# Atmospheric Effects on OFDM Wireless Links Operating in the Millimeter Wave Regime
## 3 Propagation Factors of the Atmosphere  
The millimeter wave propagation model (MPM) is used for calculation of the atmospheric frequency response taken into consideration of weather conditions. Contributions of dry air, water vapor, suspended water droplets (haze, fog, cloud), and rain are all addressed in the model. For completeness and clarity of notations, the model [7; 9] is hereby reviewed.  
In general, the propagation of millimeter waves through the atmospheric medium along a distance \(d\) is described by the transfer function:  
\[H(jf)=\sqrt{T_{FS}(d)}e^{-j\frac{2\pi f}{c}\cdot\pi(f)\cdot d} \tag{6}\]where \(n(f)\) is the _refractive index_ of the atmospheric medium and \(c\) is the speed of light in vacuum. \(T_{FS}(d)\) is the free-space power transmission along a distance \(d\) excluding the atmospheric transmittance:  
\[T_{FS}(d)=G_{R}\bigg{(}\frac{c}{2\pi f_{c}d}\bigg{)}^{2}G_{T} \tag{7}\]  
where \(G_{R}\) and \(G_{T}\) are the directivity gains of the receiving and transmitting antennas, respectively. The refractive index \(n(f)\) is a frequency dependent quantity that can be expressed in terms of dispersive complex refractivity \(N(f)\) (given in ppm):  
\[n(f)=1+\underbrace{[N_{0}+N^{\prime}(f)-jN^{\prime\prime}(f)]}_{N(f)}\times 1 0^{6} \tag{8}\]  
where the nondispersive part \(N_{0}\) is real and positive constant, \(N^{\prime}(f)\) and \(N^{\prime\prime}(f)\) are the frequency dependent real and imaginary refractivity terms, respectively. The millimeter wave propagation model (MPM) is used for calculation of these refractivity terms in the frequency domain [13; 14].  
Figure 2: Attenuation factor (**a**,**c**,**e**) and incremental phase dispersion factor (**b**,**d**,**f**) in different weather conditions: (**a**,**b**) show the relative humidity RH [%] effect (**c**,**d**) show the effect of suspended droplets via the liquid water content W [g/m\({}^{3}\)] and (**e**,**f**) are for rainy conditions, where R is the rainfall rate in [mm/h].  
Substituting Equations (7) and (8) into Equation (6), the transfer function of the atmospheric medium can be presented as:  
\[H(jf)=\sqrt{T_{FS}(d)}e^{-[\alpha(f)+j\beta(f)]d} \tag{9}\]  
where \(\alpha(f)\) is the attenuation factor, given in terms of the imaginary part of the atmospheric refractivity:  
\[\alpha(f)=\frac{2\pi f}{c}N^{\prime\prime}\left(f\right)\times 10^{-6} \tag{10}\]  
where \(\beta(f)\) is the wavenumber of the propagating field, expressed in terms of the real part of the atmospheric refractivity:  
\[\beta(f)=\frac{2\pi f}{c}+\underbrace{\frac{2\pi f}{c}[N_{0}+N^{\prime}(f)] \times 10^{-6}}_{\Delta\beta(f)} \tag{11}\]  
The term \(\Delta\beta(f)\) is defined as the incremental phase dispersion caused by the dielectric substances composing the air.  
The frequency dependent attenuation and incremental phase dispersion are illustrated in Figure 2, for different weather conditions characterized by relative humidity \(RH\) [%], suspended liquid water content \(W\) [g/m\({}^{3}\)], and rainfall rate \(R\) [mm/h].  
The results reveal absorption peaks at 22, 183, and 325 GHz, where resonance absorption of water (H\({}_{2}\)O) occurs, as well as absorption peaks at 60 and 119 GHz, due to absorption resonances of oxygen (O\({}_{2}\)). Between these frequencies, minimum attenuation is obtained at 35 GHz (Ka-band), 94 GHz (W-band), 130 GHz (D-band), and 220 GHz, which are known as atmospheric transmission \"windows\". Increasing the water content in the air in different forms (vapor (humidity), suspended water droplets (mist and fog), and drops (rain)) significantly increases the attenuation and incremental phase dispersion.  
The model was tested experimentally for various foggy conditions in a controlled environment demonstrating good agreement with measured results [9; 10; 11].