Free-Space Propagation and Friis Formula

Cellular Networking and Mobile Radio Channel Characterization

2.3 Cellular Radio Channel Characterization

2.3.2 Path Loss Computation and Estimation

2.3.2.1 Free-Space Propagation and Friis Formula

The simplest path loss model is corresponding to the case in which the signal travels along an interference-free unobstructed line-of-sight (LOS) path from the transmit antenna to the receive antenna, that is, the free-space propagation scenario. In this case the precise value of the path loss may be calculated from a simple deterministic equation, known in physics as Friis Formula, expressed in Equation 2.22.

Pr…d† ˆ g_rg_tλ² 4π

… †²d²Pt (2.22)

Here d is the TR distance in meters, g_r and g_t are directional gain of receiver and transmitter antennas, respectively,λ is the wavelength of the carrier signal in meters, P_r and P_t are received and transmitted power, respectively, in Watts. Hardware andfilter losses have been ignored in this equation and if existed should be accounted for separately. Furthermore,

it should be noted that this is a valid equation only if distance d falls in the

“far-field” region of the transmitting antenna. Cases in which free-space model may be applied includes satellite repeater-to-earth station link and line-of-sight terrestrial microwave transmission systems.

Path loss for this deterministic model is computed from Equation 2.22, as expressed in Equation 2.23.

Pℓ ˆ 10 log^P10^t 10 log^P₁₀^r ˆ 20 log^4π₁₀‡ 20 log^d₁₀ 20 log^λ₁₀ 10 log^g₁₀^r 10 log^g₁₀^t ≅ 22 ‡ 20 log^d₁₀ 10 log ^g₁₀^r ‡ log^g₁₀^t

(2.23) If the Friis equation is expressed in terms of frequency, that is, P_r… † ˆ gd _rg_tc²= 4πdf… †², the free-space path loss may be expressed in the form given in Equation 2.24 [12].

Pℓ ˆ 20 log₁₀f ‡ 20 log₁₀d 10 log₁₀g_t 10 log₁₀g_r 147:6 dB (2.24) Note that Equation 2.23 ensures that path loss is expressed as a positive quantity by defining it as 10 log^P₁₀^t 10 log^P₁₀^rand not as 10 log^P₁₀^r 10 log^P₁₀^t. 2.3.2.2 The Key Mechanisms Affecting Radio Wave Propagation

In the majority of land mobile radio networks, including AeroMACS, LOS component may not even reach the receiver antenna, and even if it does, it may not be the only ray arriving at the receiver. In the absence of LOS ray, the propagation mechanisms, in particular, reflection, diffraction, and scattering, provide means for radio communications over the wireless channel.

Reflection occurs when a traveling electromagnetic wave collides with an object whose dimensions are much larger than the wavelength of the radio wave. In land mobile radio networks, reflections off the surface of the earth, buildings, and stationary and moving objects produce addi-tional echoes that may reach the receiving antenna and create the

“multipath effect” that may have constructive or destructive effect on the received signal. In general, when an electromagnetic wave impinges upon another medium with different dielectric constant, part of the wave’s energy transmits through and part of it is reflected back. Propor-tion of transmitted and reflected energies depend upon medium’s electro-magnetic properties. For instance, if a plain wave in air collides with a perfect conductor perpendicularly, the entire energy of the radio signal is reflected back and no energy will be absorbed by the conductor.

Diffraction happens when radio waves strike a large obstructing object that has sharp edges. Diffraction, in effect, is the bending of the electro-magnetic waves around sharp edges of the colliding object. As a result,

secondary waves are generated that radiate throughout the space and even behind the obstruction, producing diffracted echoes that infiltrate into the shadowed areas created by the obstruction. Often times the diffracted copies of the radio wave provides a strong enough signal for MS in shadowed areas. Theoretically, the phenomenon of diffraction can be explained by the Huygens–Fresnel diffraction principle. Huygens–

Fresnel principle states that all points on a wave front can be considered as point sources for the production of secondary wavelets, and these wavelets combine to produce a new wave front in the direction of propagation. Estimation of signal attenuation caused by diffraction is mathematically very difficult, if not impossible. The general method for the estimation of diffraction loss is to make theoretical approximation modified by necessary empirical corrections. In simple cases in which a single obstruction (such as a building or a hilltop) is involved, the attenuation caused by diffraction can be estimated by treating the obstruction as a diffracting knife-edge. For more complicated scenarios, multiple knife-edge diffraction models may be used. Approximate equa-tions are developed based on knife-edge diffraction model that leads to estimation of attenuation caused by diffraction [4].

Scattering takes place when electromagnetic waves collide with objects that are smaller than the wavelength of the radio wave. Consequently, the reflected electromagnetic energy is spread out in all directions. Common objects that cause scattering in wireless channels are trees, lampposts, traffic signs, traffic lights, and so on. Since scattering spreads the signal energy in multiple directions, it can provide additional radio energy at a receiver, and in some occasions, it may be the only mechanism that brings the signal to the MS receiving antenna. Flat surfaces that have much larger dimensions than a wavelength may be modeled as reflective surfaces;

however, the roughness of such surfaces often induces propagation effects that are different from that of just reflection. This can be explained by observing that the surface not only has a reflective effect but it also has a scattering impact [4].

Most radio propagation models that are used in practice to predict large-scale path loss are based on the three mechanisms of reflection, diffraction, and scattering. Although empirical models, to be discussed next, are based on measured data, curvefitting, analytical expressions, or the combination thereof, they indirectly account for reflections, diffrac-tions, and scatterings that take place in the wireless channel.

In summary, the MS antenna, in general, receives several copies of the transmitted waveform that may or may not include the LOS component.

This is the multipath propagation, which is caused by a large number of reflection, diffraction, and scattering processes in the wireless channel.

Clearly, it becomes too complicated to account for the effect of each

individual occurrence on the received signal. The most important sought-after parameter is the received signal power that determines the range of BS antenna for an acceptable QoS. Probabilistic description of the range of values assumed by the received signal power and empirical methods that estimate the statistical averages of this parameter are all that the wireless network planners and designers have at their disposal.