1 Introduction

Recent advancements in mobile technologies have exponentially increased the demand for more radio spectrum bandwidths [1, 2]. The growth of the internet and its enabling technologies, such as the internet of things (IoT) [3,4,5], healthcare management systems [6] and big data networks [7,8,9], have contributed to the surge in demands for higher frequencies that will accommodate broadband communication technologies (IMT-2020) as described in International Telecommunication Union Recommendation—ITU-R M.2083-0 [10]. However, outdoor propagations of millimetre-wave in the tropospheric stratum of the atmospheric layers are extremely sensitive to terrestrial objects and environmental conditions (rain, dust, oxygen molecules, water vapour, vegetation, snow, and fog). Besides the normal free space path loss, the environmental conditions also cause atmospheric gaseous losses. These environmental conditions include water vapour and oxygen molecule absorptions, rain scattering or absorption, foliage, snow, and fog losses [11]. These impeding factors usually cause high signal path loss, signal degradation, signal fading, penetration loss, multipath propagation, and other signal impairments [11, 12].

Various outdoor propagation measurement studies have examined the feasibility of using millimetre-wave to deploy 5G networks at frequencies ranging from 24 to 100 GHz in some developed nations [7, 13]. Ben-Dor et al. [14] used a channel sounder that operates at 38 and 60 GHz with a passband bandwidth of 1.9 GHz to carry out propagation measurements in cellular peer-to-peer outdoor environments and in-vehicle scenarios. Results showed that some antenna orientations can exploit beamforming to create links. NLOS channels provided sub-nanosecond RMS delay spreads and an average path loss exponent of 2.23. Also, the measurements using rotating directional antennas in NLOS antenna pointing scenarios found links with up to 36.6 ns RMS delay spread and an average propagation path loss exponent of 4.19. In contrast, LOS channels provided sub-nanosecond RMS delay spreads and an average path loss exponent of 2.23 (close to free space).

Samimi et al. [15] presented the world’s first empirical measurements for 28 GHz outdoor cellular propagation in New York City. Measurements were made from 400 mega chip-per-second channel sounder and directional horn antennas at three different base station locations and 75 receiver locations with different antenna gains and 36 pointing angles over distances up to 500 m. The results showed several resolvable multipath components in both NLOS and LOS environments, with multipath excess delay spreads as great as 1388.4 ns and 753.5 ns, respectively. Also, the results showed that the path loss exponent varied in different NLOS environments, with a 5.73 overall path loss exponent without the beam steering.

MacCartney et al. [16] observed through a channel sounder system in Manhattan, New York City, that the omnidirectional path loss exponent (n) and the large-scale shadow fading standard deviation (σ dB) were 2.1 and 3.6 dB for LOS condition, while 3.4 and 9.7 dB for NLOS, respectively, at 28 GHz. Similarly, at 73 GHz, the n and σ were 2.0 and 4.8 dB for the LOS condition, while 3.4 and 7.9 dB for NLOS, respectively. Measurements with a spherical scanning 28 GHz channel sounder system in urban street-canyon settings in Daejeon, Korea, and the NYU campus in Manhattan, for three urban scenarios covering urban microcell (UMi) and urban macrocell (UMa) cases, were compared with 3D ray-tracing simulations made for the exact locations [17].

The results from measurements done with the channel sounder system showed that the omnidirectional path loss exponent is 2.66, and the large-scale shadow fading standard deviation is 7.16 dB [18].

Outdoor measurement studies of frequencies from 27 to 40 GHz were carried out at the PUC-RIO campus [19, 20]. The transmitter was positioned on a building’s roof, and 23 receiver points were selected for measurements. The received power level at each point and each frequency was measured. Results indicated that foliage caused weak signals at some points, and lower frequencies had a better coverage range. The study found that most receiver points had a path loss exponent of 2.35 to 2.65 at 27 GHz, and at 40 GHz, the path loss exponent values ranged from 2.20 to 2.55. This suggests that millimetre waves could be a feasible option if the cell radius is kept between 200 and 300 m.

Majed et al. [21] presented propagation path loss, outdoor coverage, and link budget measurements for frequencies above 6 GHz (mm-wave bands) using directional horn antennas at the transmitter and omnidirectional antennas at the receiver. The data showed that unobstructed LOS channels obeyed free space propagation path loss at 28 GHz, 38 GHz, and 60 GHz, while NLOS channels had large multipath delay spreads and could utilise various pointing angles. At 60 GHz, path loss increased, and delay spreads reduced. The mean RMS delay spread varied between 7.2 ns and 74.4 ns for 60 GHz and 28 GHz, respectively. Outdoor studies showed that consistent coverage could be achieved by having base stations with a cell radius of 200 m. Similarly, outdoor channel characterisation and modelling were conducted on 26, 28, 36, and 38 GHz frequency bands in Malaysia, representing a tropical region environment [22]. The results showed that the proposed model was simple and accurate regarding frequency and environment signal attenuation. The path loss exponent values were 1.54 and 3.05 for the 20 and 30 GHz bands, respectively. The study was part of a series of studies aiming to realise a general path loss model for mmWave frequency bands. The measurements were conducted in a specific outdoor environment in Malaysia, which had different construction buildings, providing the signal degradation of 5G channels in such diverse regions.

Outdoor wireless channels’ LOS and NLOS propagation characteristics at 28, 39, 60, and 73 GHz were studied by Muttair et al. [23]. It was seen that LOS had a higher receiving capacity and fewer path losses than NLOS. Additionally, the study revealed that high frequencies were more affected than low frequencies, with increasing path loss and decreasing received power.

Also, several computer-based simulations have been carried out to examine the possibility of using millimetre-wave in 5G network deployment. Li et al. [24] examined the application of a ray-tracing technique with simplified input parameters to predict outdoor channel properties. The results showed that delay profiles and average propagation losses can differ significantly between simulations and wideband measurements. The study also revealed that the deviation of input parameters affects simulation results, with simplified layouts causing significant differences in delay profiles. The paper highlighted the need for more measurements to verify the effectiveness of the ray-tracing technique in predicting propagation properties in practical microcell environments. Guo et al. [25] used a ray-tracing simulation of millimetre-wave at 28 GHz to investigate various path loss fitting techniques that combine antenna patterns and outage thresholds. It was discovered that the close-in-reference model obtained a more robust attenuation exponent than the floating intercept model. Furthermore, its shadow fading deviation is lower than that of the floating intercept method for shorter transmit distances and only slightly more than 2 dB for longer transmit distances. To simulate 3-D statistical channel models for mmWave NLOS communications in Boise City, Idaho, Khatun et al. [26] used 3-D ray tracing. The outcomes included statistics of the power lobe for the angle of arrival profile (AOA), angle of departure (AOD) spectra, power delay profiles, and path loss models for both the 28 and 73 GHz bands. Hassan et al. [27] obtained path loss exponent (n) to be 2.1, 2.0, 2.5 and 2.0 for 28 GHz, 38 GHz, 60 GHz, and 73 GHz, respectively, when a typical Umi scenario with a LOS environment was used in NYUSIM. Da Silva et al. [28] conducted 3D ray-tracing simulations for the Manaira and Jardim Oceania areas of the city of Joao Pessoa, Brazil. Their results showed that the path loss exponent for LOS scenarios was close to the free-space path loss exponent, n = 2.0, with a maximum shadow fading standard deviation of 4.01 dB. On the other hand, the path loss exponent for NLOS scenarios was higher, reaching a maximum value of n = 3.58 and a shadow fading standard deviation of 21.99 dB. Ullah et al. [29] presented a radio propagation algorithm for 5G mmWave in a populated city, simulating a smart 3D ray tracing algorithm in MATLAB. The research considered a 28 GHz operating frequency and showed moderate outdoor-to-indoor service. The model deals with radio propagation using efficient resource use and considers factors like transmitter–receiver separation, building structure, and RF wave obstructions.

The development, deployment, and reliable performance of cellular communications systems depend on the correct comprehension of the characteristics of radio signals when propagating through a different range of radio frequency channels. Therefore, it is imperative to perform comprehensive on-field measurement studies or computer simulations [30, 31] to understand the characteristics of outdoor propagation of millimetre-wave at 28, 38, 60, and 73 GHz in the West African region [32,33,34]. The essential channel parameters include the received signal strength (RSS), largescale path loss exponent, large-scale shadow factor (Xσ), large-scale shadow fading standard deviation, angular spread, power delay profile, angle of arrival profile and angle of departure, decay factor, delay spread, signal large-scale path loss, and the maximum radio coverage radius. These parameters are necessary for the design of a millimetre-wave cellular network and its equipment that will adapt across various global environmental conditions and physical infrastructures [35].

It is common to use ray-tracing simulation to model physical channels, as it provides directionally resolved spatial characteristics and allows for the investigation of wireless channel propagation mechanisms such as diffraction and reflection. Most theoretical and empirical models may not accurately represent specific characteristics of mmWave channel that can affect the system's overall performance. These characteristics include the evolution of the channel’s MultiPath Components (MPCs), each of which represents a distinct planar wavefront propagating between the Transmitter (TX) and the Receiver (RX) [36, 37].

Although conducting a real-world measurement campaign remains the most effective way of modelling the millimetre-wave channel parameters, simulation results from various studies have shown good agreement with the measured data [38]. To predict the ray paths from transmitters to receivers, ray tracing models utilise numerical simulations for a specific 3-D environment. These models use multipath to determine the angles and times of arrival from the multipath and also calculate the path loss and phase change for each ray.

Ray-tracing simulations are a powerful tool for studying the propagation characteristics of mmWaves. Such simulations take into account specific atmospheric conditions, terrain, building materials, structures, and geographical layouts of each site. In this research, ray-tracing simulations were conducted to generate data for path loss modelling of outdoor propagation of mmWaves at 28, 38, 60, and 73 GHz for 5G mobile systems in four Nigerian cities, namely Abuja, Lagos, Ibadan, and Port Harcourt. The study aims to formulate three empirical large-scale path loss models: the close-in free-space reference distance, the floating intercept and the alpha–beta-gamma, and afterwards compare their performances. The simulations will be performed for both LOS and NLOS cases with a receiver threshold (sensitivity) of − 100 dBm at each frequency [39].

The necessity of this study is underscored by the rapid technological advancements and increasing demand for high-frequency spectrum bandwidths, which require precise and reliable models for network planning and implementation. Therefore, the current study makes significant contributions to the field of 5G mobile systems by providing detailed path loss models specifically tailored to the sub-Saharan tropical environment, which has been largely underexplored in existing literature. By addressing the unique atmospheric conditions, terrain, building materials, and urban layouts of four major Nigerian cities, the research fills a critical gap in understanding how millimetre-wave signals propagate in these regions. This is particularly essential for the effective deployment and optimisation of 5G networks in similar tropical environments globally. Moreover, this research lays a foundational framework for future studies, providing a benchmark for further empirical investigations and simulations in similar climatic and geographical contexts. By modelling the path loss of mmWaves outdoor propagation, the study contributes to the advancement of wireless communication technologies and paves the way for innovative applications and services that rely on high-speed, low-latency networks.

The research paper is organised as follows: Section II discusses the ray tracing model, while Sections III and IV provide the composite raytracing-image-method setup and mmWaves omnidirectional path loss modelling, respectively. In Section V, the performance metrics are presented, followed by the results analysis in Section VI. Finally, Section VII presents the conclusions of this research work.

2 The ray tracing model

2.1 Composite raytracing-image-method propagation model

The image method is a ray tracing technique used in wireless communications to calculate propagation paths. It works by locating the image of the transmitter (Tx) relative to a planar reflection surface, creating an image transmitter (Tx'). This image is then connected to the receiver (Rx) with a line segment. A valid propagation path from Tx to Rx is established if this segment intersects the reflection surface. The method can handle paths with multiple reflections by recursively creating higher-order image sources and tracing rays from each to the receiver. This technique calculates an exact number of propagation paths with precise geometric accuracy, accounting for reflections but not diffraction, refraction, or diffuse scattering effects. For each path, it supports up to two reflections [35]. However, the computational complexity increases exponentially with the number of reflections, making it less efficient for complex geometries or scenarios requiring numerous reflections. Despite this limitation, the image method remains valuable for its accuracy in modelling high-frequency wave propagation in simpler environments.

The Composite Raytracing-Image-Method Propagation Model was adopted from the ray-tracing model used in [39, 40]. It employs the method of images, including surface reflections only (with a maximum of two reflections) and does not account for effects from refraction, diffraction, or scattering. This model is valid for a frequency range of 100 MHz to 100 GHz. The model calculates total path loss by summing free space loss (PL0), interaction loss (PLIL), and atmospheric loss (PLAT) together [39]. The free space path loss is determined using a specific formula involving the transmit wavelength and free space path distance. The interaction loss, incorporating polarization and reflection loss, is calculated by transforming electromagnetic fields into local coordinates, applying reflection coefficients, and then converting them back to global coordinates [39, 40]. Meanwhile, the atmospheric loss is derived from gaseous and rain attenuations [41,42,43].

The composite raytracing-image-method path loss, PLCRTI is formulated as [39]:

$$PL_{CRTI} = \,PL_{0} \, + \,PL_{IL} \, + \,PL_{AT} \,\left( {{\text{dB}}} \right)$$
(1)

where the free space path loss, PL0, is

$$P{L}_{0}=10{log}_{10}{\left(\frac{4\pi {d}_{0}}{\lambda } \right)}^{2}$$
(2)

\(\lambda\) represents the transmit frequency’s wavelength and \({d}_{0}\), is the free space path distance.

The interaction loss, denoted as PLIL, is commonly known as polarisation plus reflection loss:

$${PL}_{IL}=-20{\text{log}}_{10}\left|{J}_{rx}^{-1}R{J}_{tx}\right|$$
(3)

To calculate polarisation and reflection loss, the model employs the following steps:

1. By computing the propagation matrix, P, it tracks the movement of a ray in three dimensions. The matrix is a recursive product where i represents the number of reflection points.

$$P={\prod }_{i}{P}_{i}$$
(4)

This method involves converting the global coordinates of the incident electromagnetic field into the local coordinates of the reflection plane. The resulting values are then multiplied by a reflection coefficient matrix. Finally, the coordinates are transformed back into their original global coordinate system to determine Pi for each reflection [39, 44]. The equations for Pi and Po are:

$${P}_{i}={[ {\text{s}}_{out} {\text{p}}_{out} {\text{k}}_{out} ]}_{i}{\left[\begin{array}{ccc} {R}_{V(\alpha )}& 0& 0\\ 0& {R}_{H(\alpha )}& 0\\ 0& 0& 1\end{array}\right]}_{i}{[\begin{array}{ccc}{\text{s}}_{in}& {\text{p}}_{in}& {\text{k}}_{in}\end{array}]}_{i}^{-1}$$
(5)
$${P}_{o}=\left[\begin{array}{ccc} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$
(6)

where:

  • The plane of incidence is built on the variables s, p, and k. The s and p are orthogonal and parallel, respectively, to the plane of incidence.

  • \({\text{k}}_{in}\) and \({\text{k}}_{out}\)= incident and exiting rays’ directions in global coordinates, respectively.

  • \({\text{s}}_{in}\) and \({\text{s}}_{out}\)= the incident and exiting rays’ horizontal polarization directions in global coordinates, respectively.

  • \({\text{p}}_{in}\) and \({\text{p}}_{out}\) = horizontal polarizations for the incident and exiting rays, respectively, in global coordinate directions.

  • RH and RV = Fresnel reflection coefficients for the two polarisations, horizontal and vertical, respectively.

  • The incident angle of the ray is represented by α, and the complex relative permittivity of the material is denoted by εr..

The type of material on a surface can affect the amount of reflection loss when a ray interacts with it. By considering the complex relative permittivity of both the building and surface materials, a ray-tracing model can be used to estimate propagation losses. To calculate the relative permittivity (ϵr) for different frequencies, ITU-R P.527–5 [45] and ITU-R P.2040–1 [46] recommendations provide formulas, values, and procedures. The equations used to calculate ϵr are included in these recommendations as follows:

$$\varepsilon_{{\text{r}}} = \varepsilon^{\prime}_{r} + {\text{j}}\varepsilon^{\prime\prime}_{r}$$
(7)
$$\varepsilon^{\prime\prime}_{r} = \frac{{\upsigma }}{{2{\uppi }\varepsilon_{0} { }f}}$$
(8)

where:

\({\upvarepsilon }_{r}^{\prime}=\) real relative permittivity.

\(\upsigma =\) conductivity in S/m.

\({\upvarepsilon }_{0}=\) permittivity of free space (electric constant).

\(f=\) frequency in Hz.

The ray tracing model determines \({\upvarepsilon }_{r}^{\prime}\) and \(\upsigma\) for building materials as follows:

$$\varepsilon^{\prime}_{r} = {\text{a }}f^{ b}$$
(9)
$$\sigma = {\text{c}} f^{d}$$
(10)

where a, b, c, and d, are constants determined by the surface material.

$$R_{{H\left( {\alpha { }} \right)}} = \frac{{\cos \left( {\alpha { }} \right) - { }\sqrt {[\varepsilon_{{\text{r}}} { } - {\text{ sin}}^{2} \left( \alpha \right)]/\varepsilon_{{\text{r}}}^{2} } }}{{\cos \left( {\alpha { }} \right) + { }\sqrt {[\varepsilon_{{\text{r}}} { } - {\text{ sin}}^{2} \left( \alpha \right)]/\varepsilon_{{\text{r}}}^{2} } }}$$
(11)
$$R_{{V\left( {\alpha { }} \right)}} = \frac{{\cos \left( {\alpha { }} \right) - { }\sqrt {\varepsilon_{{\text{r}}} { } - {\text{ sin}}^{2} \left( \alpha \right)} }}{{\cos \left( {\alpha { }} \right) + { }\sqrt {\varepsilon_{{\text{r}}} { } - {\text{ sin}}^{2} \left( \alpha \right)} }}$$
(12)

A 2-by-2 polarization matrix R is created by projecting the propagation matrix P into it. The model also rotates the transmitter and receiver’s coordinate systems to place them in global coordinates.

$$\text{R}=\left[\begin{array}{cc}{\text{H}}_{\text{i}n} .{\text{H}}_{\text{r}x} & {\text{V}}_{\text{i}n} .{\text{H}}_{\text{r}x}\\ {\text{H}}_{\text{i}n} .{\text{V}}_{\text{r}x}& {\text{V}}_{\text{i}n} .{\text{V}}_{\text{r}x}\end{array}\right]$$
(13)
$${\text{H}}_{\text{i}n}=\text{P}({\text{V}}_{\text{t}x} \times {\text{ K}}_{\text{t}x})$$
(14)
$${\text{V}}_{\text{i}n}={\text{PV}}_{\text{t}x}$$
(15)

where:

\({\text{H}}_{\text{r}x}\) & \({\text{V}}_{\text{r}x}\) represent the receiver’s directions of the horizontal (Eθ) and vertical (Eϕ) polarisations in global coordinates, respectively.

\({H}_{in}\) & \({V}_{in}\) represent the propagated horizontal and vertical polarisations’ directions in global coordinates, respectively.

\({V}_{tx}\) represents the transmitter’s nominal vertical polarisation’s direction for the departing ray in global coordinates.

\({K}_{tx}\) represents the direction of the transmitter’s departing ray in global coordinates.

It specifies the normalised horizontal and vertical polarisations of the electric field at the transmitter and receiver by using the 2-by-1 Jones polarisation vectors \({J}_{tx}\) and \({J}_{rx}\), respectively. If either the transmitter or the receiver is unpolarised, then the model assumes:

$${\text{J}}_{\text{t}x}={\text{J}}_{\text{r}x}=\frac{\sqrt{2}}{2}\left[\begin{array}{c}1\\ 1\end{array}\right]$$
(16)

The atmospheric loss, \(P{L}_{AT}\) is derived from losses due to atmospheric gas and rain droplets:

$$P{L}_{AT}={\gamma }_{g}+ {\gamma }_{r}$$
(17)

the specific gaseous attenuation [47], \({\upgamma }_{g}\) is given by:

$${\upgamma }_{g}= {\upgamma }_{0}+{\gamma }_{w}=0.1820f\left({N}_{ Oxygen}^{{\prime}{\prime}}\left(f\right)+{N}_{ Water\, Vapour}^{{\prime}{\prime}}\left(f\right)\right)$$
(18)

where, \({\upgamma }_{0}\) (dB/km) is the specific attenuations due to dry air (oxygen, pressure-induced nitrogen, and non-resonant Debye attenuation), \({\upgamma }_{w}\) (dB/km) is the attenuation due to water vapour, \(f\) (GHz) is the radio frequency, \({N^{{\prime}{\prime}}}_{Oxygen}(f)\) and \({N^{{\prime}{\prime}}}_{Water\,Vapour}(f)\) are the imaginary parts of the frequency-dependent complex radio refractivity.

The specific rain attenuation, \({\upgamma }_{r}\) (dB/km), is given by [48,49,50]:

$${\upgamma }_{r}=k{R}^{\alpha }$$
(19)

where \(k\) and α are the frequency-dependent coefficients for linear polarisations (horizontal – H, vertical – V) and horizontal paths, respectively, and R is the rain rate (mm/h).

3 Composite ray tracing-image-method setup

The composite raytracing-image-method (CRTI) considers the site-specific atmospheric conditions, terrain and building materials, building structures, and the city geographical layouts of each site during simulations. Studies have shown that the outdoor propagation of mmWaves is mainly due to direct path and, single and two-time reflections [51, 52]. Also, Khawaja et al. [52] confirmed that, for the mmWaves propagation, the third and higher-order reflections would have a negligible contribution to the overall received power. Moreover, considering higher-order reflections increases the complexity of the model unnecessarily compared to their contribution to the received power [52]. Therefore, to avoid high computational complexity due to limited resources and time, only the direct path and reflected paths with a maximum of two reflections are considered in this configuration. However, it has also been established that refraction, scattering, and diffraction in outdoor propagation of mmWave bands contribute minimally to the overall path loss [41, 53].

For mmWaves, the base station antenna is usually located well below the building skyline; hence, it is unlikely that diffraction will happen across building rooftops. The short wavelengths of mmWave signals result in low diffraction [42]. According to Samimi [34], the impact of diffraction varies with the carrier frequency, being more pronounced at lower frequencies (below 6 GHz) and diminishing as frequencies rise. Also, based on research work on mmWaves by Thomas et al. [43], reflection and scattering dominate over diffraction. Furthermore, Senic et al. [41] reported that the prevailing assumption is that diffraction plays a lesser role in the mmWave region and that in NLOS, the receiver will rely instead on ambient reflected paths.

For effective transmission and reception of mmWaves for 5G mobile networks, the channel and antenna parameters for UMi scenario were selected based on ITU-R M.2412 [54] and 3GPP 5G NR recommendations [55] as displayed in Table 1. Typical 3-sector transmission site with 16 by 16 uniform rectangular array antenna per sector and receive site (UE) with 4 by 4 uniform rectangular array antenna were used. The transmit power used was 44 dBm with both transmit and receive antennas’ HPBW of Azimuth = 10◦ and Elevation = 10◦. Also, the 2021 annual average meteorological parameters used for Abuja City, Ibadan City, Lagos Island, and Port Harcourt City, as well as the electrical characteristics of surface materials, are shown in Tables 2 and 3, respectively. The meteorological parameters and electrical characteristics of the surface materials were used as input variables in the MATLAB codes accordingly.

Table 1 Channel and antenna parameters for urban microcell scenario
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Table 2 Meteorological parameters [56,57,58]
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Table 3 Electrical characteristics of surface materials [54]
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MATLAB application software, with the accessible online real-time 3D building and terrain data of Abuja City (Geo. 9.07◦ N, 7.49◦ E), Lagos Island (Geo. 6.46◦ N, 3.39◦ E), Ibadan City (Geo. 7.45◦ N, 3.89◦ E), and Port Harcourt City (Geo. 4.78◦ N, 7.01◦ E), from OpenStreetMap website, [59] was used for the simulations as shown in Figs. 1, 2, 3 and 4.

Fig. 1
figure 1

Three-dimensional (3D) maps of Abuja a buildings and b rays from the Tx site to Rx sites (UEs), where the red balloon corresponds to the Tx location and blue balloons correspond to Rx locations

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Fig. 2
figure 2

Three-dimensional (3D) maps of Lagos Island a buildings and b rays from the Tx site to Rx sites (UEs), where the red balloon corresponds to the Tx location and blue balloons correspond to Rx locations

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Fig. 3
figure 3

Three-dimensional (3D) maps of the University of Ibadan a buildings and b rays from the Tx site to Rx sites (UEs), where the red balloon corresponds to Tx location and blue balloons correspond to Rx locations

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Fig. 4
figure 4

Three-dimensional (3D) maps of Port Harcourt a buildings and b rays from the Tx site to Rx sites (UEs), where the red balloon corresponds to the Tx location and blue balloons correspond to Rx locations

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In each simulation, a fixed transmitter site (TX) and 200 to 350 randomly generated receiver sites (RXs) with a distance range of 10 m to 400 m were used for all four frequencies (28, 38, 60, and 73 GHz). The simulation was conducted multiple times per location using TX azimuth angles rotated from 0◦ to 360◦ and elevation angles from − 45◦ to + 45◦ with a stepsize of 10◦ to produce omnidirectional path loss data from all the possible angles for 2800, 3050, 3100, and 2800 randomly generated receiver sites in Abuja City (ABV), Lagos Island (LAG), Ibadan City (IBA), and Port Harcourt City (PHC), respectively. The received power values for all the various TX and RX azimuth and elevation angle combinations (with the antenna gains already factored out) were summed together in linear scale (milliwatts). This summation yielded an omnidirectional received power figure, which was then used to calculate the omnidirectional path loss data. The choice of these four locations was based on the teledensity, topographical [60], geographical [50], and climatic factors [61] of the study sites.

4 mmwaves omnidirectional path loss modeling

Many empirical models have been suggested to predict path loss. Path loss modelling entails calculating all attenuation losses, which are typically expressed in a logarithmic formula and factor in both distance-dependent attenuation and losses from all signal propagation mechanisms [28, 62, 63]. The close-in free space reference distance model [16, 54], the floating-intercept model [64], and the alpha–beta-gamma model [65, 66] are the three most typical large-scale path loss models that can be generated from the simulation results.

4.1 The close-in free-space reference distance large-scale path loss model

The close-in free-space reference distance large-scale path loss model [67] is a single slope model with physical relevance that is tied to the free-space loss at a specific reference distance, typically, 1 m for mmWave applications. Close-in free space reference distance large-scale path loss, (PLCI) is given as:

$${PL}^{CI} \left(f, d\right)\left[dB\right]=P{L}_{0}+10n{log}_{10}\left(\frac{d}{{d}_{0}}\right)+{\upchi }_{\sigma }^{CI}$$
(20)

where the free space path loss, \(P{L}_{0}\), is

$$P{L}_{0}=10{log}_{10}{(\frac{4\pi {d}_{0}}{\lambda } )}^{2}$$
(21)

\(d\) represents the distance between the transmit antenna and the receive antenna, do is the close-in free space reference distance of 1 m, \(\lambda\) represents the transmit frequency’s wavelength, n denotes the single model parameter, the path loss exponent (PLE) for a particular frequency band or environment, and \({\upchi }_{\sigma }^{cl}\) is the large-scale shadow fading with a standard deviation,\({\upsigma }^{\text{CI}}\).

4.2 The floating intercept path loss model

Floating Intercept large-scale path loss,\({PL}^{FI}\) is given as [68, 69]:

$$PL^{FI} \left( {\text{d}} \right)\left[ {{\text{dB}}} \right] = \alpha^{FI} + 10\beta^{FI} log_{10} \left( {\text{d}} \right) + \chi_{\sigma }^{FI}$$
(22)

where \(d\) is the distance between the transmit antenna and the receive antenna, \({\alpha }^{FI}\) denotes the floating intercept in dB, \({\upbeta }^{FI}\) represents a coefficient known as line slope (path loss exponent) that characterises the distance dependency of the path loss, \({\upchi }_{\sigma }^{FI}\) is a zero mean Gaussian random variable that stands for shadow factor, and \({\sigma }^{FI}\) denotes the shadow factor standard deviation in decibels.

4.3 The alpha–beta-gamma path loss model

The ABG reference distance and frequency large-scale path loss, \({PL}^{ABG}\) is given as:

$$PL^{ABG} \,(f,\,d)\,)\left[ {{\text{dB}}} \right]\, = \,10\,{\upalpha }^{ABG} log_{10} \,\left( {\frac{d}{1\, m}} \right)\, + \,{\upbeta }^{ABG} \, + \,10{\upgamma }^{ABG} {\text{log1}}0\,\left( {\frac{f}{1\, GHz}} \right)\, + \,{\upchi }_{\sigma }^{ABG}$$
(23)

where \(\alpha^{ABG}\) and \({\upgamma }^{ABG}\) are the coefficients displaying the dependence of path loss on distance and frequency, respectively. \({\upbeta }^{ABG}\) represents the optimised offset value for path loss in dB, d is the 3D transmitter–receiver (T-R) separation distance in meters, f denotes the carrier frequency in GHz, and \({\upchi }_{\sigma }^{ABG}\) is the Shadow Fading with standard deviation, \({\sigma }^{ABG}\) describes large-scale signal fluctuations about the mean path loss over distance.

5 Performance metrics

The path loss models’ accuracy was evaluated using performance matrices such as mean absolute error (MAE), mean square error (MSE), and root mean square error (RMSE).

$$MAE= \frac{1}{N}{\sum }_{k=0}^{N}(P{L}_{S}-P{L}_{P})$$
(24)
$$MSE= \frac{1}{N}{\sum }_{k=0}^{N}{\left(P{L}_{S}+P{L}_{P}\right)}^{2}$$
(25)
$$RMSE=\sqrt{\frac{1}{N}{\sum }_{k=0}^{N}{\left(P{L}_{S}+P{L}_{P}\right)}^{2}}$$
(26)

where N denotes the total number of samples, \(P{L}_{S}\) is the simulated path loss in dB, and \(P{L}_{P}\), is the predicted path loss in dB.

The simulated path loss values were obtained from the Composite Raytracing-Image-Method Propagation Model’s simulations, while the predicted path loss values were mathematically derived from the three empirical models (CI, FI and ABG), respectively. The simulated path loss serves as the reference point in this study, highlighting the model's accuracy and any deviations observed under varying scenarios, frequencies and locations.

6 Results and discussion

The mathematical derivations of the average values of n, αFI, βFI, αABG, βABG, and γABG parameters in the CI, FI, and ABG omnidirectional path loss models were computed in Table 4., utilising closed-form solutions that minimise the SF standard deviations (σCI, σFI, σABG) through the RMSE fit on all of the combined path loss data from the simulation distances at 28, 38, 60, and 73 GHz, respectively, as defined by earlier studies [64, 68, 69].

Table 4 Omnidirectional channel parameters
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According to Table 4., Port Harcourt City has the highest average n values of 3.554, 3.689, and 3.739 for LOS conditions at 28, 38, and 73 GHz, respectively. These values differ from the standard free space path loss exponent of 2 [28]. On the other hand, Lagos Island has the highest average n values of 3.819 for LOS conditions at 60 GHz, 4.026 and 4.268 for NLOS conditions at 28 and 38 GHz, respectively. However, Ibadan City has the highest average n values of 5.033 and 4.659 for NLOS conditions at 60 and 73 GHz, respectively. The results show that under the same channel conditions, the omnidirectional path loss increases marginally as the frequency increases, except for 60 GHz, which has a higher path loss per distance than 73 GHz. This is due to additional signal attenuation caused by oxygen absorption at 60 GHz.

For the LOS scenarios in the four cities, the n values range from 2.712 to 3.819. Also, the σCI values range from 11.778 to 37.199, while in the NLOS scenarios, the n values range from 3.375 to 5.033 and the σCI values from 12.441 to 44.181 at the four frequencies.

From Table 4., it can be observed that at the four frequencies for both LOS and NLOS conditions in the four cities, the αFI values range from -14.976 to 50.998 and -0.441 to 114.277, respectively. Also, in the LOS scenarios in the four cities, the βFI values range from 3.519 to 7.961, and σFI values range from 10.566 to 35.129 at the four frequencies, while in the NLOS scenarios, the βFIvalues range from 1.653 to 8.250 and σFI values range from 12.119 to 42.732.

Table 4. also shows that αABG values at the four frequencies for both LOS and NLOS circumstances in the four cities are the same as values of βFI as discussed above. For LOS scenarios at the four frequencies, the values of βABG, γABG and σFI span -103.250 to 73.754, -4.771 to 7.029, and 11.474 to 35.158, respectively. However, the values of βABG, γABG and σABG for the NLOS scenarios span from -9.080 to 143.842, -3.167 to 5.198, and 12.119 to 42.738, respectively.

It can be observed that the values of the three path loss models differ significantly between cities and between the LOS and NLOS scenarios. This is due to factors such as terrain, weather conditions, and the presence of buildings and other obstacles, which can affect the propagation of radio signals. Also, it is worth noting that for some cities, the ABG path loss model has negative values for γABG, which is not physically possible, indicating that the model is not the best fit for all scenarios and frequencies in these cities.

Figures 5, 6, 7 and 8 (in Appendix I) show the omnidirectional path losses at each frequency per location for the CI, FI, and ABG models in the UMi street canyon (SC) environments for both LOS and NLOS conditions, respectively.

Fig. 5
figure 5figure 5

Omnidirectional Path Loss for Abuja City (ABV) at a 28 GHz, b 38 GHz, c 60 GHz, and d 73 GHz

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According to Table 5., Guo et al. [25] provided results for LOS and NLOS scenarios at 28 GHz. In LOS, they reported a lower path loss exponent (2) compared to this study, which ranged from 2.789 to 3.554. However, their NLOS path loss exponent (5.2) is higher than that of this study (ranging from 3.375 to 4.026). This suggests that Guo et al. [25] observed stronger signal attenuation in NLOS conditions compared to this study. In another study, Hassan et al. [27] only provided LOS results at different frequencies (28, 38, 60, and 73 GHz), with path loss exponents ranging from 2 to 2.5. Their results are all less than the lower end of the range observed in this study. Hindia et al. [22] reported LOS results at 28 and 38 GHz with path loss exponents of 2.72 and 3.12, respectively. These values are within the range reported by this study, but they lack data for NLOS environments. Xing and Rappaport [70] provided results for both LOS and NLOS scenarios at 28, 38 and 73 GHz. Their path loss exponents (for LOS: 2.1, 1.9 & 1.9, and NLOS: 3.4, 2.7 & 2.8, respectively) are generally lower than this study, indicating less signal attenuation in both LOS and NLOS conditions. Sun et al. [62] documented path loss exponent values of 2.1 for LOS scenarios and 3.4 for NLOS scenarios at 28 GHz. At 73 GHz, LOS scenarios exhibited a path loss exponent of 2, while NLOS scenarios maintained a value of 3.4. In contrast, this investigation unveils a dynamic spectrum of path loss exponent values across various frequency bands and scenarios. For instance, at 28 GHz, the path loss exponent ranges from 2.789 to 3.739 for LOS and 3.375 to 4.659 for NLOS scenarios. Similarly, at 73 GHz, the path loss exponent varies from 2.955 to 3.739 for LOS and 3.61 to 4.659 for NLOS scenarios.

Table 5 Comparative analysis with relevant studies on path loss modelling parameters
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However, further findings from Sun et al. [62] reveal αFI values of 31.8 for LOS scenarios and 80.6 for NLOS scenarios at 28 GHz. At 73 GHz, αFI values are 115.6 for LOS and 80.6 for NLOS scenarios. Similarly, the βFI values are 3.9 for LOS scenarios and 2.5 for NLOS scenarios at 28 GHz, and -0.8 for LOS scenarios and 2.9 for NLOS scenarios at 73 GHz. These results align closely with the outcomes of this study, where αFI values range from -13.253 to 45.058 for LOS scenarios and from 15.962 to 97.378 for NLOS scenarios at 28 GHz. At 73 GHz, αFI values range from -6.821 to 50.998 for LOS scenarios and 20.348 to 114.277 for NLOS scenarios. Similarly, βFI values range from 3.519 to 7.34 for LOS scenarios and from 1.653 to 6.265 for NLOS scenarios at 28 GHz, while spanning -4.107 to 7.618 for LOS scenarios and 2.373 to 6.659 for NLOS scenarios at 73 GHz.

Sun et al. [64] presented findings encompassing both LOS and NLOS scenarios spanning 2 to 73.5 GHz. Their reported path loss exponents of 2 for LOS and 3.1 for NLOS scenarios generally exhibit lower values than the results observed in this study. A trend of reduced signal attenuation in both LOS and NLOS conditions was observed in their findings. Furthermore, Sun et al. [64] reported αABG values of 2 for LOS scenarios and 3.5 for NLOS scenarios, alongside fixed values for βABG (31.4 for LOS and 24.4 for NLOS) and γABG (2.1 for LOS and 1.9 for NLOS). Notably, these values fall within the range reported in this study.

Overall, there is variability among the results presented by different authors. However, there is a consistent trend of higher path loss in NLOS environments compared to LOS, which tends to increase with frequency. These differences highlight the importance of considering various factors and conducting thorough measurements when designing mmWave communication systems.

For each path loss model, frequency, location, and scenario, Table 6. provides the Mean Absolute Error and Root Mean Square Error as compared to the simulated path loss results. The lower the values of these metrics, the better the performance of the path loss models. The performance of the path loss models varies across various cities, frequencies, scenarios, and evaluation metrics, as the table demonstrates. For instance, at 38 GHz in Abuja, the ABG Path Loss Model performs best for the NLOS scenario, whereas the CI Path Loss Model performs better for the LOS scenario. The ABG Path Loss Model, on the other hand, performs best in Lagos Island at 28 GHz for both LOS and NLOS scenarios.

Table 6 Performance metrics of the models
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7 Conclusion

This research utilised three omnidirectional large-scale path loss models, which are Close-In Free-Space Reference Distance, Floating Intercept, and Alpha–Beta-Gamma, to simulate the performance of a 5G network across four different cities in Nigeria. The study examines millimetre-wave frequencies at 28, 38, 60, and 73 GHz using channel parameters and outdoor millimetre-wave propagation antenna properties for UMi scenarios.

The results show that the path loss exponent and standard deviation differ among cities, frequencies, and scenarios. Line-of-sight conditions generally have lower path loss exponent and standard deviation values than non-line-of-sight ones. The ABG model exhibits negative values for the frequency-dependent path loss exponent in some cases, which is not possible in practice. The implication is that the model is not suitable for those scenarios. Path loss increases as the frequency increases, except for 60 GHz, which has a higher path loss per distance than 73 GHz due to additional signal attenuation caused by oxygen absorption at 60 GHz.

In all the cities and conditions, the path loss values increase with a corresponding increase in frequency, indicating that higher frequencies experience greater path losses compared to lower frequencies. The four cities chosen for this study—Abuja, Lagos, Ibadan, and Port Harcourt—were selected due to their distinct topographical and atmospheric conditions, which are representative of the varied environments found in Nigeria. Abuja, the capital city, features a mix of urban and suburban landscapes with significant elevation changes. As the most populous city and a major economic hub, Lagos presents a dense urban environment with a complex mix of high-rise buildings and coastal conditions. Ibadan, with its sprawling and less densely populated area, offers a different urban setting, while Port Harcourt, a complete coastal area and an important industrial and oil-producing city, combines industrial and residential zones with high humidity and heavy rainfall.

Furthermore, significant impacts on the results were noted due to the environmental conditions and structural characteristics of these cities. For example, Port Harcourt exhibited the highest path loss values across all frequencies and scenarios, attributed to its suburban environment, high humidity, heavy rainfall, and coastal conditions. Lagos followed closely with substantial path loss values due to its coastal conditions, dense urban environment, and abundance of high-rise buildings, contributing to increased signal reflection and absorption. In contrast, the varied elevations in Abuja and Ibadan were observed to influence signal propagation differently, affecting both path loss and fading characteristics much less than in the other two cities.

According to the findings of this research, the ABG path loss model consistently outperformed the other two models in terms of mean absolute error and root mean square error values across the four frequencies and the two scenarios. Hence ABG model is recommended for sub-Saharan tropical environments.

The study was conducted at four different frequencies—28 GHz, 38 GHz, 60 GHz, and 73 GHz. These specific frequency bands were chosen due to their relevance to current and near-future 5G deployments and their varying propagation characteristics. Lower mmWave bands, like 28 GHz and 38 GHz, are being widely adopted for initial 5G deployments due to their balance between capacity and coverage. The higher bands, such as 60 GHz and 73 GHz, are significant for ultra-high-speed communication but are more susceptible to atmospheric absorption and blockage by obstacles. From the authors' perspective, these results could potentially be scaled up to 100 GHz. Although higher frequencies tend to have higher path loss and are more susceptible to environmental factors, the fundamental propagation mechanisms remain similar. By understanding the characteristics of signals at 60 GHz and 73 GHz, which already exhibit significant challenges in terms of attenuation and absorption, it is plausible to extend these findings to 100 GHz with appropriate adjustments [7, 10, 13]. This would require further empirical validation but provides a promising foundation for future research in even higher frequency bands.

These measurements and models can be used in designing and analysing millimetrewave communication systems in these four cities and similar environments, specifically to predict the path loss of radio waves under LOS and NLOS conditions. The study provides baseline data for semi-urban and urban cities at 28, 38, 60 and 73 GHz to develop or validate propagation models and outage analyses, particularly in sub-Saharan Africa and similar tropical regions.