Propagation Models

Path loss models, free space propagation, and terrain effects.

Introduction: Predicting the Invisible Journey

In our journey into wireless communications, we have established that antennas act as gateways, launching information into space as electromagnetic waves. Once a signal leaves the transmitting antenna, it begins a complex journey through the environment. Unlike the predictable path down a copper wire or through a fiber-optic cable, the wireless channel is chaotic and unpredictable. The signal does not simply travel in a straight line; it interacts with everything in its path, from the ground and buildings to trees and even the atmosphere itself.

The most fundamental consequence of this journey is that the signal gets weaker as it travels away from the transmitter. This reduction in signal power is known as or attenuation. For a wireless system to work, the signal that arrives at the receiver must be strong enough to be clearly distinguished from the background noise that is ever-present in the radio spectrum. Therefore, being able to predict or estimate the amount of path loss between a transmitter and a receiver is one of the most critical tasks in designing a wireless network.

This is where come in. A propagation model is a mathematical formula or a set of algorithms that attempts to predict the signal strength at a receiver. These models are the essential tools for radio network planners. They are used to determine how many cell towers are needed to cover a city, where to place a Wi-Fi access point in an office for optimal coverage, how far a satellite link can reliably transmit, and how much power a transmitter needs to use. The process of using these models to ensure a reliable connection is called creating a link budget, which is essentially an accounting of all the gains and losses a signal experiences along its path.

The Ideal Case: The Free Space Propagation Model

To begin to understand path loss, we must start with the simplest, most idealized scenario imaginable: free space. The assumes a perfect environment with a transmitter and receiver floating in a vacuum, with absolutely nothing else around them. There are no obstacles, no ground to reflect off, no buildings to block the signal, and no atmosphere to absorb it. This scenario is a perfect, unobstructed Line-of-Sight (LOS) path. While this seems unrealistic for terrestrial communication, it is an excellent approximation for satellite-to-satellite links or deep-space communication.

The Spreading of Energy

In free space, the only source of signal loss is the natural spreading of the wave. Imagine a lightbulb in a vast, dark space. As you move away from it, the light becomes dimmer. This is not because the light is being absorbed, but because the same amount of energy is being spread out over the surface of an ever-expanding sphere. The power density (power per unit area) decreases as the surface area of the sphere ( $4 \pi d^2$ ) increases. This is the fundamental reason for free space path loss.

The Friis Transmission Equation

The mathematical relationship describing this loss is captured by the Friis transmission equation. From it, we can derive the formula for free space path loss $(L_{fs})$ . It shows how the loss depends on two key factors: the distance between the antennas and the frequency of the signal.

First, let's look at it as a simple power ratio. This formula tells us how many times weaker the received power is compared to the transmitted power.

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)

Let's analyze the components of this crucial formula:

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 serves as the absolute best-case scenario and is the starting point for almost all other, more complex propagation models.

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

Once we bring our antennas down to Earth, the situation becomes vastly more complicated. The idealized vacuum of free space is replaced by a complex environment filled with objects that interact with the radio wave. These interactions are broadly categorized into three mechanisms:

Reflection: This occurs when the radio wave impinges upon an object that is very large compared to the wavelength of the wave. The surface acts like a mirror. The most significant reflector in most terrestrial environments is the Earth's surface itself. Other examples include the walls of buildings, bodies of water, and large vehicles. A reflected wave travels a longer path to reach the receiver than the direct wave. At the receiver, these multiple waves combine. If they arrive in phase, they strengthen the signal (constructive interference), but if they arrive out of phase, they can cancel each other out, creating a weak signal spot or "dead zone" (destructive interference). This phenomenon is the primary cause of fading.
Diffraction: This phenomenon allows radio waves to bend around sharp edges or corners of obstacles. According to Huygens' principle, every point on a wavefront can be considered a source of secondary wavelets. When a wave hits an obstacle, these secondary wavelets propagate into the shadowed region behind it. This bending is more pronounced for lower frequencies (longer wavelengths). Diffraction is a crucial mechanism that enables communication even when there is no direct, clear line-of-sight path between the transmitter and receiver. It's the reason you can receive a radio signal even when you are behind a large hill or building that completely obstructs the view of the transmitter.
Scattering: Scattering occurs when a radio wave hits an object whose size is on the order of, or smaller than, the wavelength. Instead of simply reflecting or being blocked, the object acts like a small secondary radiator, scattering the energy in many different directions. Common scattering objects include foliage on trees, street signs, lampposts, and even irregularities on a rough surface. Scattering causes the transmitted energy to spread out, resulting in a weaker, more diffuse signal arriving at the receiver from many angles.

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

Modeling every single reflection and diffraction event in a city is computationally impossible. Therefore, engineers rely on . These models are derived from extensive real-world measurements and use statistical methods to predict the average path loss in a given type of environment.

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, these two waves combine. The crucial insight from this model is that for distances that are large relative to the antenna heights, the path loss increases much more rapidly than in free space. Instead of being proportional to the square of the distance ( $d^2$ ), the received power becomes proportional to the fourth power of the distance ( $d^4$ ). In decibels, this means the path loss increases by 40 dB per decade of distance, compared to only 20 dB per decade in free space. This result accurately reflects that signals get much weaker much faster when operating close to the ground.

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 address the shortcomings of Okumura's graphical approach, Masaharu Hata developed a set of mathematical formulas that approximate the Okumura charts. The Hata model became the industry standard for predicting path loss in urban, suburban, and rural environments for frequencies up to about $1.5 \text{ GHz}$ . The standard formula for an urban area 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 incorporates frequency $(f)$ in MHz, distance $(d)$ in km, the effective height of the transmitter antenna $(h_{te})$ , and a correction factor for the receiver antenna height $(a(h_{re}))$ . Variants of the Hata model, like the COST-231 Hata model, extended its applicability to higher frequencies for planning 2G and 3G networks.

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 is a very common and versatile model that generalizes the path loss relationship. It states that the average path loss increases logarithmically with distance.
$PL(d) \text{ (dB)} = PL(d_0) + 10n \log_{10}\left(\frac{d}{d_0}\right)$
The key parameter here is $n$ , the path loss exponent. This exponent captures the effect of the environment:
- 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: The ITU provides standardized values for the log-distance model, which also include a factor for losses when the signal passes through floors $(L_f)$ . For example, in an office environment at $2.4 \text{ GHz}$ , the model might look like this:
$PL \text{ (dB)} = 20 \log_{10}(f) + 30 \log_{10}(d) + L_f(N_{floors}) - 28$
Here, the exponent is set to 3 $(30 \log_{10}(d))$ , and $(L_f)$ adds a specific loss in dB for each floor $(N_{floors})$ the signal must penetrate.