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Path loss
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Path loss
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Path loss, or path attenuation, is the reduction in power density (attenuation) of an electromagnetic wave as it propagates through space.[1] Path loss is a major component in the analysis and design of the link budget of a telecommunication system.

This term is commonly used in wireless communications and signal propagation. Path loss may be due to many effects, such as free-space loss, refraction, diffraction, reflection, aperture-medium coupling loss, and absorption. Path loss is also influenced by terrain contours, environment (urban or rural, vegetation and foliage), propagation medium (dry or moist air), the distance between the transmitter and the receiver, and the height and location of antennas.

Overview

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In wireless communications, path loss is the reduction in signal strength as the signal travels from a transmitter to a receiver, and is an application for verifying the loss. There are several factors that affect this:

  • Free-space path loss: This is the fundamental loss that occurs due to the spreading of the radio wave as it propagates through space. [2] It follows an inverse square law, meaning the signal strength decreases proportionally to the square of the distance between the transmitter and receiver.
  • Diffraction: When a radio wave encounters an obstacle, it can be diffracted, or bent around the edge of the obstacle. This can cause additional signal loss, especially in urban environments with many buildings. [3]
  • Absorption: Certain atmospheric gases and obstacles like buildings and foliage can absorb radio waves, reducing their strength.[4]
  • Reflection and scattering: Radio waves can be reflected off surfaces like buildings and the ground, and scattered by objects like trees and lampposts. This can lead to multipath propagation, where the receiver receives multiple copies of the signal that may interfere with each other. [5]

In understanding path loss and minimizing it, there are four key factors to consider in designing a wireless communication system:

1) Determining the required transmitter power: The transmitter must have enough power to overcome the path loss in order for the signal to reach the receiver with sufficient strength.

2) Determine the appropriate antenna design and gain: Antennas with higher gain can focus the waves in a specific direction, reducing the path loss.

3) Optimize modulation scheme: The choice of modulation scheme can affect the robustness of the signal to path loss.

4) Set the receiver sensitivity appropriately: The receiver must be sensitive enough to detect weak signals.

Causes

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Path loss normally includes propagation losses caused by the natural expansion of the radio wave front in free space (which usually takes the shape of an ever-increasing sphere), absorption losses (sometimes called penetration losses), when the signal passes through media not transparent to electromagnetic waves, diffraction losses when part of the radiowave front is obstructed by an opaque obstacle, and losses caused by other phenomena.

The signal radiated by a transmitter may also travel along many and different paths to a receiver simultaneously; this effect is called multipath. Multipath waves combine at the receiver antenna, resulting in a received signal that may vary widely, depending on the distribution of the intensity and relative propagation time of the waves and bandwidth of the transmitted signal. The total power of interfering waves in a Rayleigh fading scenario varies quickly as a function of space (which is known as small scale fading). Small-scale fading refers to the rapid changes in radio signal amplitude in a short period of time or distance of travel.

Loss exponent

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Main article: Log-distance path loss model

In the study of wireless communications, path loss can be represented by the path loss exponent, whose value is normally in the range of 2 to 4 (where 2 is for propagation in free space, 4 is for relatively lossy environments and for the case of full specular reflection from the earth surface—the so-called flat earth model). In some environments, such as buildings, stadiums and other indoor environments, the path loss exponent can reach values in the range of 4 to 6. On the other hand, a tunnel may act as a waveguide, resulting in a path loss exponent less than 2.

Path loss is usually expressed in dB. In its simplest form, the path loss can be calculated using the formula

where is the path loss in decibels, is the path loss exponent, is the distance between the transmitter and the receiver, usually measured in meters, and is a constant which accounts for system losses.

Radio engineer formula

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Radio and antenna engineers use the following simplified formula (derived from the Friis Transmission Formula) for the signal path loss between the feed points of two isotropic antennas in free space:

Path loss in dB:

where is the path loss in decibels, is the wavelength and is the transmitter-receiver distance in the same units as the wavelength. Note the power density in space has no dependency on ; The variable exists in the formula to account for the effective capture area of the isotropic receiving antenna.[6]

Prediction

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Calculation of the path loss is usually called prediction. Exact prediction is possible only for simpler cases, such as the above-mentioned free space propagation or the flat-earth model. For practical cases the path loss is calculated using a variety of approximations.

Statistical methods (also called stochastic or empirical) are based on measured and averaged losses along typical classes of radio links. Among the most commonly used such methods are Okumura–Hata, the COST Hata model, W.C.Y.Lee, etc. These are also known as radio wave propagation models and are typically used in the design of cellular networks and public land mobile networks (PLMN). For wireless communications in the very high frequency (VHF) and ultra high frequency (UHF) frequency band (the bands used by walkie-talkies, police, taxis and cellular phones), one of the most commonly used methods is that of Okumura–Hata as refined by the COST 231 project. Other well-known models are those of Walfisch–Ikegami, W. C. Y. Lee, and Erceg. For FM radio and TV broadcasting the path loss is most commonly predicted using the ITU model as described in P.1546 (successor to P.370) recommendation.

Deterministic methods based on the physical laws of wave propagation are also used; ray tracing is one such method. These methods are expected to produce more accurate and reliable predictions of the path loss than the empirical methods; however, they are significantly more expensive in computational effort and depend on the detailed and accurate description of all objects in the propagation space, such as buildings, roofs, windows, doors, and walls. For these reasons they are used predominantly for short propagation paths. Among the most commonly used methods in the design of radio equipment such as antennas and feeds is the finite-difference time-domain method.

The path loss in other frequency bands (medium wave (MW), shortwave (SW or HF), microwave (SHF)) is predicted with similar methods, though the concrete algorithms and formulas may be very different from those for VHF/UHF. Reliable prediction of the path loss in the SW/HF band is particularly difficult, and its accuracy is comparable to weather predictions.[citation needed]

Easy approximations for calculating the path loss over distances significantly shorter than the distance to the radio horizon:

  • In free space the path loss increases with 20 dB per decade (one decade is when the distance between the transmitter and the receiver increases ten times) or 6 dB per octave (one octave is when the distance between the transmitter and the receiver doubles). This can be used as a very rough first-order approximation for (microwave) communication links;
  • For signals in the UHF/VHF band propagating over the surface of the Earth the path loss increases with roughly 35–40 dB per decade (10–12 dB per octave). This can be used in cellular networks as a first guess.

Examples

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In cellular networks, such as UMTS and GSM, which operate in the UHF band, the value of the path loss in built-up areas can reach 110–140 dB for the first kilometer of the link between the base transceiver station (BTS) and the mobile. The path loss for the first ten kilometers may be 150–190 dB (Note: These values are very approximate and are given here only as an illustration of the range in which the numbers used to express the path loss values can eventually be, these are not definitive or binding figures—the path loss may be very different for the same distance along two different paths and it can be different even along the same path if measured at different times.)

In the radio wave environment for mobile services the mobile antenna is close to the ground. Line-of-sight propagation (LOS) models are highly modified. The signal path from the BTS antenna normally elevated above the roof tops is refracted down into the local physical environment (hills, trees, houses) and the LOS signal seldom reaches the antenna. The environment will produce several deflections of the direct signal onto the antenna, where typically 2–5 deflected signal components will be vectorially added.

These refraction and deflection processes cause loss of signal strength, which changes when the mobile antenna moves (Rayleigh fading), causing instantaneous variations of up to 20 dB. The network is therefore designed to provide an excess of signal strength compared to LOS of 8–25 dB depending on the nature of the physical environment, and another 10 dB to overcome the fading due to movement.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Path loss

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Path loss, also known as path attenuation, refers to the reduction in power density of an electromagnetic wave as it propagates from a transmitter to a receiver in wireless communication systems, primarily due to the spreading of the wavefront and absorption by the medium. This phenomenon is quantified as the ratio of transmitted power to received power, often expressed in decibels (dB), and is a fundamental aspect of radio propagation that influences signal strength, coverage range, and system performance in technologies such as cellular networks, Wi-Fi, and satellite communications. Path loss increases with distance, frequency, and environmental factors like obstacles or terrain, and is typically modeled using empirical or theoretical equations to predict signal degradation in line-of-sight (LOS) or non-line-of-sight (NLOS) scenarios. Key models for path loss include the model, which assumes an unobstructed LOS path and follows the where received power PrP_r decays proportionally to the square of the distance dd, given by Pr=PtGtGr(λ4πd)2P_r = P_t G_t G_r \left( \frac{\lambda}{4\pi d} \right)^2, with PtP_t as transmitted power, GtG_t and GrG_r as antenna gains, and λ\lambda as . In more realistic environments, the is widely used, expressed as PL(d)=PL(d0)+10nlog⁡10(d/d0)PL(d) = PL(d_0) + 10n \log_{10}(d/d_0), where PL(d0)PL(d_0) is the path loss at a reference distance d0d_0, and nn is the path loss exponent (typically 2 for free space, 2.7–3.5 for urban areas, and up to 6 for obstructed indoor settings). These models account for large-scale effects, distinguishing path loss from small-scale caused by multipath interference or shadowing from buildings and foliage. Understanding and mitigating path loss is crucial for optimizing design, as it directly impacts link budgets, required transmit power, and receiver sensitivity thresholds; for instance, in systems, advanced models like the alpha-beta-gamma (ABG) or close-in (CI) formulations incorporate dependence for millimeter-wave bands where path loss can exceed 100 dB over short distances. Factors such as antenna height, carrier (e.g., higher frequencies like 28 GHz experience greater loss), and propagation environment (free space vs. urban microcells) further modulate path loss, necessitating site-specific measurements and simulations for accurate predictions.

Introduction

Definition and Basics

Path loss refers to the reduction in of an electromagnetic wave as it propagates from a transmitter to a receiver through space or a medium. This attenuation occurs due to the spreading of the wave's energy over an increasing area and interactions with the environment, resulting in a decrease in the received signal power compared to the transmitted power. Path loss is typically expressed in decibels (dB) for convenience in system analysis, where it quantifies the of transmitted power PtP_t to received power PrP_r as PL=10log⁡10(Pt/Pr)PL = 10 \log_{10} (P_t / P_r). It can also be represented in linear units as a power , but the is preferred because it converts multiplicative effects into additive ones, simplifying calculations in link budgets and system design. Importantly, path loss represents the mean or average signal over a and is distinct from other effects such as , which describes rapid fluctuations due to multipath interference, and shadowing, which accounts for location-specific obstructions. Fundamentally, path loss depends on factors including the between transmitter and receiver, the operating of the signal, and the characteristics of the medium, such as free space or terrestrial environments. In free space, it increases with the square of the and the square of the , illustrating the geometric spreading and wavelength-dependent nature of wave . The concept of path loss originated in early radio engineering during the development of wireless communication systems in the early , with its quantitative formulation in free space first provided by Harald T. Friis in 1946 through a simple transmission formula that relates received and transmitted powers under ideal conditions.

Significance in Communications

Path loss fundamentally determines the received signal strength in wireless communications by attenuating the transmitted power over distance and through environmental obstacles, directly influencing the calculation that balances gains and losses to ensure viable connectivity. In the equation, path loss subtracts from the effective isotropic radiated power, reducing the (SNR) and thereby limiting the coverage range of communication systems; for instance, in free-space scenarios, path loss increases quadratically with distance, constraining the maximum operable distance to maintain an adequate SNR for reliable data transmission. This attenuation effect is particularly pronounced in higher-frequency bands, where even modest distance increases can degrade SNR by orders of magnitude, necessitating precise budgeting to avoid link failures. In system design, path loss compels engineers to incorporate compensatory measures such as increased transmitter power, enhanced antenna gains, and improved receiver sensitivity to offset expected losses and achieve desired thresholds. For example, antenna designs with higher can help counteract path loss, while improved receiver sensitivity ensures marginal signals remain detectable, all calibrated against predicted path loss to optimize the overall link margin. These adjustments are essential across diverse applications, including cellular networks where path loss models inform placement for urban coverage, systems that rely on it to extend indoor ranges, and communications where extreme distances amplify losses, demanding high-gain antennas to sustain low-Earth orbit links. By dictating these design choices, path loss shapes the reliability and interference profiles of networks, as unmitigated losses exacerbate and reduce in multi-user environments. The broader implications of path loss extend to system capacity and economic considerations, where excessive curtails throughput by lowering achievable SNR and thus modulation orders, while also influencing interference management in dense deployments like cellular and Wi-Fi spectra. In systems, path loss dominates the due to vast distances, directly impacting global coverage reliability and requiring robust error correction to maintain . Technologically, mitigating path loss through elevated transmitter power or advanced introduces trade-offs, such as increased power consumption that drains batteries in mobile devices or exceeds regulatory emission limits, in array-based systems. These balances highlight path loss as a pivotal factor in sustainable wireless infrastructure, where overcompensation can lead to inefficient resource use, while underestimation compromises network viability.

Fundamental Concepts

Free Space Path Loss

Free space path loss represents the theoretical signal attenuation experienced by an electromagnetic wave propagating in a without any obstacles, reflections, or absorptions. This ideal scenario assumes between isotropic radiators, which are hypothetical antennas that radiate power uniformly in all directions, and operates under far-field conditions where the distance dd is much greater than the λ\lambda (typically d≫λd \gg \lambda). These assumptions simplify the model to focus solely on the geometric spreading of the , ignoring atmospheric effects or multipath interference. The concept originates from the , developed by H.T. Friis in 1946, which relates the power received by an antenna to the power transmitted by another in free space. For isotropic antennas with unity gain, the equation simplifies to express the path loss directly. The derivation begins with the power at a distance dd from an isotropic transmitter radiating power PtP_t, given by the surface area of a : Pt4πd2\frac{P_t}{4\pi d^2}. The received power PrP_r is then this density multiplied by the effective AeA_e of the receiving antenna, where Ae=λ24πA_e = \frac{\lambda^2}{4\pi} for an isotropic receiver. Substituting yields Pr=Pt(λ4πd)2P_r = P_t \left( \frac{\lambda}{4\pi d} \right)^2, so the power ratio PrPt=(λ4πd)2\frac{P_r}{P_t} = \left( \frac{\lambda}{4\pi d} \right)^2. Thus, the PLfsPL_{fs} is the reciprocal: PLfs=(4πdλ)2PL_{fs} = \left( \frac{4\pi d}{\lambda} \right)^2. Since λ=cf\lambda = \frac{c}{f} where cc is the and ff is , this becomes PLfs=(4πdfc)2PL_{fs} = \left( \frac{4\pi d f}{c} \right)^2. In decibels, for practical calculations, the path loss is expressed logarithmically as PLfs(dB)=20log⁡10(d)+20log⁡10(f)+20log⁡10(4πc)PL_{fs} (dB) = 20 \log_{10} (d) + 20 \log_{10} (f) + 20 \log_{10} \left( \frac{4\pi}{c} \right), where dd is in meters, ff in Hz, and c=3×108c = 3 \times 10^8 m/s. This form highlights the quadratic dependence on both and , meaning signal strength diminishes as the square of the propagation and the square of the operating . For example, doubling the quadruples the path loss, a critical consideration in high-frequency systems like millimeter-wave communications. This model is limited to ideal free space and far-field approximations, failing to account for near-field effects or real-world propagation impairments, which can significantly alter actual losses. It serves as a baseline for more complex models but underscores that path loss inherently scales with d2f2d^2 f^2, establishing the fundamental geometric and frequency-induced attenuation in unobstructed environments.

Propagation Mechanisms

Electromagnetic waves used in communications propagate through various physical mechanisms that determine the extent of path loss between transmitter and receiver. These mechanisms describe how waves travel from source to destination, often deviating from ideal conditions due to interactions with the environment. In the absence of obstacles, propagation occurs primarily via direct waves, but real-world scenarios involve additional processes like reflection, , , and , each contributing to signal . Direct wave propagation refers to the line-of-sight (LOS) transmission of electromagnetic waves from the transmitter to the receiver without interruption. In this mechanism, the wave spreads spherically from the source, following the , where decreases proportionally to the square of the distance due to geometric spreading. This is the dominant mode in free space or unobstructed environments, serving as the baseline for path loss calculations. Reflection occurs when electromagnetic waves encounter smooth surfaces, such as buildings or the ground, causing the wave to bounce off at an angle equal to the angle of incidence, as governed by the laws of adapted for radio frequencies. This can lead to , where multiple reflected paths interfere at the receiver, potentially causing additional loss through destructive interference. , on the other hand, allows waves to bend around edges of obstacles, such as hills or structures, enabling in non-line-of-sight (NLOS) scenarios; this bending arises from the wave's interaction with the obstacle's boundary, resulting in secondary wavelets that propagate into shadowed regions, though with significant . Together, reflection and mitigate complete signal blockage but introduce extra path loss compared to direct . Scattering involves the interaction of waves with small particles, irregularities, or rough surfaces—such as foliage, raindrops, or urban clutter—much smaller than the , causing the wave to disperse in multiple directions like . This mechanism leads to a diffused pattern, where is spread over a wide area, reducing the signal strength at any specific receiver location due to the loss of coherence. describes the bending of waves as they pass through media with varying densities, such as atmospheric layers with differing refractive indices due to temperature, , or gradients. In the , for instance, super-refraction can curve waves downward, extending beyond the horizon, while sub-refraction increases path loss by straightening trajectories; in the , it affects higher-frequency signals like HF radio. These variations alter the effective path length and contribute to fluctuating loss. All these propagation mechanisms are fundamentally described by the wave equation derived from , which model electromagnetic fields as coupled partial differential equations governing wave behavior in space and time. The scalar , a time-independent form, captures how waves propagate, reflect, diffract, scatter, and refract under different boundary conditions and media properties.

Causes and Factors

Attenuation Mechanisms

Path loss in propagation arises from several fundamental mechanisms that reduce signal power as the electromagnetic wave travels from transmitter to receiver. One primary mechanism is free space spreading loss, which occurs even in an ideal without obstacles or absorbing media. In free space, the transmitted wave emanates from the antenna as a spherical , diluting the power density over the surface of an expanding whose radius equals the propagation distance dd. This geometric spreading, combined with the dependence in the , results in the received power being inversely proportional to (df)2(d f)^2, where ff is the , assuming fixed antenna gains. Absorption loss represents another key mechanism, where the propagating wave's energy is dissipated as within the medium through molecular interactions. In the atmosphere, this primarily involves absorption by oxygen and molecules, which exhibit resonant spectral lines leading to frequency-dependent . For instance, oxygen causes significant absorption near 60 GHz due to a broad band from merged rotational lines, while peaks at discrete frequencies such as 22.235 GHz and 183.31 GHz; these effects intensify at higher and millimeter-wave frequencies, with specific varying by , , and . Foliage and introduce similar absorption, where in leaves and branches scatters and absorbs energy, particularly above 1 GHz, resulting in higher losses for denser or wetter media. Building and terrain penetration loss occurs when the signal passes through obstructing materials, weakening it via absorption, reflection, and multiple internal scattering. Walls, floors, and building materials like concrete, brick, or glass absorb and reflect portions of the wave, with losses depending on material thickness, composition, and frequency; for example, at microwave frequencies around 5.8 GHz, penetration through typical urban structures can add 10-30 dB of excess loss compared to free space. Terrain features such as soil or rock similarly attenuate signals through dielectric absorption and conduction currents, especially in non-line-of-sight scenarios where the wave must diffract or refract around obstacles. Polarization mismatch introduces additional when the transmitting and receiving antennas are not aligned in polarization orientation. Electromagnetic waves carry polarization (e.g., linear horizontal/vertical or circular), and any misalignment—due to antenna tilt, propagation-induced like in the , or scattering—reduces the coupled power; for orthogonal polarizations, the loss can reach 20-30 dB, though typical mismatches yield 3 dB for random orientations. This mechanism is particularly relevant below 10 GHz where ionospheric effects dominate. Quantitatively, many attenuation mechanisms, especially absorption in media, are modeled using an exponential decay form for the electric field amplitude E∝e−αdE \propto e^{-\alpha d}, where α\alpha is the attenuation coefficient (in nepers per unit distance) dependent on frequency, medium properties, and environmental conditions; the corresponding power loss follows e−2αde^{-2\alpha d}. This form captures the cumulative dissipative effect over distance dd without accounting for spreading.

Environmental Influences

Environmental influences on path loss arise from the physical characteristics of the surrounding medium and structures, which can amplify beyond fundamental free-space conditions by introducing , absorption, and blockage effects. In urban settings, dense buildings and infrastructure create significant and shadowing, leading to higher path loss compared to rural areas where open terrain allows for more direct signal paths. Measurements indicate excess path loss on the order of 25 dB in urban environments, decreasing to under 10 dB in suburban or rural areas due to fewer obstructions. This disparity is primarily attributed to the increased density of scatterers in cities, which cause signal reflections and diffractions that degrade the direct line-of-sight component. Frequency dependence plays a critical role in how environmental factors affect path loss, with higher bands experiencing greater overall . For instance, millimeter-wave (mmWave) frequencies above 24 GHz suffer enhanced path loss relative to sub-6 GHz bands, not only from the quadratic increase in free-space loss with but also from heightened atmospheric absorption by oxygen and , as well as stronger interactions with obstacles like foliage and buildings. Empirical measurements confirm that path loss at 28 GHz can exceed that at 2.9 GHz by 20-30 dB over similar distances in urban microcells, limiting mmWave range to shorter links unless mitigated by . This frequency-induced sensitivity makes higher bands more vulnerable to environmental variability, though path loss exponents may remain comparable across bands in line-of-sight scenarios. Atmospheric conditions introduce dynamic variations in path loss, particularly through weather-related phenomena. Rain fade, caused by scattering and absorption of signals by raindrops, can add 5-20 dB or more of attenuation on slant paths, with severity increasing at frequencies above 10 GHz and during heavy precipitation rates exceeding 50 mm/h. Fog and clouds contribute additional gaseous and particulate absorption, typically 1-5 dB in dense conditions, while tropospheric scintillation—rapid fluctuations due to refractive index variations in the lower atmosphere—induces signal fading of up to 3-5 dB in 0.1% of time for microwave links. These effects are more pronounced in satellite or long-range terrestrial communications, where path length through the troposphere amplifies the impact. Indoor environments impose substantial additional path loss compared to outdoor due to penetration losses from walls, floors, and furnishings. Building materials such as or walls can attenuate signals by 10-20 dB per penetration at frequencies, while lighter partitions add 3-6 dB; furniture and other clutter further contribute 5-10 dB through diffuse and absorption. Overall, indoor path loss often exceeds outdoor by 10-30 dB for equivalent distances, depending on layout density, with multi-floor scenarios incurring extra floor of 15-20 dB. This containment effect necessitates specialized considerations for in-building wireless systems. Terrain features like hills and elevation changes significantly alter path loss by causing line-of-sight blockages and inducing over irregular profiles. Elevated terrains, such as hills or plateaus, can create shadowing zones where signals are obstructed, increasing path loss by 10-40 dB in non-line-of-sight regions behind rises, while varying altitudes affect the effective height and ground reflection contributions. In forested or cluttered hilly areas, additional and exacerbate these effects, leading to higher variability in signal strength compared to flat terrains. Such topographic influences are particularly relevant for rural or suburban deployments spanning undulating landscapes.

Modeling Approaches

Deterministic Models

Deterministic models predict path loss by applying principles of electromagnetism and geometry to simulate signal propagation in a precisely defined environment, offering exact calculations without reliance on statistical averaging. These approaches typically solve approximate forms of Maxwell's equations using ray optics, accounting for phenomena like reflection, diffraction, and direct transmission based on the site's topography, buildings, and antenna positions. Unlike broader propagation models, they require detailed environmental data, such as 3D maps, to trace signal paths accurately. Ray-tracing models form a of deterministic , simulating multiple paths—including direct line-of-sight, reflections off surfaces, and diffractions around obstacles—between the transmitter and receiver. Reflections are modeled using to determine the angle of incidence and reflection, while diffractions invoke Huygens' principle to treat wavefronts as sources of secondary wavelets, often incorporating the uniform theory of diffraction for edge effects. These models launch rays from the transmitter in various directions, trace their interactions with the environment via image theory or shooting and bouncing methods, and sum the contributions at the receiver to compute total path loss. A seminal demonstrated their utility in urban microcells by integrating building databases and , achieving predictions within 6-8 dB of measurements. The two-ray ground reflection model simplifies deterministic analysis for open terrains, considering only the direct path and a single reflection from a flat surface. It assumes perfect reflection and neglects atmospheric effects, leading to constructive or destructive interference depending on distance. For large separation distances where the direct and reflected paths interfere destructively, the path loss approximates PL=(d2hthr)2PL = \left( \frac{d^2}{h_t h_r} \right)^2 in linear units, where dd is the transmitter-receiver distance and hth_t, hrh_r are the respective antenna heights above ground. This model builds on for the direct component but incorporates the to capture the d4d^4 distance dependence observed beyond the critical distance dc=4hthr/λd_c = 4 h_t h_r / \lambda. The formulation originates from early analyses of over reflective surfaces. The knife-edge diffraction model addresses signal blockage by a single sharp , such as a building edge or hill crest, treating it as an ideal wedge that bends waves around the obstruction. loss is derived from the Fresnel-Kirchhoff diffraction theory, expressed through the complex F(v)=∫v∞ejπt2/2 dt,F(v) = \int_v^\infty e^{j \pi t^2 / 2} \, dt, where the parameter v=h2(d1+d2)/(d1d2[λ](/page/Lambda))v = h \sqrt{2 (d_1 + d_2) / (d_1 d_2 [\lambda](/page/Lambda))}
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