Propagation Models | Teleinf Edu

Wireless Propagation and Path Loss

An antenna launches a modulated electromagnetic wave into the propagation environment. After leaving the transmitting antenna, the wave interacts with the ground, buildings, vegetation, vehicles, and the atmosphere. A wireless channel therefore has a path that varies with location, frequency, antenna height, and time, unlike the guided path inside a copper or fibre-optic cable.

A signal normally becomes weaker as the separation from the transmitter grows. The reduction between transmitted and received power for a defined link is called or attenuation. The received signal must exceed noise and interference by the margin required by the modulation and coding scheme. Estimating path loss between a transmitter and receiver is therefore a basic input to coverage, link-budget, and interference calculations.

estimate received field strength or transmission loss from distance, frequency, antenna heights, terrain, building geometry, and measured statistics. Planners use them to estimate the number of cell sites for a city, place Wi-Fi access points inside an office, determine a satellite link margin, and select transmitter power. A link budget records the transmit power, antenna gains, feeder losses, propagation loss, receiver sensitivity, and required margin for one link.

The Ideal Case: The Free Space Propagation Model

The simplest reference case is free space. The treats the path as ideal, unbounded, homogeneous, and lossless, with no ground, buildings, or other objects to reflect or block the wave. It is an unobstructed line-of-sight (LOS) reference. The assumption is useful for satellite-to-satellite and deep-space links, while a terrestrial link normally needs additional propagation terms.

The Spreading of Energy

In free space, the only source of signal loss is the natural spreading of the wave. A lightbulb in a large, dark space illustrates this relationship. As the distance increases, the light becomes dimmer because the same amount of energy spreads over the surface of an ever-expanding sphere rather than being absorbed. The power density (power per unit area) decreases as the surface area of the sphere ( $4 \pi d^2$ ) increases. This relationship explains free-space path loss.

The Friis Transmission Equation

The Friis transmission equation describes the received power for a free-space point-to-point link. For isotropic antennas, it gives the free-space basic transmission loss $(L_{fs})$ as a function of antenna separation and wavelength, or equivalently frequency. Antenna gains are included separately in a practical link budget.

Written as a power ratio, the expression tells us how much smaller the received power is than the transmitted power when both antennas are treated as isotropic:

L_{fs} = \left(\frac{4 \pi d f}{c}\right)^2

where $d$ is distance, $f$ is frequency, and $c$ is the speed of light.

In engineering, it is more convenient to work with a logarithmic scale called decibels (dB). The formula for free space path loss in dB is:

L_{fs} \text{ (dB)} = 20 \log_{10}(d) + 20 \log_{10}(f) + 20 \log_{10}\left(\frac{4\pi}{c}\right)

The numerical form depends on the units. ITU-R P.525-5 gives $L_{bf} \text{ (dB)} = 32.4 + 20 \log_{10}(f) + 20 \log_{10}(d)$ when $f$ is in MHz and $d$ is in km. This basic transmission loss assumes a free-space path between isotropic antennas. Feeder, mismatch, polarization, atmospheric, and antenna gain terms are entered separately in a practical link budget.

The formula has three useful parts:

Dependence on Distance ( $d$ ): The loss increases by $20 \log_{10}(d)$ . This is a logarithmic expression of a square-law relationship. This means that if you double the distance, the path loss increases by $20 \log_{10}(2)$ , which is approximately $6 \text{ dB}$ . A 6 dB increase in loss means the received power is four times weaker. This "inverse square law" is a fundamental characteristic of free space propagation.
Dependence on Frequency ( $f$ ): The loss also increases by $20 \log_{10}(f)$ . This means that if you keep the distance the same but double the frequency, the path loss also increases by $6 \text{ dB}$ . This can be counter-intuitive. It does not mean higher frequency signals are "weaker". Instead, it is a consequence of antenna properties. For a given antenna design, its "effective aperture" (its ability to capture energy) is related to the square of the wavelength. Since higher frequencies mean shorter wavelengths, a high-frequency antenna of the same type has a smaller effective capture area, thus receiving less power.
The Constant Term: The last part of the formula, $20 \log_{10}(4\pi/c)$ , is a constant that depends on the units used for distance and frequency. If distance is in meters and frequency is in Hertz, this constant is approximately $-147.55 \text{ dB}$ .

The free-space model is a low-loss reference for an unobstructed path. Other models add terrain, buildings, atmospheric effects, antenna heights, or measured variation to this starting point.

Propagation in the Real World: Reflections, Diffraction, and Scattering

A terrestrial path contains surfaces and objects that interact with the radio wave. The resulting mechanisms are commonly grouped into reflection, diffraction, and scattering:

Reflection: This occurs when a wave reaches a surface whose dimensions are large relative to its wavelength and whose roughness is small enough for a reflected component to form. Common reflectors include the ground, building walls, water surfaces, and large vehicles. The reflected path is longer than the direct path, so the components can arrive with different phases. In-phase components strengthen the received signal, while out-of-phase components can cancel it and form a weak spot. This is a common source of fading.
Diffraction: Radio waves can spread into the shadow region behind an edge or rounded obstacle. Huygens' principle models each point on a wavefront as a source of secondary wavelets. The effect is generally stronger for lower frequencies with longer wavelengths, although the actual loss depends on obstacle geometry, path clearance, and surface properties. Diffraction can preserve a usable component when a hill or building blocks the direct line of sight.
Scattering: Scattering occurs when a wave encounters objects or surface irregularities with dimensions comparable to its wavelength, or a rough surface that prevents a single specular reflection. Foliage, street signs, lampposts, and rough walls can act as secondary radiators. The receiver then sees components arriving from several angles with different amplitudes and phases.

In any real-world environment, a receiver rarely picks up just a single, clean signal. It receives a complex superposition of the direct line-of-sight wave (if available) and a multitude of reflected, diffracted, and scattered waves, all arriving at slightly different times with different amplitudes and phases. Understanding these mechanisms is the first step toward building models that can predict their collective effect on signal strength.

Practical Models for Terrestrial Environments

Explicitly modeling every reflection and diffraction event in a city requires detailed geometry and high computational cost. Engineers therefore use for planning, then use site-specific measurements or deterministic methods where the link requires more detail. Their predictions describe an average or distribution for a defined environment, frequency, distance, and antenna configuration.

1. Two-Ray Ground Reflection Model

This is one of the simplest models that goes beyond free space. It is more realistic for predicting signal strength over relatively flat terrain. The model considers two paths the signal can take from the transmitter to the receiver:

A direct, line-of-sight (LOS) path.
A second path that reflects off the ground.

At the receiver, the direct and reflected components combine. When the distance is large relative to the antenna heights and the model assumptions hold, the received power decreases approximately with the inverse fourth power of distance in the far region, $d^{-4}$ , rather than the inverse square relationship in the free-space reference, $d^{-2}$ . The corresponding path-loss slope is 40 dB per decade of distance, compared with 20 dB per decade in free space. The transition distance and the interference pattern depend on antenna heights, wavelength, ground reflection, and terrain.

2. Macrocell Models for Outdoor Coverage (Cellular)

These models are used for planning large-area cellular networks, where the base station antenna is mounted high on a tower, above most surrounding buildings.

Okumura Model: This is a classic empirical model based on extensive measurements taken in and around Tokyo. It works by taking the free space path loss and adding a median attenuation factor $(A_{mu})$ that is determined from a series of charts. These charts account for the type of terrain (urban, suburban, open), frequency, distance, and antenna heights. While influential, its reliance on charts makes it cumbersome to use in modern automated planning tools.
Hata Model: To replace the graphical Okumura curves with calculations, Masaharu Hata developed formulas that approximate them. The original urban formulation was fitted for approximately 100 to 1500 MHz, distances of 1 to 20 km, base-station antenna heights of 30 to 200 m, and mobile antenna heights of 1 to 10 m. It is commonly used with urban, suburban, and open-area corrections within those assumptions. The standard urban formula is:
$L_{urban} \text{ (dB)} = 69.55 + 26.16 \log_{10}(f) - 13.82 \log_{10}(h_{te}) - a(h_{re}) + (44.9 - 6.55 \log_{10}(h_{te})) \log_{10}(d)$
This formula uses frequency $(f)$ in MHz, distance $(d)$ in km, the effective transmitter antenna height $(h_{te})$ , and the receiver-height correction $(a(h_{re}))$ . Variants such as COST-231 Hata extend the method for selected higher-frequency planning cases. Their validity range and correction terms must be checked before applying them to a 2G, 3G, or newer network.

3. Indoor Propagation Models (Wi-Fi, Small Cells)

Propagation inside a building is a completely different challenge. Walls, floors, ceilings, furniture, and people create a dense, highly reflective environment where a direct line-of-sight path is often unavailable.

Log-distance Path Loss Model: This model represents the median path loss as a logarithmic function of distance. It is useful for an initial estimate, while the exponent and reference loss must be selected for the measured environment.
$PL(d) \text{ (dB)} = PL(d_0) + 10n \log_{10}\left(\frac{d}{d_0}\right)$
The key parameter is $n$ , the path loss exponent. It summarizes the distance dependence for a specified environment. The following values are illustrative ranges, not universal constants:
- In free space, $n = 2$ .
- In an office with a clear line-of-sight path, $n$ might be $1.6 - 1.8$ (waveguiding effect).
- In an office with cubicles and soft partitions (NLOS), $n$ is typically $3 - 4$ .
- In a factory with heavy machinery or a building with thick concrete walls, $n$ can be as high as $5 - 6$ .
ITU Indoor Model: ITU-R P.1238 provides site-general and site-specific indoor models for 300 MHz to 450 GHz, while noting that further measurements are needed to validate the models above 100 GHz. A site-specific form includes the reference loss, the distance coefficient, and a floor-penetration term $(L_f)$ :
$L_{\text{total}} \text{ (dB)} = L(d_0) + N \log_{10}\left(\frac{d}{d_0}\right) + L_f(n)$
Here $L(d_0) = 20 \log_{10}(f) - 28$ dB when $f$ is in MHz and the reference distance $d_0$ is 1 m. The coefficient $N$ depends on the environment. An office example with $N = 30$ gives $20 \log_{10}(f) + 30 \log_{10}(d) + L_f(n) - 28$ . The floor term $L_f(n)$ is selected from measurements for the building and the number of penetrated floors, so it is not a fixed loss for every indoor link.