A Channel Loss Model for THz Networks From 100–600 GHz Considering Both Molecular and Water Vapor Continuum Absorptions

6G networks are expected to use the terahertz (THz) frequency spectrum in some way. However, relatively little is known about the transmission channel in this part of the spectrum. In this article, we propose a novel and accurate channel loss model that considers both molecular and water vapor continuum absorptions for THz networks based on the frequency range between 100 GHz and 600 GHz. The proposed channel model assumes that the atmospheric absorption is accurately accounted for by the molecular absorption coefficient and the water vapor continuum absorption coefficient. The new model is based on equations developed to accurately match the channel response given by a spectroscopic database. This makes the channel model robust and useful for evaluating the performance of THz networks. The model is validated against a spectroscopic database defined as a benchmark. The performance of the new channel loss model is evaluated against different models, including several error metrics. Numerical results show the new channel model is as accurate as the spectroscopic database by correctly reproducing the provided experimental dataset. Therefore, the proposed channel model is a reliable, robust and simple tool that can be successfully used to carry out performance evaluation of THz networks with accuracy.


I. INTRODUCTION
Over the past few years, there has been a growing demand for broadband services offered by mobile networks, taking data traffic volumes to unprecedented levels [1], [2]. Mobile networks are expected to handle an increase in global traffic volume that is currently doubling every two years [3]. Importantly, this constant raising in broadband services demand and data traffic generation will pose challenges on power consumption. Regardless of the sector, the latter is becoming a paramount concern mainly due to both climate change and limitation of renewable energy resources. The current levels of greenhouse gas emissions and their environmental impact are unsustainable and pose a real challenge to industries, even putting in risk their capacity to achieve net zero emission by 2050 [3]. One eventual way to reduce the gas emissions and environmental impact, is dematerialization. The telecom industry has the potential to lessen the need for material usage by substituting physical products with digital ones. Dematerialization [3], which leads to reduced consumption of natural resources, can be achieved for example via VR/AR, holograms, and metaverse. Holograms not only can enable many applications, but also allow for a true virtual representation of complex entities of the physical world, helping reduce the total quantity of exemplars of several goods. This would potentially lead to a successful dematerialization and consequently environmental impact reduction. Notwithstanding, holograms, VR/AR, and metaverse [4], [5] are broadband applications requiring larger bandwidth and higher data rates (as well as lower latency) than offered by current networking solutions. In fact, these emerging applications will most likely not be fully satisfied by current fifth-generation (5G) technologies [6], but, instead, will require support to ultra-high data rates up to terabits per second (Tb/s) and highly accurate positioning [7]. Higher frequencies within the terahertz (THz) region is envisioned as a key technology to satisfy this demand by alleviating the spectrum scarcity and capacity limitations of current mobile systems. Following this trend, Tb/s wireless links will become a reality within the upcoming years.
As societal needs evolves, next-generation networks such as 6G will require higher frequencies to satisfactorily serve these demanding applications. Accordingly, 6G networks based on the THz spectrum will provide a significant improvement in bandwidth availability, data rates, and full symmetric duplex operation [8]. Eventually, in future systems, both uplink and downlink will have similar capabilities, especially in terms of bandwidth and data rates. In this way, the THz bands can grant 6G networks the feasibility to support wider bandwidth and higher data-rates. Not only can this be the key to supporting future data traffic demands, but also a way to enable dematerialization. 6G networks based on THz bands can further enable different networking scenarios like for example vehicle-to-everything (V2X) communications [9], [10], environment virtualization [11], and holograms [12], to name but a few. Such a scenario is illustrated in Fig. 1, where 6G-based THz networks enable a new range of applications and several devices are simultaneously served by multiple access points through multi-point transmissions.
6G networks may use part of the spectrum defined between 0.1-10 THz. While the frequency regions immediately below and above this range (the microwaves and the far infrared) have been extensively investigated, this is still one of the least explored frequency bands for communication, with several open challenges. The THz Interest Group IEEE 802.15 [13] responsible for standardization of THz communications was created and the IEEE 802.15.3d-2017 became the world's first wireless standard at 300 GHz [14]. Despite standardization efforts, there are many open challenges for leveraging THz communications and its capabilities [15], [16], [17]. Among them, the channel propagation model renders THz-based networks to be misleading in its performance and future potential. Therefore, a new channel model for 6G/THz networks is addressed here. A reliable THz channel loss model is key for helping developing as well as evaluating mobile systems. In this way, a reliable and accurate channel model must include the molecular absorption loss as well as the water vapor continuum absorption loss [18], [19].
The molecular absorption loss can be obtained via two different ways. The first way is the line-by-line functions which account for the molecular attenuation based on the HI-TRAN (high-resolution transmission) spectroscopic database in combination with the Van Vleck-Weisskopf (VVW) line shape functions [18], [20], [21]. For example, in the ITU-R P.676-13 Recommendation [22] the atmospheric attenuation in the range between 1 and 1000 GHz is defined. Such a recommendation is based on a numerical method that calculates the individual summation of spectral lines (i.e. line-by-line calculation) by using 553 spectroscopic data points from the HITRAN database in combination with the full Lorentz line shape functions in order to finally be capable of evaluating the atmospheric attenuation. The usage of the spectroscopic data along with line-by-line calculations not only adds complexity to the evaluation, rendering the same an arduous procedure, but also is an approximate method which may not provide accurate results.
In contrast, the second way to address the molecular absorption loss is via closed-form expressions [23], [24], [25], [26]. Accordingly, a set of equations to evaluating the molecular attenuation considering water vapor is reported in the ITU-R P.676-10 Recommendation [23]. This model is based on full Lorentz line shape functions with the coefficients fitted to experimental measurements [27], [28], but with low accuracy and limited for the 1-350 GHz frequency range.
Furthermore, a simplified channel model for the frequency range of 275-400 GHz is reported in [24]. Such a model is based on polynomials and an equalization factor for fitting purposes along with Van-Vleck Huber line shape functions using the parameters from the HITRAN database. This simplified model was subsequently extended in order to cover the frequency range of 200-450 GHz in [25]. Later on, the 200-450 GHz channel loss model was further developed by including new polynomials to cover the frequency range between 100 and 450 GHz in [26]. Despite being capable of covering the D-band, such a channel loss model is not accurate for its entire frequency range and has high average error. Lastly, the water vapor continuum attenuation, which contributes to the atmospheric attenuation, was not taken into account by the model [26].
By its turn, the water vapor continuum attenuation can be calculated via empirical equations fitted for the THz region [19], [29], [30], [31], [32]. The contribution of the water vapor continuum absorption in the evaluation of the atmospheric absorption coefficient is highlighted in [30]. The absence of the continuum spectrum can lead to discrepancies around hundreds of dB for transmission distances over a kilometer. The attenuation of the water vapor continuum absorption up to 1 THz is investigated in [32]. The latter defines the empirical coefficient parameters that describe the most accurate attenuation due to the water vapor continuum absorption spectrum.
Within this context, in this article a new and accurate channel loss model for the frequency range between 100 and 600 GHz is proposed. The new channel model developed here accounts for the total atmospheric absorption by taking into account both the molecular absorption coefficient and the water vapor continuum absorption coefficient. This allows, for the first time, to the best of our knowledge, the THz transmission channel loss to be accurately accounted for. The newly developed channel model is based on closed-form equations given by absorption line shape functions and a fitting parameter. The new model is validated against the HITRAN database, which is defined as a benchmark. The mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error (MAPE), maximum absolute error, and adjusted R-squared for the proposed channel loss model are calculated and analyzed. Moreover, the performance of the proposed channel loss model is evaluated against many different models. Numerical results show that the new channel model matches the performance behaviour of experimental dataset from the HITRAN spectroscopic database for the designed frequency range. This renders the proposed channel model to be an accurate and reliable tool for performance evaluation of THz networks based on 100 -600 GHz. Lastly, the newly developed accurate model enables the evaluation of key figures-of-merit (that are dependent on the channel losses) in THz networks such as free-space path-loss (FSPL), signal-noise-rate (SNR), and bit error rate (BER), as analyzed here.

II. MATHEMATICAL FRAMEWORK
Communication systems are expected to transmit information from one point to another. They are normally made up of transmitters, a channel, and receivers. Briefly, at the transmitter the modulation of signals occurs, and then they are sent through a channel to the receiver. Two features in a communication system can be highlighted: the transmitted power and the channel bandwidth [33], [34]. The transmitted power defines the SNR, and depends on the transmitter's design. On the other hand, the channel bandwidth is defined as the frequency range allocated for signal transmission, and can be obtained by modeling the channel.
In addition, the THz band is formed by multiple absorption lines that are unique for each molecule present in the atmosphere. The energy associated with the change of quantum levels determines the frequency of the absorption line, and this frequency can be altered by some environmental factors, such as the change of temperature, pressure, and the quantity of water vapor in the atmosphere [18].
Thus, the THz channel can be modeled as [35], [36] where x is the transmitted signal, y is the received signal, n is the noise, and h is the channel state given by where h p and h at m represent the propagation gain and the atmospheric absorption fading. According to the Friis equation, h p in the free space can be given by [34] with c, G T x , G Rx , d, and f denote the speed of light, the transmitter antenna gain, the receiver antenna gain, the transmission distance, and the THz wave frequency, respectively. Based on the Beer-Lambert Law, h at m is given by [18] where k( f ) is the total absorption coefficient of the THz channel, which is composed of the sum of the molecular absorption coefficient, k a ( f ), and the water vapor continuum absorption coefficient, k c ( f ) [18], [19]. Next, the channel noise power is an important system metric, and it can be calculated as follows [18] P where k B is the Boltzmann constant, f is the channel available bandwidth and T noise is the channel noise temperature which is given by with T sys is the system electronic noise temperature, T at m is the atmospheric absorption noise, and T ot her considers any additional noise source. Thus, T sys is given by where NF is the noise factor, whose value in dB represents the system noise figure, and T is the system temperature. T at m is obtained by with τ ( f , d ) is the channel transmittance in the medium, and it can be related to the transmitter (P T x ) and receiver (P Rx ) powers as For the communication systems considered here, it is assumed that T sys + T at m T ot her , and then, (6) can be rewritten as and from (10), we can determine the channel noise power given by (5). It is worth mentioning that the phase noise of the signal is adequately compensated by advanced digital signal processing techniques at the receiver side [37]. In this way, from (3) and (4), one can obtain the total line-of-sight (LOS) path loss (PL) of a THz wave propagating in the atmosphere as [33], [36] PL [dB] = 20 log 10 4π df c where e is Euler's number. The system transmits on-off pulses employing on-off keying (OOK) modulation. Complex modulation formats such as pulse-amplitude modulation [38] will be considered as future work. Lastly, the SNR of the system can be calculated as and the BER of the system is given by [39] where erfc (x) is the complementary error function. Next, the methods for calculating the molecular absorption and water vapor continuum absorption coefficients are presented.

A. MOLECULAR ABSORPTION COEFFICIENT
The molecular absorption coefficient, k a ( f ), can be obtained via two different approaches, namely, the numerical and simplified models. First, the numerical modeling is based on radiative transfer theory with information supplied by the HI-TRAN database [18], [21]. Thus, k a ( f ) can be obtained by the following expression [18] k num where k i,g a is the individual absorption coefficient for the isotopologue i of gas g, N A is the Avogadro constant, R is the universal gas constant, p and T are the system pressure and temperature, respectively, p 0 and T STP are the standardpressure-temperature values. In addition, q i,g , S i,g , and F i,g ( f ) are the mixing ratio, spectral line intensity, and line shape functions for the isotopologue i of gas g, respectively. The quantities S i,g and F i,g ( f ) are obtained from the HITRAN database [21]. More details for obtaining k a ( f ) numerically can be found in [18], [40].
Next, the molecular absorption coefficient can be alternatively obtained via simplified models [26]. Accordingly, the model presented in [26] is based on fitted equations with numerical results determined from the radiative transfer theory in combination with the HITRAN spectroscopic database, but for the frequency range of 100 to 450 GHz. The coefficients of the equations depend on the temperature, pressure (fixed), and relative humidity (RH) of varied water vapor in the atmosphere. Furthermore, the parametric model is characterized by the molecular absorption coefficients y j on absorption lines j. The simplified molecular absorption coefficient is given by [26] k simpl with f is the frequency of interest, g( f , μ) is a fitting equation, and μ is the water vapor mixing ratio that depends directly on RH [26], and it is given by where RH(%) is the relative humidity given in percentage and p s is the saturated water vapor partial pressure, and is described by Buck's equation [26] p s = 6.1121 1.0007 where the pressure p is given in hectopascals and T is given in degree Celsius. Subsequently, we carry on the development of the channel model but assuming an accurate molecular absorption coefficient and the frequency range of 100 to 600 GHz.

1) NOVEL PROPOSED CHANNEL MODEL
A new channel loss model is developed here considering polynomials for the absorption coefficient as well as a frequency range of 100-600 GHz. To do so, new equations describing the absorption peaks are developed. In addition, a fit equation parameter is also developed and included in the g( f , μ) in order to accurately fit (the new model behaviour) with (experimental) spectroscopic database. Hence, the molecular absorption coefficient is calculated as and, the polynomial equations representing the strong absorption lines in the 100 to 450 GHz range are [26] y Then, without loss of generality, we new develop here the resonant absorption lines for the center frequencies at 425, 475, 487, and 557 GHz as follows whose coefficients are given by w 9 (μ) = 3.40115 × 10 9 + 4.56867 × 10 10 μ S(μ) = 2w 9 (μ) π 1.72001 × 10 7 − 3.96597 × 10 8 μ − 2.13925 × 10 10 μ 2 + 5.23041 × 10 11 μ 3 T (μ) = α π w 9 (μ) 2 −9.00089 × 10 5 + 1.00072 × 10 9 μ w 10 (μ) = 6.23187 × 10 9 + 1.46195 × 10 10 μ, Then, without loss of generality, the fitting equation g( f , μ) can be developed as follows Note that the resonance absorption lines y 5 , y 9 and y 10 , in (23), (27) and (28), respectively, are based on the pseudo-Voigt peak function [41]. Such a function is defined by the convolution of the Gaussian and Lorentzian functions, and it is generally used to obtain experimental spectral line shapes [41]. In contrast, the coefficients from (23), (27) and (28) are obtained by the help of the HITRAN database and VVW functions along with fitting procedure. The fitting procedure is carried out to determine in an accurate way the new expressions of the absorption line coefficients according to the numerical model from [18], [21], [40].
Finally, the (56) is developed via the fitting procedure described earlier. Hence, (56) depends on the optimization parameter, θ ad j , which has an optimal value for each frequency band in the 100-600 GHz range. Then, once the optimized θ ad j is obtained, the mean absolute error (MAE) can be minimized. Next, we address the formalism to obtain the water vapor continuum absorption coefficient.

B. WATER VAPOR CONTINUUM ABSORPTION COEFFICIENT
The water vapor is one of the main elements responsible for the atmospheric attenuation of electromagnetic waves propagating in the THz band [19], [29], [30], [31], [32]. There are two components of atmospheric attenuation related to water vapor which are resonant absorption lines and continuum absorption [19]. The contribution of the water vapor absorption lines is accounted for in (18). The continuum absorption coefficient rises quadratically as the frequency increases and decreases inversely as the temperature increases in the THz region (up to ∼3 THz) [19], [29], [30], [31], [32]. Therefore, according to [19], [31], the water vapor continuum absorption coefficient can be expressed as where f is the frequency in GHz, P W and P A are the partial pressures of the water vapor and foreign gases in the atmosphere, respectively, C 0 W and C 0 A are the water vapor and foreign-continuum parameters at 300 K, in units of (dB/km)/(hPa GHz) 2 , respectively. θ = 300/T is the dimensionless temperature factor, and T is the system temperature in K. n W and n A are the temperature exponents. And then, are the water vapor and foreign-continuum parameters as a function of temperature, respectively. As the water vapor continuum absorption coefficient is usually obtained empirically, the values of the parameters are given by experimental works, such as [19], [29], [30], [31], [32]. For the channel loss model proposed herein, we consider the empirical parameters from [29], which gives C W = 4.39 × 10 −8 (dB/km)/(hPa GHz) 2 and C A = 4 × 10 −9 (dB/km)/(hPa GHz) 2 for the room temperature.
Concerning the partial pressures P W and P A , they are determined as follows respectively, recalling that μ is the water vapor mixing ratio and p 0 is the standard pressure given in hPa. From the parameters P W , P A , C A and C W , and using (57), the water vapor continuum absorption coefficient can be calculated. Lastly, the total atmospheric absorption coefficient, k( f ), is composed of the sum of the molecular absorption coefficient, k a ( f ) given by (18), and the water vapor continuum absorption coefficient, k c ( f ) given by (57), for a specified temperature value.

III. RESULTS AND DISCUSSION
The thoroughly described channel loss model is now applied for the performance evaluation of THz networks based on 100-600 GHz. Its performance and accuracy is assessed against several models.

A. NUMERICAL VALIDATION OF THE MOLECULAR ABSORPTION COEFFICIENT
The numerical model based on the Radiative Transfer Theory, using the VVW line shapes and the spectroscopic data obtained from the HITRAN database [18], [21], [40] is considered here as the benchmark. Then, the contributions to the molecular absorption of each isotopologue i of each gas g present in the atmosphere is calculated. Afterwards, the molecular absorption coefficient of the THz channel is determined by (14). Subsequently, by using the numerical approach, the molecular absorption coefficients of each type of molecule available in the atmosphere is calculated. To do so, the percentages of molecules in the atmosphere are considered as: N 2 = 78.084%, O 2 = 20.946% and CO 2 = 0.0417%. Furthermore, the atmospheric pressure of 1 atm (1.01325 × 10 3 hPa), temperature equal to 298 K (25 • C), and RH ranging from 10% to 90% (with a 20% step) are considered.
The numerical molecular absorption coefficient results (VVW + HITRAN) for RH = 10% and 90%, in the 100 to 450 GHz range, are shown in Fig. 2(a) and (b), respectively. It can be noted from Fig. 2 that for higher RH values, the larger the molecular absorption coefficients are, which suggests that water molecules are responsible for the major absorption loss in the atmosphere. It can also be seen from Fig. 2 that there are well-defined peaks at some frequencies. These are the resonance peaks given by the line-by-line absorption model [18], [21], [40]. However, between two adjacent peaks there are some lower values of the absorption coefficient. This region between adjacent peaks can be regarded as potential transmission windows for THz wireless communication systems.
Moreover, Fig. 2 shows the performance comparison among the channel model given by [26], the numerical ITU-R model referred here as Full-ITU-R [22], the numerical VVW + HITRAN model (that is considered as benchmark) [18], [21], and the new proposed model. One can observe from Fig. 2(a), that the new proposed model (black lines) is the only one to converge and complete agree/overlap with the benchmark model (red lines). The performance estimation difference, among the channel models, becomes even more evident for the strong resonance peaks with center frequencies at 119 GHz and 425 GHz, where the remaining channel models clearly cannot reproduce the peaks according to the bechmark. Moreover one can further notice that the regions between peaks 1 and 2 (119-183 GHz), 2 and 3 (183-325 GHz), and 3 and 4 (325-380 GHz) show some divergence between the absorption lines in these transmission windows, for the model from [26], misleading the performance of the network. For example, under 119 GHz and RH = 10%, while the proposed model is as accurate as the benchmark, the models left underestimate the molecular absorption losses in an order of magnitude. Furthermore, analyzing the top inset in Fig. 2(a), the D-band (110-170 GHz) is highlighted, and it can be infered that the peak around 119 GHz for the benchmark and proposed models have 13 dB/km, whereas the models from [26] and Full ITU-R have 1.4 dB/km only. This difference among models, undestimates the network performance in an order of magnitude, which may be unaccetable in THz systems where signal power transmission is key for the smooth network operation. Also, the range from 400 to 450 GHz is highlighted in the bottom inset from Fig. 2(a), once again, it can be realized that for the peak around 425 GHz, the proposed and benchmark models have 30.3 dB/km, whereas the Full-ITU-R model has 9.1 dB/km, and [26] simply misses the peak.
The molecular absorption coefficient versus the frequency is plotted in Fig. 2(b) for RH = 90%. In such a figure, one can note that the both benchmark (red) and our proposed channel model (black) peaks at 425 GHz, whereas the remaining model (blue) and ITU-R (pink) are smoothly throughout the region. As a result, the performance of the THz system is hindered. In contrast, it can be further noted from Fig. 2(b) that all models peak at 119 GHz, while our model (black) excellently agrees with the benchmark (red) instead, the remaining model (blue) and ITU-R (pink) peak shorter, resulting in a short tail and consequently underestimating the channel loss estimation for a wider range of frequencies. Surprisingly, the ITU-R model (pink curve) has a heavy tail than do the all remaining models for the entire frequency range, but even more substantial in the frequency window between peaks 2 and 3 (183-325 GHz), as can be seen in Fig. 2(b). As a consequence, the ITU-R model predicts higher molecular absorption losses in the THz channel than one would actually expect. To circumvent this issue, someone should eventually use the experimental/spectroscopic database solution, or the channel loss model proposed in this work. Additionally, in such top inset from Fig. 2(b), on one hand, both benchmark and proposed model have around 14.1 dB/km at 119 GHz, and on the other hand, the Full-ITU-R (pink) and remaining (blue) models have around 3.6 dB/km at 119 GHz. Lastly, in such bottom inset from Fig. 2(b), while the proposed and benchmark models have 75.2 dB/km at 425 GHz, the previous model (blue) does not peak, and has an absorption coefficient value around 46 dB/km, representing a 61.2% underestimation in the performance loss.
Furthermore, the previous model, described in [26], covers up to 450 GHz, whereas the channel loss model herein proposed covers up to 600 GHz. The same is possible thanks to the development of three additional peaks, namely, at 475 GHz, 487 GHz, and 557 GHz. This renders, for the first time to the authors' knowledge, the estimation of the molecular absorption coefficient for the Y-band (325-500 GHz). Given the large bandwidth available, THz signals can be substantially impacted by molecular absorption, while a robust channel model can be fruitful and ease that challenge regarding performance analysis.
Accordingly, the molecular absorption coefficient curves as a function of the frequency covering up to 600 GHz and for RH = 10% is plotted in Fig. 3(a). All the four models are considered for the sake of analysis consistency. It can be seen that the Full-ITU-R model does not correctly describe the peaks at 425 GHz and 487 GHz. This unconformity becomes more clear in the zoom inset from Fig. 3(a). For example, the Full-ITU-R model shows 17 dB/km at the 487 GHz peak, whereas the proposed and benchmark models suggest approximately 27 dB/km. Despite seeming relatively small the divergence, this renders a ∼63% prediction difference between the models, which can lead to misleading performance analysis. In addition, the herein developed model is easier to handle and apply rather than the Full-ITU-R model as the former only employes simple mathematical functions (such as Lorentz and Pseudo-Voigt) that can be easily simulated with very low computational complexity and computing power demand. Subsequently, the molecular absorption coefficient versus the frequency range up to 600 GHz is plotted in Fig. 3(b) for RH = 90% and for all the three pertinent models. Overall, all the three models have good agreement, showing similar performance levels from 450 GHz to 600 GHz. Despite that, one can clearly realize from the inset in Fig. 3(b) that the new proposed model (black curve) shows better agreement with a great deal of accuracy than the Full-ITU-R model (pink curve). Next, in order to further quantify the difference among the pertinent channel models, the mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error (MAPE), maximum absolute error (max.AE), and adjusted R-squared (adj.R 2 ) [42] are calculated and analyzed, assuming the VVW + HITRAN as the benchmark. Then, the MAE values are summarized in Table 1. Bear in mind that θ ad j was optimized to minimize the MAE for the following frequency bands: 110 to 300 GHz, 220 to 325 GHz, 275 to 450 GHz, 100 to 450 GHz, 325 to 500 GHz, 450 to 600 GHz, and 100 to 600 GHz. Hence, one must choose the desired frequency band and set θ ad j as indicated in Table 1 in order to reproduce the results. Then, by analyzing the MAE values (dB/km) from Table 1, one can note that the new proposed model has lower MAE's values than do the remaining models [22], [26], for all frequency bands, and considering RH ranging from 10% to 90%. For example, in the 275 to 450 GHz range, for θ ad j = 3.8 × 10 −6 , our model yields a MAE of 0.32 dB/km, 1.04 dB/km, and 3 dB/km for RH = 10%, 50%, and 90%, respectively. In contrast, the previous model [26] resulted in a MAE of 1.55 dB/km, 2.3 dB/km, 3.6 dB/km for RH = 10%, 50%, and 90%, respectively. Thus, such an error difference is substantially relevant and may poorly estimate the performance analysis of THz networks. Another error metric analyzed here is the RMSE. This mechanism penalizes larger errors more than small errors. In this manner, for the 275 to 450 GHz range, and under RH = 90%, the proposed channel model suggests an RMSE of 4.5 dB/km, whereas the previous model [26] results in an RMSE of 6.6 dB/km. Hence, the latter envisages more extreme errors than the new proposed model. Afterwards, for the 100 to 450 GHz range, under the most critical condition such as RH = 90%, a MAPE around 15.7% is obtained for the proposed model. In comparison, the previous model [26], achieved a MAPE of 15.2%, which is slightly less than the MAPE of the proposed model. Still concerning the error evaluation, the max.AE and adj.R 2 metrics yield values of 17.5 dB/km and 0.9992, respectively, for the proposed channel model. In contrast, 28.9 dB/km and 0.9986 are obtained for the model in [26]. As a result, the new proposed model can be deemed as a promising tool for supporting and advancing the performance of THz networks in a more accurate way.
Next, let us consider the 450-600 GHz range, and θ ad j = 5.0 × 10 −7 , then, the proposed channel model yields a MAE around 10.1 dB/km, 48.2 dB/km, and 86.5 dB/km for RH = 10%, 50%, and 90%, respectively. Nonetheless, the Full-ITU-R model obtained a MAE of 26.8 dB/km, 158.6 dB/km, and 325 dB/km for RH = 10%, 50%, and 90%, respectively. This strongly suggests that former is a more robust way for estimating the molecular absorption coefficients in this region, with a substantial MAE difference between the models, especially for RH = 90%. Additionally, under the same range and θ ad j , the metrics such as RMSE, MAPE, max.AE and adj.R 2 result in 190 dB/km, 6.8%, 1,143 dB/km, and 0.9994, respectively, for the new proposed model. The Full-ITU-R model whereas yields 861 dB/km, 23.5%, 5,714 dB/km, and 0.9882, respectively, for the same error metrics. All the results are summarized in Table 1. Consequently, the Full-ITU-R model clearly shows a larger discrepancy against the benchmark. Thus, the new model is a more reliable alternative for evaluating the molecular absorption coefficient under the range of 450 to 600 GHz.

1) NOVEL PROPOSED MODEL WITH REDUCED TERMS FOR THE 450-600 GHZ RANGE
The new proposed channel model, based on closed-form equations, for estimating the molecular absorption coefficient in the 100 to 600 GHz range is mainly given by (18). In order to reduce the number of equations to describe only the 450 to 600 GHz band, which was never addressed before in the literature, the new model with reduced terms is herein developed. Accordingly, the reduced absorption coefficient can be expressed as where y 6 ( f , μ) to y 10 ( f , μ), and g( f , μ) are given by (24) to (28), and (56), respectively. In order to assess the performance of the new model with reduced terms (y 6 to y 10 ), we compare the former with its own complete version with all terms (y 1 to y 10 ) and also against the benchmark model (VVW + HITRAN), under the range 450 to 600 GHz, and for RH = 90%. Then, the molecular absorption coefficient versus the frequency range (450-600 GHz) in plotted in Fig. 4. In addition, also shown in Fig. 4 is the results for the difference among the complete proposed model will all terms, the reduced proposed model, and finally the benchmark (VVW + HITRAN). Overall, it can be observed from Fig. 4 an excellent agreement between the proposed model and benchmark, showing complete convergence up to 580 GHz. In addition, it can be seen from Fig. 4 that in the 450 to 500 GHz range, the proposed model with reduced terms (green dashed curve) has slightly better performance than do the complete model with all terms (pink curve). However, both models yield virtually the same performance for the 500-600 GHz range. The results for different error metrics are summarized in Table 2.
Moreover, the error metrics for the new developed models with all terms as well as reduced terms compared against the benchmark in the 450 to 600 GHz range for RH = 10% and 90%, is summarized in Table 2. Bear in mind that g( f , μ) remains the same for both models, but θ ad j = 5 × 10 −7 for the  complete model with all terms, and θ ad j = 3.05 × 10 −3 for the reduced terms model. One can realize that the error metrics are similar for both complete and reduced terms models. Besides that, only a small reduction in terms of MAE and MAPE can be seen for the reduced terms model in Table 2, which is owing to the optimization of θ ad j . As a conclusion, both complete and reduced terms models are accurate, predicting with success the actual absorption losses. The experimental validation of the total atmospheric absorption coefficient is carried out in the next Section.

B. EXPERIMENTAL VALIDATION OF THE TOTAL ATMOSPHERIC ABSORPTION COEFFICIENT
The total atmospheric absorption coefficient is defined as the sum of the molecular absorption coefficient, k a ( f ) given by  (18), and the water vapor continuum absorption coefficient, k c ( f ), given by (57). Then, let us analyse k c ( f ) and the experimental data [29] used here for validation purposes of the new developed formalism model.
The total absorption coefficient was measured using FTIR (Fourier transform infrared spectrometer) for room temperature [29]. The measurements were performed for the frequency range of 300 GHz to 1.5 THz, varying the amount of water vapor (RH) in a chamber for different transmission paths. To calculate the contribution of k c ( f ) to the total absorption coefficient, (57) is used, whose C W = 4.39 × 10 −8 (dB/km)/(hPa GHz) 2 and C A = 4 × 10 −9 (dB/km)/(hPa GHz) 2 [32].
Accordingly, adding k c ( f ) to k a ( f ) results in the total absorption coefficient, whose performance versus the frequency range from 300 to 600 GHz are show in Fig. 5, for RH = 36.78% (Fig. 5(a)) and for RH = 70.84% (Fig. 5(b)). The experimental data of the total absorption coefficient, which can be found in [29], is also plotted in Fig. 5, considering shown. This limitation is due to saturation effects of the measurement equipment in [29]. The average errors of measurement for RH = 36.78% and RH = 70.84% are 4% and 2.87%, respectively [29]. Remarkably, both experimental data and new developed model show very good overall agreement, suggesting a great deal of accuracy. One can clearly realize from Fig. 5 that the new proposed model shows convergence with the experimental data.
Additionally, the simulated atmospheric transmittance was calculated by (9), which depends on the total absorption coef- where k a ( f ) is given by (18) and k c ( f ) given (57). Then, the transmittance performance considering a 6-meter transmission path for RH = 36.78% and 70.84% are illustrated in Fig. 6(a) and (b), respectively. Those conditions were used in the measurement of the experimental data analysis of the transmittance in [29]. Hence, the experimental transmittance versus the frequency range 300 to 600 GHz is also plotted in Fig. 6. Once again, the channel

C. LINK BUDGET CALCULATIONS CONSIDERING THE TOTAL ATMOSPHERIC ABSORPTION COEFFICIENT
In this section, link budget calculations are carried out in order to employ the new proposed model for performance analysis of THz networks from 100-600 GHz. The goal is to show that the new developed tool is simpler and faster to implement than numerical models like the VVW + HITRAN model. Accordingly, the developed tool can be used to evaluate different THz scenarios with line-of-sight supporting several applications as illustrated in Fig. 1. In this way, the performance evaluation is done by means of link budget calculations, which estimates the power at the receiver and many other quantities related with gains and losses in the link. Then, the SNR and BER are considered and calculated for different THz links.
To do so, let us consider the antenna gain of the LOS THz link as follows [26] G T x/Rx = 20 log 10  [43], the second and third are given by [14], the fourth, fifth and sixth are taken from [33], [44], for the sake of analysis consistency. Accordingly, the atmospheric conditions for the THz channels considered are T = 296 K, p = 1 atm and RH = 50%. Next, the molecular and water vapor continuum absorption coefficients are obtained via (18) and (57), respectively. Thus the atmospheric absorption coefficient is given by the sum of (18) and (57). The optimization parameters θ ad j used in the calculation of the molecular absorption coefficient are provided in Tables 3 and 4, considering their respective frequency bands. Thus, the transmitter power and noise figure considered in the evaluation are P T x = 0 dBm and NF = 10 dB, respectively [45], [46], [47], [48], [49]. Subsequently, the P n , P Rx , PL, SNR and BER of the THz link are determined by (5), (9), (11), (12) and (13), respectively. The results obtained are shown in Tables 3 and 4. The THz link distances are always 1 km in Table 3, but they are different in Table 4 so that the remaining parameters/metrics such as SNR and BER can be further stressed. The comparison of the proposed model against the benchmark (VWW + HITRAN) in terms of several link budget parameters including key figure-of-merits such as the PL, SNR, and BER for a fixed distance (d = 1000 m) are summarized in Table 3. Noteworthy, the relative errors between the models are indeed small and can even be considered negligible. For example, as can be seen in Table 3, for f c = 157.75 GHz, RE = 0.55% only. Once again, the proposed model converges towards the benchmark. This is important as the remaining parameters/metrics are also impacted, but as shown in Table 3, the PL, SNR, and BER basically show the same performance for both models. In this manner, for both model and benchmark, the PL, SNR, and BER are 139.6 dB, 23.3 dB, and 1.3 × 10 −13 , respectively, considering f c = 157.75 GHz. Likewise, the same analysis holds valid if one considers f c = 261.36 GHz. Interestingly, one can note that the SNR for these two THz link scenarios ( f c = 157.75 GHz and f c = 261.36 GHz) is high enough to yield a BER level under the regime of error-free data transmissions. Eventually, this can enable different network scenarios and consequently support many broadband applications.
Lastly, a similar link budget analysis is carried out but for THz links at higher frequencies such as f c = 410 GHz, f c = 484 GHz, and f c = 542 GHz, as summarized in Table 4. Bear in mind that the THz link distances are different for each center frequency. Once more, even under different link distances, the relative error between the model and benchmark is small in general, which yields virtually similar levels of the key figure-of-merits, i.e., the SNR and BER. Remarkably, it can be seen in Table 4 that for f c = 542 GHz, the SNR (∼15 dB) and BER (∼ 2 × 10 −3 ) resulted in poor levels, which hinders the performance of such THz link. Thus, it can be concluded that the new channel model is robust and proved to be a useful tool for performance evaluation of THz networks.

IV. CONCLUSION
In this article, a novel mathematical formalism rendering completely accuracy for performance analysis of line-of-sight THz networks within the frequency range of 100-600 GHz was proposed. The new developed channel model takes into account both the water vapor continuum absorption loss and the molecular absorption loss. In addition, the model's formalism in based on closed-form absorption line shape equations and a fitting parameter (equalization factor), developed to accurately match the channel response given by the HI-TRAN spectroscopic database. The latter is considered as a benchmark, against which the performance can be compared. Moreover, several error metrics (such as MAE, RMSE, and MAPE) have been considered and analyzed. The new channel model was validated and compared against many different models. It was shown that the channel model accurately reproduces the experimental dataset HITRAN, which is considered the benchmark. In other words, the model shown complete convergence and is a more appropriate and accurate choice for performance assessment of THz networks due to its simplicity. Its great performance indicator sets a new benchmark for analytical performance evaluation of such networks. This new channel model can be successfully applied for accurate performance evaluation of THz networks within a large frequency range, including the prominent D-band. Moreover, the model can be further applied for evaluating the link transmission distance of THz networks and their performance in terms of key figures-of-merit such as SNR and BER. Therefore, this robust and easy-to-use channel loss model is deemed as a unique tool and a new benchmark to be added to the portfolio for the performance evaluation of THz networks.