Two-ray ground-reflection model

Sources: From the figure the received line of sight component may be written as :r_{los}(t)=Re \left\{ \frac{ \lambda \sqrt{G_{los}} }{4\pi}\times \frac{s(t) e^{-j2\pi l/\lambda}}{l} \right\} and the ground reflected component may be written as :r_{gr}(t)=Re\left\{\frac{\lambda \Gamma(\theta) \sqrt{G_{gr}}}{4\pi}\times \frac{s(t-\tau) e^{-j2\pi (x+x')/\lambda}}{x+x'} \right\} where s(t) is the transmitted signal, l is the length of the direct line-of-sight (LOS) ray, x + x' is the length of the ground-reflected ray, G_{los} is the combined antenna gain along the LOS path, G_{gr} is the combined antenna gain along the ground-reflected path, \lambda is the wavelength of the transmission (\lambda = \frac{c}{f}, where c is the speed of light and f is the transmission frequency), \Gamma(\theta) is ground reflection coefficient and \tau is the delay spread of the model which equals (x+x'-l)/c. The ground reflection coefficient is Note that the power decreases with as the inverse fourth power of the distance in the far field, which is explained by the destructive combination of the direct and reflected paths, which are roughly of the same in magnitude and are 180 degrees different in phase. G_t P_t is called "effective isotropic radiated power" (EIRP), which is the transmit power required to produce the same received power if the transmit antenna were isotropic. ==In logarithmic units==

In logarithmic units : P_{r_\text{dBm}}=P_{t_\text{dBm}}+ 10 \log_{10}(G h_t ^2 h_r ^2) - 40 \log_{10}(d) Path loss : PL\;=P_{t_\text{dBm}}-P_{r_\text{dBm}}\;=40 \log_{10}(d)-10 \log_{10}(G h_t ^2 h_r ^2) ==Power vs. distance characteristics==

When the distance d between antennas is less than the transmitting antenna height, two waves are added constructively to yield bigger power. As distance increases, these waves add up constructively and destructively, giving regions of up-fade and down-fade. As the distance increases beyond the critical distance dc or first Fresnel zone, the power drops proportionally to an inverse of fourth power of d. An approximation to critical distance may be obtained by setting Δφ to π as the critical distance to a local maximum. ==An extension to large antenna heights==

The above approximations are valid provided that d \gg (h_t+h_r), which may be not the case in many scenarios, e.g. when antenna heights are not much smaller compared to the distance, or when the ground cannot be modelled as an ideal plane . In this case, one cannot use \Gamma \approx -1 and more refined analysis is required, see e.g. ==Propagation modeling for high-altitude platforms, UAVs, drones, etc.==

The above large antenna height extension can be used for modeling a ground-to-the-air propagation channel as in the case of an airborne communication node, e.g. an UAV, drone, high-altitude platform. When the airborne node altitude is medium to high, the relationship d \gg (h_t+h_r) does not hold anymore, the clearance angle is not small and, consequently, \Gamma \approx -1 does not hold either. This has a profound impact on the propagation path loss and typical fading depth and the fading margin required for the reliable communication (low outage probability). ==As a case of log distance path loss model==

The standard expression of Log distance path loss model in [dB] is : PL\;=P_{T_{dBm}}-P_{R_{dBm}}\;=\;PL_0\;+\;10\nu\;\log_{10} \frac{d}{d_0}\;+\;X_g, where X_g is the large-scale (log-normal) fading, d_0 is a reference distance at which the path loss is PL_0 , \nu is the path loss exponent; typically \nu = 2...4. and is also used for indoor communications, see e.g. and references therein. The path loss [dB] of the 2-ray model is formally a special case with \nu = 4: :PL\;=P_{t_{dBm}}-P_{r_{dBm}}\;=40 \log_{10}(d)-10 \log_{10}(G h_t ^2 h_r ^2) where d_0=1 , X_g = 0, and : PL_0 =-10 \log_{10}(G h_t ^2 h_r ^2) , which is valid the far field, d > d_c = 4\pi h_r h_t/\lambda = the critical distance. ==As a case of multi-slope model==

The 2-ray ground reflected model may be thought as a case of multi-slope model with break point at critical distance with slope 20 dB/decade before critical distance and slope of 40 dB/decade after the critical distance. Using the free-space and two-ray model above, the propagation path loss can be expressed as L =\max \{G, L_{min},L_{FS},L_{2-ray}\} where L_{FS}=(4\pi d/\lambda)^2 and L_{2-ray}=d^4/(h_t h_r)^2 are the free-space and 2-ray path losses; L_{min} is a minimum path loss (at smallest distance), usually in practice; L_{min} \approx 20 dB or so. Note that L \ge G and also L \ge 1 follow from the law of energy conservation (since the Rx power cannot exceed the Tx power) so that both L_{FS}=(4\pi d/\lambda)^2 and L_{2-ray}=d^4/(h_t h_r)^2 break down when d is small enough. This should be kept in mind when using these approximations at small distances (ignoring this limitation sometimes produces absurd results). ==See also==