Measurement-based Close-in Path Loss Modeling with Diffraction for
Rural Long-distance Communications

Path loss models that are applicable to frequencies below 6 GHz and various heights have been released by the 3rd Generation Partnership Project (3GPP) and International Telecommunication Union (ITU) [1, 2]. Furthermore, based on the previous models developed for frequencies below 6 GHz, 3GPP has developed new path loss models for frequencies above 6 GHz [3, 4] and 0.5 GHz [5]. These path loss models from 3GPP and ITU suggest modeling for line-of-sight/non-line-of-sight (LOS/NLOS) scenarios, applying to urban microcell, urban and rural macrocell, as well as indoor office environments, with LOS models applying for the breakpoint distance. Close-in (CI) path loss models have been widely used in diverse environments and have demonstrated good accuracy over a wide range of frequencies [6]. In contrast to other models such as the alpha-beta-gamma model, which include floating-offset parameters that are far removed from the physical principles of outdoor channel propagation, the CI models are based on the fundamental physical principles of signal propagation presented by Friis and Bullington [7]. The path loss exponent (PLE) included in the CI model offers physical insights into path loss based on the propagation environment.

Consequently, numerous researchers have been drawn to the development of close-in (CI) path loss models. Among them, MacCartney and Samimi et al. have produced omnidirectional CI path loss models from measurements at 28 and 73 GHz in Downtown Manhattan, where the path loss exponents (PLEs) for line-of-sight (LOS) at those frequencies were 2.1 and 2.0, respectively, and for non-line-of-sight (NLOS) at both frequencies were 3.4 [7, 8]. Additionally, MacCartney et al. have conducted field measurements at 73 GHz in rural Virginia, deriving CI models (LOS: PLE 2.16, NLOS: PLE 2.75) and height-dependent path loss exponent (CIH) models (LOS: PLE 2.31, NLOS: PLE 3.07), which indicate that the PLE may not be frequency-dependent [4]. Sun et al. have also provided CI path loss models, along with alpha-beta-gamma (ABG) and frequency-weighted path loss exponent (CIF) models, for frequencies ranging from 2 to 73 GHz and for urban macrocell (UMa), urban microcell (UMi), and indoor hotspot (InH) scenarios. The CI models for the UMa scenario had a LOS PLE of 2.0 and an NLOS PLE of 2.9, both of which were frequency-independent [9].

For a more accurate path loss analysis, ITU introduced diffraction loss models such as single knife-edge (sKE), and delta-Bullington (DB) [10, 11]. The sKE diffraction loss model treats obstacles as a knife-edge with negligible thickness [10], while the DB diffraction loss model combines the Bullington part of diffraction with spherical-Earth diffraction [11]. Based on the review of literature, it is reasonable to assume that the inclusion of diffraction effect in the CI path loss model, which was neglected in previous studies [4, 7, 12], can improve the accuracy of the path loss model. Additionally, it is plausible to assume that in outdoor settings, the path loss exponent of the CI models might be frequency-independent beyond the initial meter of propagation loss.

This letter presents modified CI path loss models for the rural macrocell (RMa) scenario [13, 14, 15], incorporating a diffraction component to the general form. To the best of our knowledge, our research team is the first to take into account the diffraction component in the CI path loss model. These models are based on measurements conducted at 1400 and 2250 MHz in northern Seoul. As the ITU is investigating new frequency bands with strong radio properties for International Mobile Telecommunications (IMT), including those below 10 GHz [16] and in the 1$\sim$2 GHz range [17], our tested frequencies of 1400 and 2500 MHz can serve as valuable examples for further research into path loss modeling. Our proposed models offer improved accuracy compared to models without a diffraction component. Additionally, we observe that our proposed CI path loss model with diffraction exhibits frequency-independence beyond the first meter of propagation, unlike existing models in 3GPP [4, 13, 14]. Hence, we can conclude that our proposed model is suitable for use in upper-band spectrum ranges [18, 19], including millimeter-wave [20, 21], where diffraction loss becomes a significant issue [22].

The general CI path loss model is given by [4, 23]

where $d$ is the path distance in meters between the transmitter and receiver, $d_{0}$ is the close-in free space reference distance in meters, $f$ is the carrier frequency in Hz, $c$ is the speed of light in m/s, $n_{\text{CI}}$ is the PLE, and the Gaussian random variable $\chi_{\sigma}$ represents the shadow fading with zero-mean and STD $\sigma$ in dB [9, 12, 7, 8, 24, 25]. In (1), the first term on the right-hand side is the free-space path loss at the reference distance $d_{0}$, which is given by

Therefore, the path loss model can be rewritten as

where $f_{c}$ is the carrier frequency in GHz.

Adding the diffraction component to the general CI model, the CI path loss model with diffraction can be expressed as

where $\phi$ is the diffraction component in dB, and the model parameter for diffraction $\alpha$ is the coefficient showing the dependence of path loss on the diffraction.

We use the estimated vector

for CI path loss application. Here, $A=32.4+20\log_{10}f_{c}$, $B=n_{\text{CI}}$, $\mathbf{X_{1}}=[x_{1,1},x_{1,2},\cdots,x_{1,N}]^{\mathrm{T}}=10\log_{10}\mathbf{d}$ for the distance vector $\mathbf{d}=[d_{1},d_{2},\cdots,d_{N}]^{\mathrm{T}}$, $C=\alpha$, and $\mathbf{X_{2}}=[x_{2,1},x_{2,2},\cdots,x_{2,N}]^{\mathrm{T}}=\mathbf{\Phi}$ for the diffraction vector $\mathbf{\Phi}=[\phi_{1},\phi_{2},\cdots,\phi_{N}]^{\mathrm{T}}$. The $n$th element $y_{n}$ of the estimated vector $\mathbf{Y}$ can therefore be written as $y_{n}=A+Bx_{1,n}+Cx_{2,n}$. Based on the CI model, we set $A$ as a constant and estimate $B$ and $C$ by least-square method, which is given by following theorem:

This letter presented the large-scale CI path loss models for the RMa scenario in the central rural area of South Korea based on measurements taken during spring. The measurements were conducted at 1400 and 2250 MHz, and we collected massive data from 3,858 positions (1,262 LOS and 2,596 NLOS) and 4,957 positions (1,427 LOS and 3,530 NLOS) for 1400 and 2250 MHz, respectively. The PLE values of the original CI path loss models at 1400 and 2250 MHz were the same, resulting in 2.16 for LOS and a small difference of 0.11 for NLOS. Compared to the CI path loss models at 73 GHz [4, 13], the PLE values were the same as 2.16 for LOS, and the PLE differences were less than 0.19 for NLOS. We can therefore concluded that, along with the result that the CI model at 73 GHz leads to similar manner with the results in 3GPP [4], the CI path loss model has frequency-independent characteristics beyond the first meter of propagation, and hence we can deduce that our proposed CI models may be used for frequencies up to millimeter-wave [20, 21, 28, 29, 30].

When we extended the CI models to include diffraction loss component without regression, in case of 1400 MHz, the PLE values reduced to 2.24 and 2.08, and furthermore, the STD dramatically decreased to 4.85 and 6.00 for the sKE and DB, respectively. When we included the diffraction loss component in regression, the PLE values changed to 2.28 and 2.22, and the STD further decreased to 4.60 dB and 4.02 dB for sKE and DB, respectively. In case of 2250 MHz, the PLE and STD values show a very similar manner to those of 1400 MHz, which is also shown in Fig. 3 and 4. According to the derived results, it is noted that the PLE values approached 2 when the diffraction component is isolated from the regression of overall path loss. Therefore, we concluded that introducing the diffraction component CI model can dramatically contribute to modeling the path loss more accurately.

The cost function $\mathcal{J}$ is given by the mean-squared error (MSE) between the estimated vector $\mathbf{Y}=[y_{1},y_{2},\cdots,y_{N}]^{\mathrm{T}}$ and the measured data $\mathbf{\widehat{Y}}=[\widehat{y_{1}},\widehat{y_{2}},\cdots,\widehat{y_{N}}]^{\mathrm{T}}$:

Therefore, minimizing $\mathcal{J}$ is equivalent to the least-square problem as

Differentiating the objective function of (18) with respect to $B$ and $C$ and giving the result to zero, we can get

respectively. Since $A$ is a constant and not to be regressed, we can modify (19) into a linear system with respect to $B$ and $C$ by

Hence, by applying matrix inverse to the left-hand side of (20), the least-square solution of (18) is given by

and the theorem follows.  $\blacksquare$