# 4.6: The RF Link

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The RF link is between a transmit antenna and a receive antenna. Sometimes the RF link includes the antenna and sometimes it does not, this will be clear from the context, but usually it includes the antennas. The principle source of link loss is the spreading out of the EM field as it propagates. In the absence of any other effects (such as atmospheric loss and reflections), the power density reduces as \(1/d^{2}\), where \(d\) is distance, and this is called the** line-of-sight** (**LOS**) situation. In this section the propagation path is first described along with its impairments including propagation on multiple paths between a transmit antenna and a receive antenna. The resonant scattering is described as well as multiple fading effects due to reflections, diffraction, rain, and other atmospheric effects.

## 4.6.1 Propagation Path

When the radiated signal reflects and diffracts there are multiple propagation paths that result in what is called fading, as the paths constructively and destructively combine at the receiver. The direct path and other paths, called multipaths, which are reflected or diffracted by ground, buildings, and other objects do not arrive at the receiver in phase leading to time-varying constructive and destructive combining, called multipath interference. Of these destructive combining is much worse as it can reduce a signal level below what it would be if propagation was in free space. In urban areas, \(10\) or \(20\) paths can have significant power in them and these combine at the receive antenna [6].

When the signal on one of the paths dominates, this is usually the LOS path, fading is called **Rician fading**. With LOS and a single ground reflection, the situation is the classic Rician fading as shown in Figure \(\PageIndex{1}\)(a). This is a special situation, as the ground changes the phase of the signal upon reflection, generally by \(180^{\circ}\). When the receiver is a long way from the base station, the lengths of the two paths are almost identical and the level of the signals in the two paths are almost the same. The net result is that these two signals almost cancel, and so instead of the power falling off by \(1/d^{2}\), it falls off by \(1/d^{3}\). When there are many paths and all have similar amplitude signals, fading is called **Rayleigh fading**. In an urban area such as that shown in Figure \(\PageIndex{1}\)(b), there are many significant multipaths and the power falls off by \(1/d^{4}\) and sometimes faster.

Common paths encountered in cellular radio are shown in Figure \(\PageIndex{2}\). As a rough guide, in the single-digit gigahertz range each diffraction and scattering event reduces the signal received by \(20\text{ dB}\). The knife-edge diffraction scenario is shown in more detail in Figure \(\PageIndex{3}\). This case is fairly easy to analyze and can be used to estimate the effects of individual obstructions. The diffraction model is derived from the theory of half-infinite screen diffraction [7]. First, calculate the parameter \(\nu\) from the geometry of

Figure \(\PageIndex{1}\): Multipath propagation: (a) line of sight (LOS) and ground reflection paths only; and (b) in an urban environment.

Figure \(\PageIndex{2}\): Common paths contributing to multipath propagation.

Figure \(\PageIndex{3}\): Knife-edge diffraction.

Figure \(\PageIndex{4}\): Pine needles scattering an incoming EM signal.

the path using

\[\label{eq:1}\nu =-H\sqrt{\frac{2}{\lambda}\left(\frac{1}{d_{1}}+\frac{1}{d_{2}}\right)} \]

Next, consult the plot in Figure \(\PageIndex{3}\)(b) to obtain the diffraction loss (or attenuation). This loss should be added (using decibels) to the otherwise determined path loss to obtain the total path loss. Other losses such as reflection cancelation still apply, but are computed independently for the path sections before and after the obstruction.

## 4.6.2 Resonant Scattering

Propagation is rarely from point to point as the path is often obstructed and so non-LOS (**NLOS**). One type of event that reduces transmission is scattering. The level of the effect depends on the size of the objects causing scattering. Here the effect of the pine needles of Figure \(\PageIndex{4}\) will be considered. The pine needles (as most objects in the environment) conduct electricity, especially when wet. When an EM field is incident an individual needle acts as a wire antenna, with the current maximum when the pine needle is one-half wavelength long. At this length, the “needle” antenna supports a standing wave and will re-radiate the signal in all directions. This is scattering, and there is a considerable loss in the direction of propagation of the original fields. As well, there is loss due to a needle not being a very good conductor, so EM energy is lost as heat. The effect of scattering is frequency and size dependent. A typical pine needle is \(15\text{ cm}\) long, which is exactly \(\lambda /2\) at \(1\text{ GHz}\), and so a stand of pine trees have an extraordinary impact on cellular communications at \(1\text{ GHz}\). As a rough guide, \(20\text{ dB}\) of a signal is lost when passing through a small stand of pine trees. At \(2\text{ GHz}\), a similar impact comes from other leaves, and they do not need to look like wire antennas. Consider an oak leaf having a dimension of \(7.5\text{ cm}\). This is \(\lambda /2\) at \(2\text{ GHz}\), another dominant cellular frequency. So scattering results, but now the loss is season dependent, with the loss due to scattering being considerably smaller in winter (when trees such as oaks do not have leaves) than in summer.

## 4.6.3 Fading

Fading refers to the variation of the received signal with time or when the position of transmit or receive antennas is changed. The variation in signal strength is much greater than would be expected from changes in path

Figure \(\PageIndex{5}\): Fast and slow fading: (a) in time as a radio and obstructions move; and (b) in distance.

length. Several forms of fading are identified and associated with different physical effects. The most important fading types are flat fading, multipath fading, and rain fading.

### Flat Fading

Temperature variations of the atmosphere between the transmit antenna and receive antenna give rise to what is called flat fading and sometimes called **thermal fading**. This fading is called flat because it is independent of frequency. One form of flat fading is due to refraction, which occurs when different layers of the atmosphere have different densities and thus dielectric permittivities increasing or decreasing away from the surface of the earth. The temperature profile can increase away from the earth surface or reduce depending on whether the temperature of the earth is higher than that of the air and is commonly associated with the beginning and end of the day. **Temperature inversions** can also occur where the temperature profile in air does not uniformly increase or decrease, thus causing a layer with a higher permittivity than that of the air above or below. RF energy gets trapped in this layer, reflecting from the top and bottom of the inversion layer. This is called **ducting**. In point-to-point communication systems the transmit and receive antennas are mounted high on towers and then reflection from ground objects is often small. In such cases flat fading is the most commonly observed phenomena and minor fluctuations of several decibels in receive signal level are common throughout the day. However, when temperature variations are extreme, ducting can severely impact communications reducing signal levels by up to \(20\text{ dB}\).

### Shadow Fading

Shadow fading occurs when the LOS path is blocked by an obstruction such as a building or hill. The full signal strength returns when the obstruction or receiver has moved to restore the direct path. Shadow fading is also called slow fading and the characteristic of the channel is regarded as being relatively constant over a short time. The amplitude response varies in time and distance and is shown in Figure \(\PageIndex{5}\). This figure shows both fast fades that are \(10\) to \(15\text{ dB}\) deep and slow or shadow fades that are \(20\) to \(30\text{ dB}\) deep.

### Multipath Fading

Multipath fading is the most common fading when either the transmit antenna or the receive antenna are close to the ground, near obstructing buildings or terrains, or inside a building. In such situations there are many reflections that combine destructively and constructively. Multipath fading is also called **fast fading**, as the characteristics of the channel can change significantly in a few milliseconds. Two types of multipath fading—Rician fading and Rayleigh fading—will be considered.

Multipath is principally a problem when the line of sight between transmit and receive antennas is obscured. However, where there is line of sight the signal reflecting from the ground immediately in front of a receive antenna can sometimes largely cancel the line-of-sight signal. In time, intervening objects can move and the propagation characteristics of the various paths can change due to thermal variations. All this adds to the randomness of fast fades. Multipath fading of \(20\text{ dB}\) can occur for a small percentage of the time on time scales of many seconds when there are few propagation paths (e.g. in a rural area) to a large percentage of the time many times per second in a dense urban environment when there are many paths. Constructive combining does increase the signal level momentarily, but there is no advantage to this. Destructive combining can result in deep fades of \(20\text{ dB}\) impacting communications and forcing the communication system to accommodate either by using higher average powers or using strategies such as multiple antennas or spreading the communication signal over a wide bandwidth since fades tend to be \(500\text{ kHz}\) to \(1\text{ MHz}\) wide at all frequencies.

### Rician Fading

Rician fading occurs when there is one dominant RF path, usually the LOS path, and one or more other paths. The main situation is when there is an LOS path and a ground reflection as shown in Figure \(\PageIndex{1}\)(a). The signal received is the sum of two signals:

\[\label{eq:2}v_{r}(t)=C[\cos(\omega_{c}t)+r\cos(\omega_{c}t+\phi)] \]

where \(C\cos(\omega_{c}t)\) is the LOS signal, \(r\) is the ratio of the amplitude of the signal reflected from the ground and the LOS signal, and \(\phi\) is the relative phase. When the distance between the transmitted and received signal is large, the ground reflection will be a glancing reflection and the LOS and reflected path will be almost exactly the same length. Ground reflection will usually introduce a \(180^{\circ}\) phase rotation in the reflected signal and \(r\) could be \(0.8–0.9\). Thus there will be nearly complete cancelation of the signal [8].

### Rayleigh Fading

Rayleigh fading, or **fast fading**, in a multipath environment results from destructive cancellation as individual paths of similar amplitude drift in and out of phase as the receiver and sources of multiple reflections, diffractions, and refractions move. With multiple paths between a transmitter and a receiver, different components of the received signal arrive at different times. When the line of sight is blocked, the received signal on as many as \(20\) of these paths can have appreciable signal power. The time between when the first significant received signal is received and the last is received is called the

Figure \(\PageIndex{6}\): Two receive antennas used to achieve **space diversity **and overcome the effects of Rayleigh fading.

delay spread. The delay spread is largest inside buildings. A typical office building has a delay spread of \(30\text{ ns}\), and an office building with highly reflective walls and large open spaces has a **delay spread **of up to \(250\text{ ns}\) [9, 10, 11], similar delay spreads are obtained at all RF frequencies [12]. Rayleigh fading is the type of fading that occurs when the inverse of the delay spread is small compared to the bandwidth of the received signal. Then when the multipaths combine destructively the entire signal within a bandwidth of \(500\text{ kHz}\) to \(1\text{ MHz}\) is suppressed. This bandwidth is experimentally observed and is related to the delay spread of individual paths, each with random amplitude and phase. If a communication signal has a bandwidth that is entirely within the suppression bandwidth of a Rayleigh fade, the situation is called frequency flat fading. Some communication signals have bandwidths greater than the Rayleigh fade bandwidth, so it is possible to recover from Raleigh fades if error correction is used.

The result of Rayleigh fading is rapid amplitude variation with respect to frequency. In Figure \(\PageIndex{5}\)(a) the amplitude varies in time depending on the speed of moving objects. This graph could just as well be a plot of amplitude versus distance or amplitude versus frequency. A measured amplitude response is shown in Figure \(\PageIndex{5}\)(b). Although somewhat random, the fades occur over distances roughly \(\frac{1}{2}\lambda\) apart and \(500\text{ kHz}\) wide. The \(500\text{ kHz}\) width is almost independent of frequency from hundreds of megahertz to \(100\text{ GHz}\). The probability of receiving a signal \(x\text{ dB}\) below the time-averaged received signal power is approximately \(10^{−x}\) [13]. Rayleigh fading is named after the statistical model that describes this.

Fortunately deep Rayleigh fades are very short, last a small percentage of the time, and slight changes to the propagation environment can circumvent their effects. One strategy for overcoming fades is to use two receive antennas as shown in Figure \(\PageIndex{6}\). Here the signals received by two antennas separated by several wavelengths will rarely fade concurrently. In practice, the required separation for good decorrelation is found to be \(10\) to \(20\lambda\).

In an LOS wireless system (e.g. a point-to-point link without multipath), Rayleigh fading is due to rapidly changing atmospheric conditions, with the refractive index of small regions varying. These fades occur over a few seconds.

### Rain Fading

Rain fading is due to both the amount of rain and the size of individual rain drops and fading occurs over periods of minutes to hours. Propagation through the atmosphere is affected by absorption by molecules in the air, fog, and rain, and by scattering by rain drops. Figure \(\PageIndex{7}\) shows the attenuation in decibels per kilometer from \(3\text{ GHz}\) to \(300\text{ GHz}\). Curve (a) shows the

Figure \(\PageIndex{7}\): Excess attenuation due to atmospheric conditions showing the effect of rain on RF transmission at sea level. Curve (a) is atmospheric attenuation, due to excitation of molecular resonances, of very dry air at \(0^{\circ}\text{C}\), curve (b) is for typical air (i.e. less dry) at \(20^{\circ}\text{C}\). The attenuation shown for fog and rain is additional (in \(\text{dB}\)) to the atmospheric absorption shown as curve (b).

attenuation in dry air at \(0^{\circ}\text{C}\) with very low attenuation at a few gigahertz with several attenuation peaks as frequency increases. The first peak in attenuation is due to the resonance of water molecules in water vapor. The resonance peaks at \(23\text{ GHz}\) but loss starts increasing before that. The next absorption peak is at \(60\text{ GHz}\) due to the resonance of oxygen molecules. Two other absorption peaks are observed going up to \(300\text{ GHz}\). Curve (b) is for a less dry atmosphere at \(20^{\circ}\text{C}\). The dotted curve shows the additional effect of fog on attenuation and then a family of curves shows the effect of rain on propagation. Below \(10\text{ GHz}\) the effect of rain is very little and is safely ignored below \(5\text{ GHz}\). The attenuation due to rain increases with frequency and this derives largely from scattering and is related to the RF signal’s wavelength relative to the circumference of rain drops.

### Fading Due to Ducting

Normally the temperature and density of air, and thus its refractive index, drops with increasing height above earth. As a result, radio waves will be refracted toward the earth as shown in Figure \(\PageIndex{18}\)(a). However, during periods of stable weather characterized by a high-pressure system, the temperature can rise with increasing height before eventually falling, creating what is called temperature inversion. This can occur at heights of tens to hundreds of meters. Temperature inversion is most common in summer, but can also occur when the temperature is dropping quickly, such as at sunset. At sunset the inversion layer may occur at a meter or so above the ground as the earth cools rapidly. The denser colder air above the ground has a higher permittivity than the air above and this results in an inversion layer that has a higher refractive index than the air above and below. RF waves can become trapped in the inversion layer, as any RF energy leaving the temperature inversion layer is refracted back into the layer. This effect is called ducting. Ducting can also occur when a cold air mass is overrun by warm air.

Fading due to ducting occurs when the receiver wanders in and out of the ducting layer, as the ducting layer is not stable, increasing and decreasing in

Figure \(\PageIndex{8}\): Fading resulting from ducting: (a) normal atmospheric refraction (normally the temperature of air drops with increasing height and the lower refractive index at high heights results in a concave refraction); and (b) atmospheric ducting (resulting from temperature inversion inducing an air layer with higher dielectric constant than the surrounding air).

strength and in geometry.

### Summary of Fading

The fades of most concern in a mobile wireless system are the deep fades resulting from destructive interference of multiple reflections. These fades vary rapidly (over a few milliseconds) if a handset is moving at vehicular speeds but occur slowly when the transmitter and receiver are fixed. Fades can be viewed as deep amplitude modulation, and so so modulation is restricted to phase shift keying schemes when a transmitter and receive moving at vehicular speeds relative to each other.

## 4.6.4 Link Loss and Path Loss

With transmit and receive antennas included in the RF link, the usual case, link loss is defined as the ratio of the power input to the transmit antenna (\(P_{T}\)) to the power delivered by the receive antenna (\(P_{R}\)). Rearranging Equation (4.5.18) and taking logarithms yields the total line-of-sight (LOS) link loss, \(L_{\text{LINK, LOS}}\), between the input of the transmit antenna and the output of the receive antenna separated by distance \(d\) is (in decibels):

\[\begin{align} \label{eq:3} L_{\text{LINK, LOS}}|_{\text{dB}}&=10\log\left(\frac{P_{T}}{P_{R}}\right)=10\log\left(\frac{P_{T}}{P_{D}A_{R}}\right) \\ \label{eq:4}&=10\log\left[P_{T}\left(\frac{4\pi d^{2}}{P_{T}G_{T}}\right)\left(\frac{4\pi}{\lambda^{2}G_{R}}\right)\right] \\ \label{eq:5}&=10\log\left[\left(\frac{1}{G_{T}G_{R}}\right)\left(\frac{4\pi d}{\lambda}\right)^{2}\right] \\ \label{eq:6} &=-10\log G_{T}-10\log G_{R}+20\log\left(\frac{4\pi d}{\lambda}\right)\end{align} \]

The last term includes \(d\) and is called the LOS path loss (in decibels):

\[\label{eq:7}L_{\text{PATH, LOS}}|_{\text{dB}}=20\log\left(\frac{4\pi d}{\lambda}\right) \]

This is the preferred form of the expression for path loss, as it can be used directly in calculating link loss using the antenna gains of the transmit and receive antennas without the exercise of calculating the **effective aperture size** of the receive antenna.

An alternative definition of path loss comes directly from Equation (4.5.11) and is called the LOS path loss of the first kind:

\[\label{eq:8}^{1}L_{\text{PATH, LOS}}=4\pi d^{2} \]

but this is not commonly used.

Multipath effects result in losses that are proportional to \(d^{n}\) [14, 15] so that the general path loss, including multipath effects, is (in decibels)

\[\begin{align} L_{\text{PATH}}|_{\text{dB}}&=L_{\text{PATH, LOS}}|_{\text{dB}}+\text{excess loss}|_{\text{dB}} \nonumber \\ &=20\log\left(\frac{4\pi d}{\lambda}\right)+10(n-2)\log\left(\frac{d}{1\text{ m}}\right)\nonumber \\ &=20\log\left[\frac{4\pi (1\text{ m})}{\lambda}\right]+10(2)\log\left(\frac{d}{1\text{ m}}\right)+10(n-2)\log\left(\frac{d}{1\text{ m}}\right)\nonumber \\ \label{eq:9}&=10n\log [d/(1\text{ m})]+C\end{align} \]

where the distance \(d\) and wavelength \(\lambda\) are in meters, and \(C\) is a constant that captures the effect of wavelength. Here,

\[\label{eq:10}C=20\log [4\pi (1\text{ m})/\lambda ] \]

Combining this with Equation \(\eqref{eq:6}\) yields the link loss between the input of the transmit antenna and the output of the receive antenna:

\[\label{eq:11}L_{\text{LINK}}|_{\text{dB}}=-G_{T}|_{\text{dB}}-G_{R}|_{\text{dB}}+10n\log [d/(1\text{ m})]+C \]

As you can imagine, a few constants, here \(n\) and \(C\), cannot capture the full complexity of the propagation environment. Many models have been developed to capture particular environments better and incorporate mast height, experimental correction factors, and statistical parameters.

The path loss between two antennas is exactly the same in both directions when the frequency of the signal in each direction is the same. This is radio link reciprocity. Many communication systems use different frequencies in the two directions, and then the links are not reciprocal.

Example \(\PageIndex{1}\): Power Density

A communication system operating in a dense urban environment has a power density rolloff of \(1/d^{3.5}\) between the base station transmit antenna and the mobile receive antenna. At \(10\text{ m}\) from the transmit antenna, the power density is \(0.3167\text{ W/m}^{2}\). What is the power density at the receive antenna located at \(1\text{ km}\) from the base station?

**Solution**

\(P_{D}(10\text{ m}) = 0.3167\text{ W/m}^{2}\) and let \(d_{c} = 10\text{ m}\), so at \(d = 1\text{ km}\), the power density \(P_{D}(1\text{ km})\) is obtained from

\[\label{eq:12}\frac{P_{D}(1\text{ km})}{P_{D}(10\text{ m})}=\frac{d_{c}^{3.5}}{d^{3.5}}=\frac{10^{3.5}}{1000^{3.5}}=10^{-7} \]

so

\[\label{eq:13}P_{D}(1\text{ km})=P_{D}(10\text{ m})\cdot 10^{-7}=31.7\text{ nW/m}^{2} \]

Example \(\PageIndex{2}\): Link Loss

A \(5.6\text{ GHz}\) communication system uses a transmit antenna with an antenna gain \(G_{T}\) of \(35\text{ dB}\) and a receive antenna with an antenna gain \(G_{R}\) of \(6\text{ dB}\). If the distance between the antennas is \(200\text{ m}\), what is the link loss if the power density reduces as \(1/d^{3}\)? The link loss here is between the input to the transmit antenna and the output from the receive antenna.

**Solution**

The link loss is provided by Equation \(\eqref{eq:11}\),

\[L_{\text{LINK}}|_{\text{dB}}=-G_{T}-G_{R}+10n\log [d/(1\text{ m})]+C\nonumber \]

and \(C\) comes from Equation \(\eqref{eq:10}\), where \(\lambda =5.36\text{ cm}\). So

\[C=20\log\left(\frac{4\pi}{\lambda}\right)=20\log\left(\frac{4\pi}{0.0536}\right)=47.4\text{ dB}\nonumber \]

With \(n=3\) and \(d=200\text{ m}\),

\[L_{\text{LINK}}|_{\text{dB}}=-35-6+10\cdot 3\cdot\log (200)+47.4\text{ dB}=75.4\text{ dB}\nonumber \]

Example \(\PageIndex{3}\): Radiated Power Density

In free space, radiated power density drops off with distance \(d\) as \(1/d^{2}\). However, in a terrestrial environment there are multiple paths between a transmitter and a receiver, with the dominant paths being the direct LOS path and the path involving reflection off the ground. Reflection from the ground partially cancels the signal in the direct path, and in a semi-urban environment results in an attenuation loss of \(40\text{ dB}\) per decade of distance (instead of the \(20\text{ dB}\) per decade of distance roll-off in free space). Consider a transmitter that has a power density of \(1\text{ W/m}^{2}\) at a distance of \(1\text{ m}\) from the transmitter.

- The power density falls off as \(1/d^{n}\), where \(d\) is distance and \(n\) is an index. What is \(n\)?
- At what distance from the transmit antenna will the power density reach \(1\:\mu\text{W}\cdot\text{m}^{-2}\)?

**Solution**

- Power drops off by \(40\text{ dB}\) per decade of distance. \(40\text{ dB}\) corresponds to a factor of \(10,000 (= 10^{4})\). So, at distance \(d\), the power density \(P_{D}(d) = k/d^{n}\) (\(k\) is a constant). At a decade of distance, \(10d\), \(P_{D}(10d) = k/(10d)^{n} = P_{D}(d)/10000\), thus

\[\frac{k}{10^{n}d^{n}}=\frac{1}{10,000}\frac{k}{d^{n}};\quad 10^{n}=10,000\Rightarrow n=4\nonumber \] - At \(d=1\text{ m}\), \(P_{D}(1\text{ m})=1\text{ W/m}^{2}\). At a distance \(x\),

\[P_{D}(x)=1\:\mu\text{W/m}^{2}=\frac{k}{x^{4}}\text{m}^{2}=\frac{k}{x^{4}}\to x^{4}=\frac{1}{10^{-6}}\quad\text{and so}\quad x=31.6\text{ m}\nonumber \]

## 4.6.5 Fresnel Zones

In long-distance wireless communications from one fixed site to another, the intent is to use the LOS path and avoid reflections. Such systems are called **point-to-point links**. Thus avoiding reflections is an important consideration in design. The dominant propagation paths in a point-to-point system are shown in Figure \(\PageIndex{9}\). The refracted wave path arises because of density variations in the air producing a permittivity profile that varies with height, as shown in Figure \(\PageIndex{10}\)(a). This effect is called **beam bending**. The other important propagation path to consider is the reflected wave from the ground, which can be important if the ground is too close to the propagation path. Both of these effects will be considered in this section.

As radio waves propagate, they spread out in a plane perpendicular to the direction of propagation, the power density of the radio waves then reduces with distance from the centerline. One of the consequences of this is that an obstruction that is not in the LOS path can still interfere with signal propagation. The appropriate clearance is determined from the Fresnel zones, which are shown in Figure \(\PageIndex{11}\). The direct LOS path between the antennas has a length \(d\). If there is a reflecting object near the LOS path, then there can be a second path between the transmit and receive antennas. The path from the first antenna to the circle defined by the first Fresnel zone and then to the second antenna has a path length \(d+\lambda /2\), and so at the receive antenna this signal is \(180^{\circ}\) out of phase with the LOS signal and there will be

Figure \(\PageIndex{9}\): Three point-to-point characteristic propagation paths: line of sight, reflection, and refraction.

Figure \(\PageIndex{10}\): Beam bending by density variation in air: (a) refraction index profile with the air density reducing with height; and (b) incorporating beam bending in a curved-earth model.

Figure \(\PageIndex{11}\): Fresnel zones in a plane perpendicular to the LOS path, which has length \(d\).

cancelation. The radius of the \(n\)th Fresnel zone at a point \(P\) is

\[\label{eq:14}F_{n}=\sqrt{\frac{n\lambda d_{1}d_{2}}{d_{1}+d_{2}}} \]

where \(d_{1}\) is the distance from the first antenna to \(P\), \(d_{2}\) is the distance from the second antenna to \(P\) (so \(d = d_{1} + d_{2}\)), and \(\lambda\) is the wavelength of the propagating signal. Ninety percent of the energy in the wave is in the first Fresnel zone. A guideline is that an obstacle should be separated from the direct path by a distance more than the radius of the first Fresnel zone. Antenna heights are increased so that the beam achieves one or more Fresnel zone clearances.

The above analysis can be used even with beam bending. A convenient way of accommodating beam bending is to use a curved-earth model, as shown in Figure \(\PageIndex{10}\)(b), so that subsequent calculations can use LOS considerations [16]. The amount of beam bending to account for comes from experimental surveys. In Figure \(\PageIndex{10}\)(b) the original clearance from a hill to the first Fresnel zone is \(z\). With beam bending the clearance increases to \(y\).

Since a receive antenna is unlikely to be large enough to capture all of the energy contained in the first Fresnel zone, the effect of signal blocking by an obstruction that encroaches the first Fresnel zone is of little concern. What is of concern is the destructive combining of a reflected signal with the signal in the main LOS beam.

Example \(\PageIndex{4}\): Fresnel Zone Clearance

A transmit antenna and a receive antenna are separated by \(10\text{ km}\) and operate at \(2\text{ GHz}\).

- What is the radius of the first Fresnel zone?
- What is the radius of the second Fresnel zone?
- To ensure LOS propagation, what should the clearance be from the direct line between the antennas and obstructions such as hills and vegetation?

**Solution**

The radius of the Fresnel zone is calculated at the midpoint so that \(d_{1} = d_{2} = d/2 = 5\text{ km}\). Also \(f = 2\text{ GHz},\:\lambda = 15\text{ cm}\).

Figure \(\PageIndex{12}\)

- The radius of the first Fresnel zone is, from Equation \(\eqref{eq:14}\),

\[r_{1}=F_{1}=\sqrt{\frac{\lambda d_{1}d_{2}}{d_{1}+d_{2}}}=\sqrt{\frac{\lambda d}{4}}=\sqrt{\frac{(0.15)(10^{4})}{4}}=19.36\text{ m}\nonumber \] - At the midpoint the radius of the second Fresnel zone is

\[r_{2}=F_{2}=\sqrt{\frac{2\lambda d_{1}d_{2}}{d_{1}+d_{2}}}=\sqrt{\frac{\lambda d}{2}}=\sqrt{\frac{(0.15)(10^{4})}{2}}=27.39\text{ m}\nonumber \] - Ninety percent of the energy in the beam is contained within the first Fresnel zone of radius \(r_{1}\). Obstructions within the first Fresnel zone will result in a significant fraction of the beam being blocked. In addition, reflections from the obstruction could destructively combine with the main beam and reduce the received signal level even beyond the reduction caused by part of the beam being blocked. Two criteria are commonly used to determine the clearance required to avoid signal obstruction.

One criterion used is that the minimum clearance between the direct beam and an obstruction is \(0.6r_{1}\). So the minimum clearance is \(0.6r_{1} = 11.6\text{ m}\).

A more conservative criterion is that the minimum clearance should be \(r_{1} = 19.36\text{ m}\).

Figure \(\PageIndex{13}\)

## 4.6.6 Propagation Model in the Mobile Environment

RF propagation in the mobile environment cannot be accurately derived. Instead, a fit to measurements is often used. One of the models is the Okumura–Hata model [17], which calculates the path loss as

\[\label{eq:15}L_{\text{PATH}}|_{(\text{dB})}=69.55 + 26.16\log f − 13.82\log H + (44.9 − 6.55\log H)\cdot\log d + c \]

where \(f\) is the frequency (in \(\text{MHz}\)), \(d\) is the distance between the base station and terminal (in \(\text{km}\)), \(H\) is the effective height of the base station antenna (in \(\text{m}\)), and \(c\) is an environment correction factor (\(c = 0\text{ dB}\) in a dense urban area, \(c = −5\text{ dB}\) in an urban area, \(c = −10\text{ dB}\) in a suburban area, and \(c = −17\text{ dB}\) in a rural area, for \(f = 1\text{ GHz}\) and \(H=1.5\text{ m}\)).

More sophisticated characterization of the propagation environment uses ray-tracing models to follow individual propagation paths. The ray-tracing models are based on deterministic methods using terrain data and calculate paths accounting for obstruction and reflection analyses. Each refraction and reflection event is characterized either experimentally or through detailed EM simulations. Appropriate algorithms are applied for best compliance with radio physics. Commonly required inputs to these models include frequency, distance from the transmitter to the receiver, effective base station height, obstacle height and geometry, radius of the first Fresnel zone, forest height/roof height, distance between buildings, arbitrary loss allowances based on land use (forest, water, etc.), and loss allowances for penetration of buildings and vehicles. Such a technique was used to calculate the radio coverage diagrams on the inside front cover of this book.

There are many propagation models for different frequency ranges and different environments. The considerable effort put into developing reliable models is because being able to predict signal coverage is essential to efficient design of basestation layout.