz transmission using the measurement at 152GHz for a link length within 50m. Using a comprehensive THz measurement data at 300GHz for train-to-infrastructure and inside station [21], the authors in [20] demonstrate that the fluctuating two ray (FTR) model is a better fit for THz multi-path fading modeling than the conventional Rician and Nakagami-m distributions. Using the combined effect of the short-term FTR fading, antenna misalignment, and hardware impairments, [20] derived outage probability and ergodic capacity for RIS-aided THz systems.
The recently proposed FTR model has been extensively studied for mmWave wireless transmissions [22]- [27]. In [22]- [24], analytical performance was studied for a single-link FTR fading channel. The physical layer secrecy performance over the FTR fading channel was analyzed [25]. The authors in [26] analyzed the performance of a mixed free-space-optics (FSO)-mmWave system by modeling the mmWave and FSO channels as FTR and Gamma-Gamma distributed, respectively. In contrast to the single-antenna system, multi-antenna at the receiver can harness the spatial diversity over independent fading for improved performance [28]- [32]. In [28], the authors analyzed the performance of an equal gain combining (EGC) receiver by deriving outage probability and average BER using single-variate mathematical functions over FTR fading channels. In [29], a low complexity selection combining receiver was investigated with a performance analysis on the outage probability, average BER, and ergodic capacity in terms of multi-variate Fox's H function. In [30], the authors analyzed the optimal maximal ratio combining (MRC) receiver by deriving the PDF and CDF of the sum of arbitrarily distributed FTR variates. In [31], the outage probability and an upper bound on average BER was derived for the MRC receiver. The authors in [32] provided asymptotic and non-asymptotic expressions of the outage probability and average BER for the MRC over non-identical distributed FTR fading channels. The moment matching method was used to approximate statistics of the sum of FTR fading channel for analyzing relay-assisted radio frequency (RF)-mmWave wireless communications for high-speed trains [33].
In the light of THE above research and to the best of the author's knowledge, performance analysis of the MRC receiver with THz wireless transmissions over FTR fading channels jointly with stochastic pointing errors is not available in the open literature. Our main contributions of this paper are follows: • By deriving PDF and CDF of a single-link THz link using standard mathematical functions, we provide exact statistical characterization of the SNR for the MRC receiver under the joint effect of FTR short-term fading and zero-bore sight pointing errors considering independent and nonidentical (i.ni.d.) channel conditions in terms of multi-variate Fox's H function. • We develop exact analytical expressions of ergodic capacity, outage probability, and average BER for both singleantenna reception and MRC receiver and present asymptotic expressions for outage probability and average BER at high SNR. We derive the diversity order of the considered system to show the advantage of multi-antenna reception and the impact of pointing errors. • We evaluate multi-variate Fox's H function using Python code [34] and validate our derived analytical expressions with Monte-Carlo simulations. We also demonstrate the effect of various system and channel parameters on the performance of THz wireless communications.

II. SYSTEM MODEL
A single-antenna source communicates to an L-antenna destination over the THz spectrum. The THz link is affected by path-loss, short-term fading, pointing errors, and transceiver distortions. Assuming negligible hardware impairments [12], [35], the received signal y i at the i-th antenna is given by: where h l is the path gain, s is the transmitted signal with power P , h i denotes the fading channel coefficient, and w i is the additive white Gaussian noise with a variance σ 2 w . The deterministic path gain h l is dependent on antenna gains, frequency, and molecular absorption coefficient [6]: where c, f , and d respectively denote the speed of light, carrier frequency, link distance whereas G t and G r denote gain of the transmitting antenna and receiving antenna, respectively.
The term k(f, T, ψ, p) is the molecular absorption coefficient depends on the temperature T , relative humidity ψ, and atmospheric pressure p [36]. The compound channel coefficient is where h pi and h f i models pointing error and short term fading, respectively. We use the zero boresight model for pointing errors h pi [37]: where S 0 = erf(υ) 2 with υ = π/2 (r 1 /ω z ) and ω z is the beam-width, φ = ωz eq 2σs with ω zeq as the equivalent beam-width at the receiver, which is given as , and σ 2 s is the variance of pointing errors displacement characterized by the horizontal sway and elevation [37].
To model |h f i | 2 , we use the FTR fading channel with PDF given as [23]: where K is the ratio of the average power of the dominant component and multi-path, m is the index of fading severity, and ∆ denotes the similarity of two dominant waves. The term σ 2 represents the variance of diffused components such that σ 2 = 1 2(1+K) for the normalized averaged SNR. The factor d j is defined in [23] and recently updated with an additional factor in [38].
We denote SNR of the i-th antenna as . Assuming perfect channel state information (CSI), SNR with optimal combining for the MRC receiver is γ = L i=1 γ i . Thus, PDF and CDF for the sum of product of pointing errors and FTR random variables is required for statistical performance analysis of the MRC receiver.

III. STATISTICAL DERIVATIONS
In this section, we provide statistical results for the sum of L arbitrarily distributed FTR fading combined with stochastic pointing errors by deriving closed-form expressions of the PDF and CDF of the single THz link.
IV. PERFORMANCE ANALYSIS In this section, we analyze the performance of singleantenna THz link in terms of standard Mathematical functions and use multi-variate Fox's H and Gamma functions to provide exact and asymptotic analysis on the multi-antenna reception.

A. Single Antenna Reception (SAR)
1) Outage Probability: Outage probability is defined as the probability of instantaneous SNR less than a predetermined threshold value γ th i.e., P out = P (γ < γ th ). Thus an exact expression of the outage probability is P SAR out = F γi (γ th ), where F γi (x) is given in (6).
2) Average BER: Average BER is another important performance metric for communication systems, defined as [40] where p and q are constants to specify different modulation techniques and Ψ X is the CDF. We denote by ξ = 2γ 0 σ 2 S 2 0 . Using (6) in (14) 3) Ergodic Capacity: The ergodic capacity is defined as where ψ X denotes the PDF. Using (5) in (16) Further, we can use log(1 + γ) > log(γ) in (16) with (5), and denoting ψ (0) as the digamma function to get a simpler bound on the ergodic capacitȳ is given in (11). We use [43] to present the asymptotic outage probability in (12). Considering the dominant terms at high SNR, we get the diversity order as L l=1 min{j l + 1, φ 2 /2}. It is interesting to note that the diversity order is independent of FTR fading parameters K, m, and ∆ as also observed in earlier literature [30], and has been extensively verified for THz transmissions in numerical section V. The diversity order shows the advantage of multi-antenna reception and that the impact of pointing errors can be minimized by circumventing the multi-path fading using a sufficiently higher beam-width.

3) Ergodic
Capacity: Substituting f γ (γ) in (16) and applying the definition of multivariate Fox's H-function [44]: We use the Mellin's inverse transform ln(1 + z) = to represent the inner integral in (21) as Since the inner integral in (22) is not convergent, we use final value theorem lim t→∞ t 0 f (z)dz = lim s→0 F (s) = F ( ) with Laplace transform of integrand in (22) to get (23) Using (22) and (23) in (21), and applying the definition of multivariate Fox's H-function, we get

V. SIMULATION AND NUMERICAL RESULTS
In this section, we demonstrate the performance of the considered single-antenna and MRC receivers for THz transmissions and validate the derived analytical expressions with Monte Carlo simulations. We consider the THz channel with a distance of 50m, carrier frequency 275 GHz, and antenna gains of 50 dBi. To compute the path loss for the THz link, we consider the relative humidity, atmospheric pressure, and temperature as 50%, 101325 Pa, and 296 K, respectively. We use [37] to compute the parameters φ and S 0 of pointing errors by varying the beamwidth and the jitter variance 10cm antenna aperture radius. The AWGN noise power is considered to be −94.2 dBm. Fig. 1(a) demonstrates the impact of receiver antennas L, pointing errors parameter φ, and FTR fading m on the outage performance of the considered THz system with S 0 = 0.054, γ th = 4dB, K = 10, and ∆ = 0.5. It can be seen that improvement in the performance is significant with spatial diversity when the number of receiver antennas is increased from L = 1 to L = 4. The figure also confirms that the outage probability improves when the fading severity parameter m increases, as expected. The slope of plots clearly demonstrate the dependence of diversity order on the system parameters. It can be seen that there is a distinguishable difference in the slopes for φ = 1 and φ = 2.5 but there is a minimal change in the slope with φ = 2.5 and φ = 6. As such, the    (c) Ergodic capacity. diversity order increases linearly with L, is independent of the parameter m, depends on φ when φ 2 /2 < min j l + 1 but becomes independent otherwise. Thus, the impact of pointing errors can be mitigated using a sufficiently higher beam-width for THz transmissions. We demonstrate the average BER performance by varying the parameters ∆ and φ with K = 2, and m = 2. The figure shows that highly dissimilar specular components of FTR fading depicted through ∆ provides an improved average BER performance. Similar to the outage probability, we can observe that the average BER improves with an increase in the number of receiver antennas L and the diversity order is independent of ∆ and becomes independent of pointing errors with a sufficiently higher value of φ.
Finally, Fig. 1(c) illustrates the relationship between the ergodic capacity and the number of receiver antennas with interdependence of the various channel and system parameters. The figure shows the logarithmic scaling of the ergodic capacity with the number of receiver antennas L. It can be observed that ergodic capacity is nearly independent of the parameters K and ∆, but increases with an increase in the parameters m. The figure also shows that pointing errors significantly degrade the THz performance. However, an increase in the MRC antennas reduces the gap in performance by harnessing the spatial as compared to the single-antenna system.

VI. CONCLUSIONS
In this paper, we analyzed the performance of THz wireless transmissions under the combined effects of path loss, generalized FTR fading, and Rayleigh distributed pointing errors. We provided statistical results on the single-antenna and multiantenna receivers by deriving PDF and CDF of the resultant SNR. We analyzed the performance of the considered system by deriving closed-form expressions for the outage probability, average BER, and ergodic capacity. Using asymptotic analysis on the outage probability and average BER, we derived the diversity order of the system, which provides a design criterion of using sufficiently higher beam-width to mitigate the impact of pointing errors by circumventing the multi-path fading. We validated our derived analytical expressions with Monte-Carlo simulations to show that the impact of the number of receiver antennas L, pointing errors φ, and fading severity parameter m is higher on the THz wireless system compared with other parameters. Incorporating hardware impairment in the performance analysis may be a possible extension of the proposed work.

APPENDIX A
Using the definition of the MGF function, we apply inverse Laplace transform to find the PDF of γ = − φ 2 2 + 1, 1 − φ 2 2 + j + 1, 0 Applying the definition of Meijer's G function [42] and interchanging the sum and product, we get Thus, using f γ (z) = 1 2πi L e sz M γ (s)ds, interchanging the integral, and rearranging the terms, we get We substitute sz = −b, and apply the identity [41] (eq. 2 + ζ l (28) We substitute (28) in (27) and apply the definition of multivariate Fox's H function [44] to get (10). Finally, we use F γ (z) = L −1 N i=1 Mγ i (s) s and apply the similar steps used in the derivation of PDF to get the CDF in (11), which concludes the proof of Theorem 1.