Simplified Molecular Absorption Loss Model for Gigahertz Frequency Band

Transcription

1 Simplified Moleular Absorption Loss Model for 275 Gigahertz Frequeny Band Joonas Kokkoniemi, Janne Lehtomäki, and Markku Juntti Centre for Wireless Communiations (CWC), University of Oulu, P.O. Box, 914 Oulu, Finland Abstrat This paper fouses on giving a simplified moleular absorption loss model for a275 GHz frequeny band, whih has signifiant potential for variety of future short and medium range ommuniations. The band offers large theoretial data rates with reasonable path loss to theoretially allow even up to kilometer long link distanes when suffiiently high gain antennas are used. The moleular absorption loss in the band requires a large number of parameters from spetrosopi databases, and, thus, the exat modeling of its propagation harateristis is demanding. In this paper, we provide a simple, yet aurate absorption model, whih an be utilized to predit the absorption loss at the above frequeny band. The model is valid at a regular atmospheri pressure, it depends on the distane, the relative humidity, and the frequeny. The existing simplified model by ITU does not over frequenies above 3 GHz and has more omplexity than our proposed model. The moleular absorption loss inreases exponentially with the distane, dereasing the utilizable bandwidth in the viinity of the absorption lines. We provide a model to approximate the window widths at the above frequeny band. This model depends on the distane, the relative humidity, the frequeny, and the maximum tolerable loss. It is shown to be very aurate below one kilometer link distanes. I. INTRODUCTION The millimeter wave frequeny band (mm-wave, GHz) is of interest for future wireless ommuniations systems and appliations, suh as 5G and beyond [1]. These frequenies an be utilized in both medium and short range ommuniations, with the former requiring large antenna gains. Up to 8 meter link has been shown to be feasible at 2 GHz frequeny [2]. The main benefit of higher frequenies are the large available bandwidths that make it possible to provide extremely high data rates, or the possibility to share the spetral resoures among vast numbers of devies. The latter ase is very interesting beause of the ever inreasing numbers of internet of things (IoT) devies. One hallenge of utilizing the mm-wave frequenies and above is the proper hannel modeling. The free spae path loss and the assoiated antenna gain terms are very well known and easy to implement and adjust based on the antennas in hand. On the other hand, moleular absorption loss plays a role in the aurate hannel modeling. At short distanes (few meters) around the GHz frequeny band, the moleular absorption is not signifiant in omparison to the large free spae path loss. However, its importane beomes more evident at high distanes or high frequenies due to exponentially inreasing loss as a funtion of distane [3]. Modeling of the absorption is very well known, but requires large numbers of parameters from the spetrosopi databases, suh as HITRAN database [4]. Our aim herein is to present a simplified moleular absorption loss model that predits this loss omponent with high auray without a need for large numbers of parameters. ITU has presented aurate model for alulation of gaseous attenuation up to GHz in ITU-R P [5]. The model in [5] is line-by-line based and the results from it orrespond to those obtained by using HITRAN database, onfirming its validity (Fig. 1). The full model is not suitable for analytial alulations or for quik use, sine it requires using a signifiant number (553) of tabulated parameters and ompliated funtions. In [5], a polynomial based approximation has also been presented. It is valid up to 3 GHz. Please note that a newer version ITU-R also exists but that version does not have a speified polynomial model. We use the older version, sine the polynomial model therein is related to our work in this paper. However, ompared to our models, those models have several weaknesses. First of all, even though [5] inludes lines even at 178 GHz, it is only speified to be valid for frequenies up to 3 GHz. The simple reason is that the model gives erroneous results above 3 GHz (see Fig. 1). The simplified model in the newer version is also limited to 3 GHz. The model also (only for water absorption, oxygen absorption is handled separately) inludes 9 terms and if some of the terms are removed, they may also affet frequenies in different bands. For example, the term involving 178 GHz needs to kept or the level of the attenuation between the peaks at lower frequenies is way off. In general, the ITU model is aurate below 3 GHz, but in this paper we will extend the frequeny range and simplify the estimation for it. Our model is targeted for the frequeny range 275 GHz, whih is not overed by the simplified ITU model. Also, our model, whih uses two polynomials is muh simpler than the ITU model whih involves at least 4 to 9 polynomial funtions. The presented model is valid in the standard atmospheri pressure and is aurate at the standard atmospheri temperatures, and it only depends on the distane, the frequeny, and the relative humidity. The presented absorption model is aligned towards future standardization in WRC 19, where the use of frequeny band 275 to 4 GHz is on the agenda. Our new simplified model overs 275 GHz and an be modified easily to over also frequenies up to 4 GHz by inluding a third term. Our simplified model is

2 Speifi Attenuation (db/km) Full ITU Full simplified ITU Simplified ITU - redued terms Frequeny (GHz) Fig. 1. ITU-R models for the moleular absorption loss. funtions of frequeny and antenna orientation. The absorption oeffiient an be alulated with the help of databases, suh as the HITRAN database [4], as it will be detailed below. B. Absorption Coeffiient The main point of interest here is to model the absorption loss due to the interest in showing an easy estimate for it. First, we need to define the absorption oeffiient as a whole before showing the approximation in the next setion. The absorption oeffiient depends on pressure, temperature and moleular omposition of the hannel as κ i a(f) = µ i Nσ i (f), (3) aimed at analytial studies and simple alulations without the need of alulating the full ompliated model. It is hoped to be valuable for wireless ommuniation engineers in their researh by enabling easy approximation of the moleular absorption loss. As the moleular absorption is exponentially inreasing with distane, it has been shown that the available bandwidth of the transmission is dereasing with distane (see, e.g., [6], [7]). Utilizing a similar approah as that in the ase of moleular absorption loss, we show a simplified model to estimate the transmission window width as a funtion of the distane, the frequeny, the humidity, and the maximum tolerable loss in the viinity of the enter frequeny of the transmissions. This model estimates the available bandwidth very aurately up to about one kilometer distanes. This helps researhers to easily estimate the bandwidth of the speifi systems with known transmission distanes. The rest of this paper is organized as follows. Setion II derives the moleular absorption model and shows its validity. Setion III gives the available bandwidth model and shows its validity and Setion IV onludes the paper. II. SIMPLIFIED MOLECULAR ABSORPTION LOSS MODEL A. Path Loss Model A ommon path loss model for the higher frequeny bands is omposed of the free spae path loss and moleular absorption loss. The latter is given by the Beer-Lambert law [3], [8] τ(f,r) = P r(f) P t (f) = i e Σiκ a (f)r, (1) where τ(f,r) is the transmittane, f is the frequeny, r is the distane from transmitter (Tx) to reeiver (Rx), P t (f) and P r (f) are Tx and Rx power, respetively, and κ i a(f) is the absorption oeffiient of the ith absorbing speies at frequeny f. Combining this to the free spae path loss, we get a total path loss for the LOS paths as PL(r,f) = (4πrf)2 exp(κ a (f)r) 2 G Rx G Tx, (2) where is the speed of light and G Rx and G Tx are the antenna gains of the Rx and Tx, respetively, that further are usually where µ i is the fration of the moleules of kind i, N is the number density of all moleules, and σ i (f) is the absorption ross setion of theith moleular speies. The absorption ross setion is a produt between spetral line intensity S i (T) and spetral line shape F i (f,p,t). The absorption ross setion tells the effetive area for absorption for a single moleule. The spetral line intensity tells the strength of the absorption per absorption line and the spetral line shape tells the width and shape of the spetral lines. The enter frequeny of the absorption line i (f i ) is dependent on pressure through [3], [9] f i = f i +δ i p p, (4) where, p is the atmospheri pressure, p is the standard pressure (11325 Pa), f i is the enter frequeny of the absorption line at standard pressure, and δ i is the linear pressure shift. In the regular atmospheri pressures, the disrete absorption lines experiene pressure broadening that we model with the line shapes. This broadening an be modeled with the Lorentz half-width αl i at normal pressure onditions and that is omposed of foreign and self-broadened half-widths α f and α, i respetively, [8], [9] ( )( ) δ αl i = [(1 µ i )α f p +µ iα] i T, (5) p T where T is the temperature of the atmosphere in Kelvin, T is the standard temperature (296 K referene temperature of HITRAN atalogue [4]), and δ is temperature broadening oeffiient. Self-broadening is aused by the ollisions between moleules of the same speies, while foreign-broadening is due to the inter-moleular ollisions. The most well known line shape is the Lorentz line shape [8], [1] α i L FL(f i ±f) i = 1 π (f ±f) i 2 +(αl i (6) )2, where the minus sign is hosen when the Lorentz line shape is utilized alone. This line shape is simple, but it also overestimates the absorption at far wings and it never reahes zero [8]. This line shape was enhaned by Van Vlek and Weisskopf in

3 1945 [1]. The Van Vlek-Weisskopf line shape is defined as [1] [12] ( ) 2 f FVVW(f) i = [FL(f i f)+f i L(f i +f)]. i (7) f i The Van Vlek-Weisskopf line shape with far wing adjustments is often alled the Van Vlek Huber line shape F i VVH(f) = f f i tanh( hf 2k B T ) tanh( hfi 2k B T )[Fi L(f f i )+F i L(f +f i )], (8) where h is the Plank onstant, beause of the derivation by Van Vlek and Huber [11]. In reality, the differene of this line shape to the Van Vlek Weisskopf line shape is very small in the mm-wave frequenies. The line intensity S i an be obtained from HITRAN database for the referene temperature (296 K), but it has to be saled for the other temperatures by [13] S i (T) = S i Q(T ) Q(T) i e ( he L k B T ) e ( hei L k B T ) 1 e( hf i k B T ) 1 e ( hfi k B T ) (9) where k B is the Boltzmann onstant, EL i is the lower state energy of the transition of absorbing speies i. The partition funtion Q(T) and its definitions an be found in [13, Appendix A]. Combining the above theories, the total absorption loss beomes PL abs (f) = exp(µ i NF i (f)s i (T)). (1) Using the different line shapes in the plae of F i (f), we an ompare their differenes in Fig. 2. We an see that as predited, the Lorentz line shape overestimates the wing absorption and the Van Vlek Weisskopf and Van Vlek Huber line shapes are in pratie idential. Fortunately, the error of the Lorentz line shape is nearly linear to the more sophistiated line shapes, whih gives us tools to simplify the absorption loss based on the Lorentz line shape. C. Simplified Absorption Model The main goal here is to introdue a simplified moleular absorption model that is valid at 275 GHz frequeny band, although, the model is equally aurate from to GHz. This band has two major absorption lines, one at about 325 GHz and seond at 38 GHz. The proposed simplified hannel model starts from the assumption of the Lorentz line shape holding for the relative loss. This line shape gives the easiest way to give a simple moleular absorption model that only depends on volume mixing ratio of water (humidity) and frequeny. The absorption model is given by y 1 (f,µ H2O) =.25µ H2O(.13µ H2O +.294) ( ) 2, (11) (.93µ H2O +.925) 2 + f.835 Path loss [db/km] Van Vlek-Weisskopf Van Vlek-Huber Lorentz Differene Frequeny [GHz] Fig. 2. Comparisons of the ommon line shapes for modeling the absorption loss. Figure also shows the relative error of the Lorentz line shape to the more sophistiated line shapes. y 2 (f,µ H2O) = 2.14µ H2O(.172µ H2O +.3) ( ) 2, (12) (.537µ H2O +.956) 2 + f PL abs (f,µ H2O) = e d(y1(f,µh 2 O)+y2(f,µH 2 O)+g(f)), (13) where f is the desired frequeny grid, µ H2O is the volume mixing ratio of water vapor, whih is given in terms of relative humidity φ by µ H2O = φ p w(t,p), (14) p where φp w(t,p)/ is the partial pressure of water vapor, for whih the saturated water vapor partial pressure p w under pressure p and temperature T an be estimated, e.g., by the Buk equation [14] ( ) 17.2T p w = ( p)exp, 2.97+T (15) where p is given in hetopasals and T is given in degrees Celsius. The fator g(f) is an equalization fator given by a polynomial g(f) = p 1 f 3 +p 2 f 2 +p 3 f +p 4, (16) where oeffiients p 1 = , p 2 = , p 3 = , and p 4 = , to take into aount the differene between the Lorentz and Van-Vlek Huber line shapes. This fator is a simplified approximation of the real differene. The model above is obtained by utilizing the line-speifi data from HITRAN and by leaving only the humidity (with pressure and temperature dependenies) and frequeny grid floating. Therefore, the auray of this model is very high as the only soures of error ome from the rounding errors and a small error aused by the orretion fator g(f). The differene differene of the above simplified line shape is ompared to the fully theoretial one alulated with Van

4 Absorption loss [db] Exat theory Numerial estimate Differene at m distane Differene at meter distane meter distane meter distane Frequeny [GHz] Fig. 3. Comparison of the exat absorption loss to the numerial estimate and the error aused by the numerial estimate. Path loss [db] 1 Minimum bandwidth Absorption loss, m Absorption loss, m Absorption loss, m Maximum bandwidth Frequeny [GHz] Fig. 5. The estimated transmission window widths in omparison to the absorption loss at different distanes. Available bandwidth [GHz] Maximum bandwidth Minimum bandwidth Distane [m] Fig. 4. The estimated available bandwidth based on one sided three db loss (minimum bandwidth) and the maximum bandwidth allowed by the full transmission window. Vlek Huber line shape in Fig. 3. Two distanes are utilized, and meters. We an see that the expeted performane of the simplified model is extremely high. As it an be seen, the error is mainly at the absorption peaks, whih is not that important as the power loss is high at the absorption peaks at high distanes. Thus, they should be avoided in any ase. Instead, the performane of the numerial approximation at the wing absorption is more important where the numerial estimate is more or less perfet. III. TRANSMISSION WINDOW MODEL We saw in the previous setion that the relative loss between the frequenies is roughly the same among different line shapes, although, the absolute loss is different. Assuming that we have a ertain linear threshold γ for maximum absorption loss, i.e., to alulate available 1log 1 (γ) db bandwidth within a transmission window, we an alulate the limiting frequenies by exp(dµnf(f)s(t)) = exp(dµnf(f )S(T))γ, (17) where F( ) is the line shape and f is the frequeny at whih the left-hand side of the equation is equal to the loss at enter frequeny of the transmission f multiplied with the threshold γ. Then, F(f) = F(f )+ ln(γ) dµns(t). (18) by applying the Lorentz line shape in (6) and assuming that the line shape is roughly symmetri around the line enter, we get the frequeny deviations from the line enters ( f = f f i, where f i is absorption line enter) for the two meaningful absorption lines (325 and 38 GHz) as presented in Eqs. (19) and (). In these equations, f l is the frequeny deviation from the 325 GHz absorption line where the absorption loss has inreased by fator γ from the enter frequeny and f h is the same for the 38 GHz line, and indies l and h refer in general to the lower and higher frequeny absorption line parameters, respetively. The simplified versions are given by Eqs. (21) and (22). Then the available bandwidths beome W max = f h f h (f l + f l ), (23) W min = 2min{f h f h f,f (f l + f l )}, (24) where W max indiates the maximum bandwidth limited by the preset threshold to upper and lower frequeny band, and W min tells the minimum bandwidth by single side limitation, i.e., when either side of the transmission enter frequeny reahes the preset threshold, the bandwidth is onsidered to be limited by it. Figure 4 shows the available bandwidths aording to (23) and (24) and for three-db threshold at a transmission enter frequeny of 342 GHz. This partiular frequeny experienes the lowest loss between the two major absorption lines between 325 and 38 GHz. As it has been predited earlier, e.g., in [6], [7], the available bandwidth dereases as a funtion of frequeny beause of the exponential moleular absorption loss as a funtion of distane. Fig. 5 shows the predited available band as a funtion of frequeny. We an see that the predition is not exat due to error aused by the usage of the Lorentz line shape, similarly as in the previous setion for the numerial predition of the absorption loss. However, the error is not major and is about 2 GHz at one kilometer

5 (f ) f l = ( f l 2 (αl l )2) πlnγ + αl l dµ H 2ONS l (T) (f f h = ( f h ) 2 (α h L )2) + πlnγ αl hdµ H 2ONS h (T) f l = (f ] 2 [.842) πln(γ) µ H2O f h = (f ] 2 [ 2.679) πln(γ) µ H2O (α l L )2, (19) (α h L )2. ().98, (21).17. (22) distane. If we would inrease the distane, also the error would inrease. Thus, we see the one kilometer link distane as a maximum distane where this model an be onsidered relatively aurate. IV. CONCLUSION We have shown simple estimates for the moleular absorption loss and the expeted bandwidths within the transmission windows in the to GHz frequeny band. Despite of the simpliity, they provide an aurate way to estimate the moleular absorption loss without a need to implement omplex absorption models and obtaining values from the databases. We showed that these models are very aurate and reliable at the given frequeny range. Furthermore, the models an easily be extended to any possible frequeny band. However, the omplexity of the approximation inreases as the number of absorption lines inreases. Therefore, the links operating below one terahertz are easier to model numerially. The shown numerial models are very useful for millimeter and low terahertz band ommuniations analysis. Both being among the potential frequeny ranges for the future ommuniations. Via aurate information on the absorption loss, one an easily predit the additional loss, and the maximum utilizable and appliation speifi bandwidths. V. ACKNOWLEDGEMENT This projet (TERRANOVA) has reeived funding from Horizon, European Unions Framework Programme for Researh and Innovation, under grant agreement No [3] J. M. Jornet and I. F. Akyildiz, Channel modeling and apaity analysis for eletromagneti nanonetworks in the terahertz band, IEEE Trans. Wireless Commun., vol. 1, no. 1, pp , Ot. 11. [4] L. S. Rothman et al., The HITRAN 12 moleular spetrosopi database, J. Quant. Spetros. Radiat. Transfer, vol. 1, no. 1, pp. 4, Nov. 13. [5] ITU-R (9) Reommendation P.676-8, Attenuation by atmospheri gases, International Teleommuniation Union Radioommuniation Setor Std. [6] C. Han and I. F. Akyildiz, Distane-aware multi-arrier (DAMC) modulation in terahertz band ommuniation, in Pro. IEEE Int. Conf. Commun., 14, pp [7] C. Han, A. O. Bien, and I. F. Akyildiz, Multi-wideband waveform design for distane-adaptive wireless ommuniations in the terahertz band, IEEE Trans. Signal Proess., vol. 64, no. 4, pp , Nov. 15. [8] S. Paine, The am atmospheri model, Smithsonian Astrophysial Observatory, Teh. Rep. 152, 12. [9] Calulation of moleular spetra with the Spetral Calulator. [Online]. Available: [1] J. H. Van Vlek and V. F. Weisskopf, On the shape of ollisionbroadened lines, Rev. Mod. Phys., vol. 17, no. 2 3, pp , [11] J. H. Van Vlek and D. L. Huber, Absorption, emission, and linebreadths: A semihistorial perspetive, Rev. Mod. Phys., vol. 49, no. 4, pp , [12] I. Halevy, R. T. Pierrehumbert, and D. P. Shrag, Radiative transfer in CO2-rih paleoatmospheres, J. Geophys. Res., vol. 114, no. D18, pp. 1 18, Sep. 9. [13] L. S. Rothman et al., The HITRAN moleular spetrosopi database and HAWKS (HITRAN atmospheri workstation): 1996 edition, J. Quant. Spetros. Radiat. Transfer, vol. 6, no. 5, pp , Nov [14] O. A. Alduhov and R. E. Eskridge, Improved magnus form approximation of saturation vapor pressure, J. Appl. Meteor., vol. 35, no. 4, pp , Apr REFERENCES [1] T. S. Rappaport et al., Millimeter wave mobile ommuniations for 5G ellular: It will work! IEEE Aess, vol. 1, no. 1, pp , May 13. [2] I. Kallfass, F. Boes, T. Messinger, J. Antes, A. Inam, U. Lewark, A. Tessmann, and R. Henneberger, 64 Gbit/s transmission over 8 m fixed wireless link at 2 GHz arrier frequeny, J. Infrared Milli. Terahz. Waves, vol. 36, no. 2, pp , Feb. 15.