Ground Reflection and Double-Ray Model

Cellular Networking and Mobile Radio Channel Characterization

2.3 Cellular Radio Channel Characterization

2.3.2 Path Loss Computation and Estimation

2.3.2.4 Ground Reflection and Double-Ray Model

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.

airport, although more often than not, Rayleigh fading model has been used for characterization of this link.

Let us denote the distance between the bases of transmit and receive antennas by d0. Assume that d0is short enough to presume that the earth surface remains flat. Figure 2.10 illustrates this model with various distances and heights. The height of transmit and receive antennas are designated as htand hr, the length of the LOS path is d, and d1and d2are the distances between ground reflection point and transmitter and receiver antennas, respectively.

Since the principals of Cartesian Optics are applied, a number of statements may be made regarding this propagation model. For instance, Snell’s law is applicable to maintain that incident angle θi is equal to reflection angle θr. The objective is tofind an equation similar to Friis equation for double-ray ground reflection model. In order to facilitate this computation, the following reasonable assumptions are made that lead to a simplified approximate mathematical expression for path loss attenua-tion for this ray tracing model.

1) The distance d is much greater than the height of transmitter and receiver antennas, that is, d  ht; d  hr.

2) The incident angleθi and the reflection angles θr are very small.

3) d1‡ d2≅ d

4) Generally speaking, the earth is not a“perfect conductor” but a good conductor. The implication is that most of the energy of the incident wave is reflected and a very small fraction of it is transmitted through the earth [19]. However, it is assumed that the earth is a perfect conductor and the entire energy of the incident wave is reflected off the surface of the earth. The assumption of the perfect reflecting conductor also implies that the phase of the reflected signal is changed by 180°. It is noted that if the reflecting surface is a body of water, such

Figure 2.10 The ground reflection double-ray propagation model. This model is based on ray tracing technique that follows the principle of geometric optics. The additional geometry shown in thefigure is for understanding of the analysis that ensues.

as lakes and oceans, significant portion of the signal energy is lost to the reflecting surface and the conjecture of perfect conductor is not valid.

With these approximations in mind, we first establish the following relationships based on geometry shown in the Figure 2.10.

d≅ d ²₀‡ h… t h_r†²1=2

(2.25) d₁‡ d2≅ d ²₀‡ h… t‡ hr†²1=2

(2.26) In order to obtain an approximate value for the path length difference of the LOS ray and the ground reflection ray, we expand Equations 2.25 and 2.26 into their Taylor series and approximate each series with thefirst two terms by virtue of the assumption 1 listed already.

d≅ d0 1‡…h_t h_r†² d²₀

" #1=2

≅ d0 1‡…h_t h_r†² 2d²₀

" #

(2.27)

d₁‡ d2≅ d0 1‡…ht‡ hr†² d²₀

" #1=2

≅ d0 1‡…ht‡ hr†² 2d²₀

" #

(2.28)

Δd ≜ …d1‡ d2† …d† ≅ 2hthr

d₀ (2.29)

For the simplicity of the analysis we assume that a single-tone signal, in the form of s…t† ˆ A cos 2πf _ct

, is transmitted. The received double-path signal components may, therefore, be expressed as given in Equations 2.30 and 2.31.

rLOS…t† ˆ ALOScos 2πf _c…t d=c†

(2.30) rGR…t† ˆ AGRcos 2πf _cft …d1‡ d2†=cg 180°

(2.31) In these equations c is the speed of electromagnetic waves in free space.

The double-path signal received at the front end of the MS/SS antenna (the sum of the two signals) is given by Equation 2.32.

r…t† ˆ rLOS…t† ‡ rGR…t† ˆ ALOScos 2πf c…t d=c†

‡ AGRcos 2πf _cft …d1‡ d2†=cg 180°

 Arcos 2πf _ct‡ ϕ (2.32) The Friis equation indicates that in free-space propagation the path loss exponent is equal to 2, therefore, the signal power decays proportional to the square of the TR distance. In the case of sinusoidal signals that translates to a signal amplitude drop-off that is inversely related to TR distance. In other words, the amplitudes of the two components at the

receive antenna may be determined by Equation 2.33.

ALOS ˆA

d; AGRˆ A d₁‡ d2

(2.33) Recall that under the assumptions that were made, the ground reflection ray incurs a phase change of 180° but otherwise preserves its entire strength at the reflection point. This ray essentially propagates through the free space and traverses a path that is longer than that of the LOS ray, therefore, it is justified to assume that the amplitude of this ray at the receiving antenna be given in accordance with Equation 2.33. We now take advantage of the second assumption, that is, d≅ d1‡ d2, to further simplify Equation 2.32:

r…t† ≅A

d cos 2πf_c t d c





cos 2πf_c t d1‡ d2

c





 

(2.34) Here we have also used the trig identity: cos…α 180°† ˆ cos α… †. Apply-ing the trig identity cos… † cos βα … † ˆ 2 sin ^α‡β₂

sin ^{β α}₂

in Equation 2.34 r t… † ˆ2A

d sin πf_c d₁‡ d2

c d

c





sin 2πf_ct 2πf_cd

c‡ πf_cd₁‡ d2

c



(2.35) Substituting Equation 2.29 for…d1‡ d2† d into Equation 2.35 and using the trig identity sin… † ˆ cos α 90°α … †, Equation 2.35 can be put into the following form:

r t… † ˆ2A

d sin 2πf_chthr

cd0



cos 2πf_ct πf_cd

c‡ πf_cd1‡ d2

2 90°



(2.36) Comparing Equation 2.36 with the general form of the received signal r…t† ˆ Arcos 2πf _c‡ ϕ

, it is concluded that Ar ˆ2A

d sin 2πf_chthr

cd₀



ˆ2A

d sin 2πhthr

λd0



(2.37) The received power can, therefore, be approximated by

Pr ˆA²_r 2 ˆ2A²

d² sin² 2πhthr

λd0



(2.38) We nextfind an equation that expresses A, the amplitude of the trans-mitted single-tone signal, in terms of other parameters. Noting that if we consider just the LOS component, the power of the received signal would be calculated from the Friis equation, that is,

PLOS ˆP_tg_tg_rλ² 4πd

… †² …L ˆ 1† (2.39)

On the other hand, we know that PLOSˆ 1=2 A=d… †², equating this expression with Equation 2.39, A can be calculated from Equation 2.40.

A²ˆ2P_tg_tg_rλ² 4π

… †² (2.40)

Replacing Equation 2.40 into Equation 2.38 we obtain a general approximate equation for received power for the double-ray ground reflection model.

P_rˆ4P_tg_tg_rλ² 4πd

… †² sin² 2πhth_r λd0



(2.41) To appreciate the implications of the latter equation, we put the equation into a more familiar form by making two additional reasonable approxi-mations.

d0≅ d 2πhth_r

λ ) sin 2πhth_r λd0



≅ sin 2πhth_r λd



≅2πhth_r λd

(2.42) Replacing the latter equation into Equation 2.41, we conclude

Pr≅ Ptg_tg_rh²_th²_r

d⁴ (2.43)

Sometimes Equation 2.41 is referred to as“exact equation” (even though it is an approximation) for calculation of the received power for the double-ray ground reflection model, and Equation 2.43 is dubbed as approximate equation [4].

Comparing Equations 2.42 and 2.22, the following important observa-tions, regarding the path loss for the double-ray model, are made.

1) The path loss exponent has risen to four for the double-ray model. In other words, the received power (path loss) decays (increases) inversely proportional to distance raised to power four.

2) The received power (path loss) is independent from frequency (wave-length) of the electromagnetic wave. Equation 2.42 is an approxima-tion, nevertheless, it implies no dependency or very weak dependency of path loss on wavelength of the electromagnetic wave.

3) The height of receiver and transmitter antennas play direct roles in the intensity of the received power, whereas in Friis equation for free-space propagation indicates that the received power is independent from these parameters.

Therefore, for long distance, that is, d  ht‡ hr or d ffiffiffiffiffiffiffiffiffi hthr

p , the received power decays in accordance with the fourth power law. Referring to Equation 2.42, the term d⁴=h²_rh²_t represents the actual path loss for the

double-ray ground reflection model. Equation 2.44 shows this path loss in decibel.

Pℓ

… †_DRˆ 10 log₁₀ d⁴ h²_rh²_t

!

ˆ 40 log₁₀d 20 log₁₀hr 20 log₁₀ht (2.44)

2.3.2.5 Empirical Techniques for Path Loss (Large-Scale Attenuation)