The following may be UNCORRECTED PAGE PROOFS. PART … · 2019. 10. 12. · 100. CHAPTER 5 /...

Wireless CommunicationTechnology

CHAPTER5ANTENNAS ANDPROPAGATION

5.1 Antennas

Radiation PatternsAntenna TypesAntenna Gain

5.2 Propagation Modes

Ground Wave PropagationSky Wave PropagationLine-of-Sight Propagation

5.3 Line-of-Sight Transmission

AttenuationFree Space LossNoiseThe Expression Eb/N0Atmospheric AbsorptionMultipathRefraction

5.4 Fading in the Mobile Environment

Multipath PropagationError Compensation Mechanisms

5.5 Recommended Reading

5.6 Key Terms, Review Questions, and Problems

PARTTWO

05 StallingsIII 5/15/01 1:17 PM Page 99

Prentice Hall

The following may be UNCORRECTED PAGE PROOFS.

100 CHAPTER 5 / CONCURRENCY: MUTUAL EXCLUSION AND SYNCHRONIZATION

This chapter provides some fundamental background for wireless transmission.We begin with an overview of antennas and then look at signal propagation.

5.1 ANTENNAS

An antenna can be defined as an electrical conductor or system of conductors usedeither for radiating electromagnetic energy into space or for collecting electromag-netic energy from space. For transmission of a signal, radio-frequency electricalenergy from the transmitter is converted into electromagnetic energy by the antennaand radiated into the surrounding environment (atmosphere, space, water). Forreception of a signal, electromagnetic energy impinging on the antenna is convertedinto radio-frequency electrical energy and fed into the receiver.

In two-way communication, the same antenna can be and often is used forboth transmission and reception. This is possible because any antenna transfersenergy from the surrounding environment to its input receiver terminals with thesame efficiency that it transfers energy from the output transmitter terminals intothe surrounding environment, assuming that the same frequency is used in bothdirections. Put another way, antenna characteristics are essentially the same whetheran antenna is sending or receiving electromagnetic energy.

Radiation Patterns

An antenna will radiate power in all directions but, typically, does not performequally well in all directions. A common way to characterize the performance of anantenna is the radiation pattern, which is a graphical representation of the radiationproperties of an antenna as a function of space coordinates. The simplest pattern isproduced by an idealized antenna known as the isotropic antenna. An isotropicantenna is a point in space that radiates power in all directions equally. The actualradiation pattern for the isotropic antenna is a sphere with the antenna at the cen-ter. However, radiation patterns are almost always depicted as a two-dimensionalcross section of the three-dimensional pattern. The pattern for the isotropic antennais shown in Figure 5.1a. The distance from the antenna to each point on the radia-tion pattern is proportional to the power radiated from the antenna in that direc-tion. Figure 5.1b shows the radiation pattern of another idealized antenna. This is adirectional antenna in which the preferred direction of radiation is along one axis.

The actual size of a radiation pattern is arbitrary. What is important is the rel-ative distance from the antenna position in each direction. The relative distancedetermines the relative power. To determine the relative power in a given direction,a line is drawn from the antenna position at the appropriate angle, and the point ofintercept with the radiation pattern is determined. Figure 5.1 shows a comparisonof two transmission angles, A and B, drawn on the two radiation patterns. Theisotropic antenna produces an omnidirectional radiation pattern of equal strengthin all directions, so the A and B vectors are of equal length. For the Hertz antenna,the B vector is longer than the A vector, indicating that more power is radiated in


5.1 / ANTENNAS 101

(a) Omnidirectional

A

B

(b) Directional

B

A

Antenna location

Figure 5.1 Idealized Radiation Patterns

λ/2

λ/4

(a) Half-wave dipole

(b) Quarter-wave antenna

Figure 5.2 Simple Antennas

the B direction than in the A direction, and the relative lengths of the two vectorsare proportional to the amount of power radiated in the two directions.

The radiation pattern provides a convenient means of determining the beamwidth of an antenna, which is a common measure of the directivity of an antenna.The beam width, also referred to as the half-power beam width, is the angle withinwhich the power radiated by the antenna is at least half of what it is in the most pre-ferred direction.

When an antenna is used for reception, the radiation pattern becomes areception pattern. The longest sections of the pattern indicates the best directionfor reception.

Antenna Types

Dipoles

Two of the simplest and most basic antennas are the half-wave dipole, orHertz, antenna (Figure 5.2a) and the quarter-wave vertical, or Marconi, antenna(Figure 5.2b). The half-wave dipole consists of two straight collinear conductors ofequal length, separated by a small feeding gap. The length of the antenna is one-half



Side view (zy-plane)

(a) Simple dipole

(b) Directed antenna

z

y

Side view (zy-plane)

z

y

Top view (xz-plane)

x

z

Top view (xz-plane)

x

z

Side view (xy-plane)

x

y

Side view (xy-plane)

x

y

Figure 5.3 Radiation Patterns in Three Dimensions [SCHI00]

the wavelength of the signal that can be transmitted most efficiently. A verticalquarter-wave antenna is the type commonly used for automobile radios and port-able radios.

A half-wave dipole has a uniform or omnidirectional radiation pattern in onedimension and a figure eight pattern in the other two dimensions (Figure 5.3a).More complex antenna configurations can be used to produce a directional beam.A typical directional radiation pattern is shown in Figure 5.3b. In this case the mainstrength of the antenna is in the x direction.

Parabolic Reflective Antenna

An important type of antenna is the parabolic reflective antenna, which is usedin terrestrial microwave and satellite applications. You may recall from your pre-college geometry studies that a parabola is the locus of all points equidistant froma fixed line and a fixed point not on the line. The fixed point is called the focus andthe fixed line is called the directrix (Figure 5.4a). If a parabola is revolved about itsaxis, the surface generated is called a paraboloid. A cross section through the parab-oloid parallel to its axis forms a parabola and a cross section perpendicular to theaxis forms a circle. Such surfaces are used in headlights, optical and radio telescopes,and microwave antennas because of the following property: If a source of electro-magnetic energy (or sound) is placed at the focus of the paraboloid, and if the par-aboloid is a reflecting surface, then the wave will bounce back in lines parallel tothe axis of the paraboloid; Figure 5.4b shows this effect in cross section. In theory,this effect creates a parallel beam without dispersion. In practice, there will be somedispersion, because the source of energy must occupy more than one point. The con-verse is also true. If incoming waves are parallel to the axis of the reflecting par-aboloid, the resulting signal will be concentrated at the focus.


5.1 / ANTENNAS 103

y

a

ab

bc

f f

c

x

Dir

ectr

ix

Focus

(a) Parabola (b) Cross section of parabolic antennashowing reflective property

(c) Cross section of parabolic antennashowing radiation pattern

Figure 5.4 Parabolic Reflective Antenna

Figure 5.4c shows a typical radiation pattern for the parabolic reflectiveantenna, and Table 5.1 lists beam widths for antennas of various sizes at a frequencyof 12 GHz. Note that the larger the diameter of the antenna, the more tightly direc-tional is the beam.

Antenna Gain

Antenna gain is a measure of the directionality of an antenna. Antenna gain isdefined as the power output, in a particular direction, compared to that producedin any direction by a perfect omnidirectional antenna (isotropic antenna). For exam-ple, if an antenna has a gain of 3 dB, that antenna improves upon the isotropicantenna in that direction by 3 dB, or a factor of 2. The increased power radiated in



Table 5.1 Antenna Beamwidths for Various DiameterParabolic Reflective Antennas at f � 12 GHz [FREE97]

Antenna Diameter (m) Beam Width (degrees)

0.5 3.5

0.75 2.33

1.0 1.75

1.5 1.166

2.0 0.875

2.5 0.7

5.0 0.35

Table 5.2 Antenna Gains and Effective Areas [COUC01]

Power Gain (relative Type of Antenna Effective Area Ae (m2) to isotropic)

Isotropic �2⁄4� 1

Infinitesimal dipole or loop 1.5 �2⁄4� 1.5

Half-wave dipole 1.64 �2/4� 1.64

Horn, mouth area A 0.81 A 10 A/�2

Parabolic, face area A 0.56 A 7 A/�2

Turnstile (two crossed, 1.15 �2⁄4� 1.15perpendicular dipoles)

a given direction is at the expense of other directions. In effect, increased poweris radiated in one direction by reducing the power radiated in other directions. It isimportant to note that antenna gain does not refer to obtaining more output powerthan input power but rather to directionality.

A concept related to that of antenna gain is the effective area of an antenna.The effective area of an antenna is related to the physical size of the antenna and toits shape. The relationship between antenna gain and effective area is

(5.1)

where

G � antenna gainAe � effective area

f � carrier frequencyc � speed of light (� 3 � 108 m/s)� � carrier wavelength

Table 5.2 shows the antenna gain and effective area of some typical antennashapes.

G �4pAe

l2 �4pf2Ae

c2


5.2 / PROPAGATION MODES 105

Example. For a parabolic reflective antenna with a diameter of 2 m, operating at 12 GHz,what is the effective area and the antenna gain? We have an area of A � �r2 � � and aneffective area of Ae � 0.56�. The wavelength is � � c / f � (3 � 108) / (12 � 109) � 0.025 m.Then

G � (7A) / �2 � (7 � �) / (0.025)2 � 35,186GdB� 45.46 dB

5.2 PROPAGATION MODES

A signal radiated from an antenna travels along one of three routes: ground wave,sky wave, or line of sight (LOS). Table 5.3 shows in which frequency range each pre-dominates. In this book, we are almost exclusively concerned with LOS communi-cation, but a short overview of each mode is given in this section.

Ground Wave Propagation

Ground wave propagation (Figure 5.5a) more or less follows the contour of theearth and can propagate considerable distances, well over the visual horizon. Thiseffect is found in frequencies up to about 2 MHz. Several factors account for thetendency of electromagnetic wave in this frequency band to follow the earth’s cur-vature. One factor is that the electromagnetic wave induces a current in the earth’ssurface, the result of which is to slow the wavefront near the earth, causing the wave-front to tilt downward and hence follow the earth’s curvature. Another factor is dif-fraction, which is a phenomenon having to do with the behavior of electromagneticwaves in the presence of obstacles.

Electromagnetic waves in this frequency range are scattered by the atmos-phere in such a way that they do not penetrate the upper atmosphere.

The best-known example of ground wave communication is AM radio.

Sky Wave Propagation

Sky wave propagation is used for amateur radio, CB radio, and international broad-casts such as BBC and Voice of America. With sky wave propagation, a signal froman earth-based antenna is reflected from the ionized layer of the upper atmosphere(ionosphere) back down to earth. Although it appears the wave is reflected from theionosphere as if the ionosphere were a hard reflecting surface, the effect is in factcaused by refraction. Refraction is described subsequently.

A sky wave signal can travel through a number of hops, bouncing back andforth between the ionosphere and the earth’s surface (Figure 5.5b). With this propa-gation mode, a signal can be picked up thousands of kilometers from the transmitter.

Line-of-Sight Propagation

Above 30 MHz, neither ground wave nor sky wave propagation modes operate, andcommunication must be by line of sight (Figure 5.5c). For satellite communication,


106

Tab

le5.

3F

requ

ency

Ban

ds

Fre

e-Sp

ace

Ban

dF

requ

ency

Ran

geW

avel

engt

h R

ange

Pro

paga

tion

Cha

ract

eris

tics

Typ

ical

Use

EL

F (

extr

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to

300

Hz

10,0

00 t

o 1,

000

kmG

WP

ower

line

fre

quen

cies

; use

d by

som

e ho

me

low

fre

quen

cy)

cont

rol s

yste

ms.

VF

(vo

ice

300

to 3

000

Hz

1,00

0 to

100

km

GW

Use

d by

the

tel

epho

ne s

yste

m f

or a

nalo

g fr

eque

ncy)

subs

crib

er li

nes.

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very

3

to 3

0 kH

z10

0 to

10

kmG

W; l

ow a

tten

uati

on d

ay a

nd n

ight

; L

ong-

rang

e na

viga

tion

; sub

mar

ine

low

fre

quen

cy)

high

atm

osph

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noi

se le

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com

mun

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30

to

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10 t

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100

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107

Earth

Ionosphere

(b) Sky wave propagation (2 to 30 MHz)

Transmitantenna

Receiveantenna

Sign

alpr

opag

atio

n

Earth

(a) Ground wave propagation (below 2 MHz)

Transmitantenna

Receiveantenna

Signalpropagation

Earth

(c) Line-of-sight (LOS) propagation (above 30 MHz)

Transmitantenna

Receiveantenna

Signalpropagation

Figure 5.5 Wireless Propagation Modes



a signal above 30 MHz is not reflected by the ionosphere and therefore a signal canbe transmitted between an earth station and a satellite overhead that is not beyondthe horizon. For ground-based communication, the transmitting and receivingantennas must be within an effective line of sight of each other. The term effective isused because microwaves are bent or refracted by the atmosphere. The amount andeven the direction of the bend depends on conditions, but generally microwaves arebent with the curvature of the earth and will therefore propagate farther than theoptical line of sight.

Refraction

Before proceeding, a brief discussion of refraction is warranted. Refractionoccurs because the velocity of an electromagnetic wave is a function of the densityof the medium through which it travels. In a vacuum, an electromagnetic wave (suchas light or a radio wave) travels at approximately 3 � 108 m/s. This is the constant,c, commonly referred to as the speed of light, but actually referring to the speed oflight in a vacuum. In air, water, glass, and other transparent or partially transparentmedia, electromagnetic waves travel at speeds less than c.

When an electromagnetic wave moves from a medium of one density to amedium of another density, its speed changes. The effect is to cause a one-timebending of the direction of the wave at the boundary between the two media. Thisis illustrated in Figure 5.6. If moving from a less dense to a more dense medium,the wave will bend toward the more dense medium. This phenomenon is easilyobserved by partially immersing a stick in water. The result will look much like Fig-ure 5.6, with the stick appearing shorter and bent.

Area of lowerrefractive index

Incidentdirection

Refracteddirection

Area of higherrefractive index

Figure 5.6 Refraction of an Electromagnetic Wave [POOL98]

05 StallingsIII 5/15/01 1:17 PM 8

5.2 / PROPAGATION MODES 109

Earth

Optical horizon

Radio horizon

Antenna

Figure 5.7 Optical and Radio Horizons

The index of refraction of one medium relative to another is the sine of theangle of incidence divided by the sine of the angle of refraction. The index of refrac-tion is also equal to the ratio of the respective velocities in the two media. Theabsolute index of refraction of a medium is calculated in comparison with that of avacuum. Refractive index varies with wavelength, so that refractive effects differ forsignals with different wavelengths.

Although Figure 5.6 shows an abrupt, one-time change in direction as a signalmoves from one medium to another, a continuous, gradual bending of a signal willoccur if it is moving through a medium in which the index of refraction graduallychanges. Under normal propagation conditions, the refractive index of the atmos-phere decreases with height so that radio waves travel more slowly near the groundthan at higher altitudes. The result is a slight bending of the radio waves towardthe earth.

Optical and Radio Line of Sight

With no intervening obstacles, the optical line of sight can be expressed as

where d is the distance between an antenna and the horizon in kilometers and h isthe antenna height in meters. The effective, or radio, line of sight to the horizon isexpressed as (Figure 5.7)

where K is an adjustment factor to account for the refraction. A good rule ofthumb is K � 4/3. Thus, the maximum distance between two antennas for LOSpropagation is , where h1 and h2 are the heights of the twoantennas.

3.5712Kh1 � 2Kh2 2

d � 3.572Kh

d � 3.572h

Example. The maximum distance between two antennas for LOS transmission if oneantenna is 100 m high and the other is at ground level is

d � 3.572Kh � 3.572133 � 41 km


Now suppose that the receiving antenna is 10 m high. To achieve the same distance, howhigh must the transmitting antenna be? The result is

This is a savings of over 50 m in the height of the transmitting antenna. This example illus-trates the benefit of raising receiving antennas above ground level to reduce the necessaryheight of the transmitter.

41 � 3.5712Kh1 � 213.322Kh1 �413.57

� 213.3 � 7.84

h1 � 7.842>1.33 � 46.2 m


5.3 LINE-OF-SIGHT TRANSMISSION

With any communications system, the signal that is received will differ from the sig-nal that is transmitted, due to various transmission impairments. For analog signals,these impairments introduce various random modifications that degrade the signalquality. For digital data, bit errors are introduced: A binary 1 is transformed into abinary 0, and vice versa. In this section we examine the various impairments andcomment on their effect on the information-carrying capacity of a communicationslink. Our concern in this book is with LOS wireless transmission, and in this con-text, the most significant impairments are as follows:

• Attenuation and attenuation distortion• Free space loss• Noise• Atmospheric absorption• Multipath• Refraction

Attenuation

The strength of a signal falls off with distance over any transmission medium. Forguided media, this reduction in strength, or attenuation, is generally logarithmic andthus is typically expressed as a constant number of decibels per unit distance. Forunguided media, attenuation is a more complex function of distance and the makeupof the atmosphere. Attenuation introduces three factors for the transmission engineer:

1. A received signal must have sufficient strength so that the electronic circuitryin the receiver can detect and interpret the signal.

2. The signal must maintain a level sufficiently higher than noise to be receivedwithout error.

3. Attenuation is greater at higher frequencies, causing distortion.


5.3 / LINE-OF-SITE TRANSMISSION 111

The first and second factors are dealt with by attention to signal strength andthe use of amplifiers or repeaters. For a point-to-point link, the signal strength ofthe transmitter must be strong enough to be received intelligibly, but not so strongas to overload the circuitry of the transmitter or receiver, which would cause dis-tortion. Beyond a certain distance, the attenuation becomes unacceptably great, andrepeaters or amplifiers are used to boost the signal at regular intervals. These prob-lems are more complex when there are multiple receivers, where the distance fromtransmitter to receiver is variable.

The third factor is known as attenuation distortion. Because the attenuationvaries as a function of frequency, the received signal is distorted, reducing intelligi-bility. Specifically, the frequency components of the received signal have differentrelative strengths than the frequency components of the transmitted signal. To over-come this problem, techniques are available for equalizing attenuation across a bandof frequencies. One approach is to use amplifiers that amplify high frequencies morethan lower frequencies.

Free Space Loss

For any type of wireless communication the signal disperses with distance. There-fore, an antenna with a fixed area will receive less signal power the farther it is fromthe transmitting antenna. For satellite communication this is the primary mode ofsignal loss. Even if no other sources of attenuation or impairment are assumed, atransmitted signal attenuates over distance because the signal is being spread overa larger and larger area. This form of attenuation is known as free space loss, whichcan be express in terms of the ratio of the radiated power Pt to the power Pr receivedby the antenna or, in decibels, by taking 10 times the log of that ratio. For the idealisotropic antenna, free space loss is

where

Pt� signal power at the transmitting antennaPr�signal power at the receiving antenna�� carrier wavelengthd� propagation distance between antennasc� speed of light (3 � 108 m/s)

where d and � are in the same units (e.g., meters).This can be recast as

(5.2)LdB � 10 log

Pt

Pr� 20 loga4pd

lb � � 20 log 1l2 � 20 log 1d2 � 21.98 dB

� 20 loga4pfd

cb � 20 log 1f2 � 20 log 1d2 � 147.56 dB

Pt

Pr�14pd22l2 �

14pfd22c2



601 5 10

Distance (km)

Los

s (d

B)

f � 30 MHz

f � 300 MHz

f � 3 GHz

f � 30 GHz

f � 300 GHz

50 100

70

80

90

100

110

120

130

140

150

160

170

180

Figure 5.8 Free Space Loss

1As was mentioned in Appendix 2A, there is some inconsistency in the literature over the use of the termsgain and loss. Equation (5.2) follows the convention of Equation (2.2).

Figure 5.8 illustrates the free space loss equation.1

For other antennas, we must take into account the gain of the antenna, whichyields the following free space loss equation:

where

Gt � gain of the transmitting antennaGr � gain of the receiving antennaAt � effective area of the transmitting antennaAr � effective area of the receiving antenna

Pt

Pr�14p221d22GrGtl

2 �1ld22ArAt

�1cd22

f2ArAt



The third fraction is derived from the second fraction using the relationshipbetween antenna gain and effective area defined in Equation (5.1). We can recastthis equation as

LdB � 20 log (�) � 20 log (d) � 10 log (AtAr)��20 log (f) � 20 log (d) � 10 log (AtAt) �169.54 dB (5.3)

Thus, for the same antenna dimensions and separation, the longer the carrier wave-length (lower the carrier frequency f), the higher is the free space path loss. It isinteresting to compare Equations (5.2) and (5.3). Equation (5.2) indicates that as thefrequency increases, the free space loss also increases, which would suggest thatat higher frequencies, losses become more burdensome. However, Equation (5.3)shows that we can easily compensate for this increased loss with antenna gains. Infact, there is a net gain at higher frequencies, other factors remaining constant.Equation (5.2) shows that at a fixed distance an increase in frequency results in anincreased loss measured by 20 log(f). However, if we take into account antennagain, and fix antenna area, then the change in loss is measured by �20 log(f); thatis, there is actually a decrease in loss at higher frequencies.

Example. Determine the isotropic free space loss at 4 GHz for the shortest path to asynchronous ssttelite from earth (35,863 km). At 4 GHz, the wavelength is (3 � 108) / (4 � 109) � 0.075 m. Then

LdB � �20 log (0.075) � 20 log(35.853 � 106) � 21.98 � 195.6 dB

Now consider the antenna gain of both the satellite- and ground-based antennas. Typicalvalues are 44 dB and 48 dB, respectively. The free space loss is

LdB � 195.6 � 44 � 48 � 103.6 dB

Now assume a transmit power of 250 W at the earth station. What is the power receivedat the satellite antenna? A power of 250 W translates into 24 dBW, so the power at thereceiving antenna is 24 � 103.6 � �79.6 dBW.

Noise

For any data transmission event, the received signal will consist of the transmittedsignal, modified by the various distortions imposed by the transmission system, plusadditional unwanted signals that are inserted somewhere between transmission andreception. These unwanted signals are referred to as noise. Noise is the major lim-iting factor in communications system performance.

Noise may be divided into four categories:

• Thermal noise• Intermodulation noise• Crosstalk• Impulse noise



Thermal noise is due to thermal agitation of electrons. It is present in all elec-tronic devices and transmission media and is a function of temperature. Thermalnoise is uniformly distributed across the frequency spectrum and hence is oftenreferred to as white noise. Thermal noise cannot be eliminated and therefore placesan upper bound on communications system performance. Because of the weaknessof the signal received by satellite earth stations, thermal noise is particularly signif-icant for satellite communication.

The amount of thermal noise to be found in a bandwidth of 1 Hz in any deviceor conductor is

N0 � kT (W/Hz)

where2

N0 �noise power density in watts per 1 Hz of bandwidthk �Boltzmann’s constant � 1.3803 � 10�23 J/KT �temperature, in kelvins (absolute temperature)

Example. Given a receiver with an effective noise temperature of 294 K and a 10-MHzbandwidth, the thermal noise level at the receiver’s output is

N � �228.6 dBW � 10 log(294) � 10 log 107

� �228.6 � 24.7 � 70� �133.9 dBW

The noise is assumed to be independent of frequency. Thus the thermal noisein watts present in a bandwidth of B Hertz can be expressed as

N � kTB

or, in decibel-watts,

N � 10 log k � 10 log T � 10 log B� �228.6 dBW � 10 log T � 10 log B

Example. Room temperature is usually specified as T � 17°C, or 290 K. At this temper-ature, the thermal noise power density is

N0 � (1.3803 � 10�23) � 290 � 4 � 10�21 W/Hz � �204 dBW/Hz

where dBW is the decibel-watt, defined in Appendix 2A.

2A Joule (J) is the International System (SI) unit of electrical, mechanical, and thermal energy. A wattis the SI unit of power, equal to one joule per second. The kelvin (K) is the SI unit of thermodynamictemperature. For a temperature in degrees kelvin of T, the corresponding temperature in degreesCelsius is equal to T � 273.15.



When signals at different frequencies share the same transmission medium,the result may be intermodulation noise. Intermodulation noise produces signals ata frequency that is the sum or difference of the two original frequencies or multi-ples of those frequencies. For example, the mixing of signals at frequencies f1 and f2might produce energy at the frequency f1 � f2. This derived signal could interferewith an intended signal at the frequency f1 � f2.

Intermodulation noise is produced when there is some nonlinearity in thetransmitter, receiver, or intervening transmission system. Normally, these compo-nents behave as linear systems; that is, the output is equal to the input times aconstant. In a nonlinear system, the output is a more complex function of the input.Such nonlinearity can be caused by component malfunction, the use of excessive sig-nal strength, or just the nature of the amplifiers used. It is under these circumstancesthat the sum and difference frequency terms occur.

Crosstalk has been experienced by anyone who, while using the telephone, hasbeen able to hear another conversation; it is an unwanted coupling between signalpaths. It can occur by electrical coupling between nearby twisted pairs or, rarely,coax cable lines carrying multiple signals. Crosstalk can also occur when unwantedsignals are picked up by microwave antennas; although highly directional attennasare used, microwave energy does spread during propagation. Typically, crosstalk isof the same order of magnitude as, or less than, thermal noise. However, in the unli-censed ISM bands, crosstalk often dominates.

All of the types of noise discussed so far have reasonably predictable and rel-atively constant magnitudes. Thus it is possible to engineer a transmission system tocope with them. Impulse noise, however, is noncontinuous, consisting of irregularpulses or noise spikes of short duration and of relatively high amplitude. It is gen-erated from a variety of causes, including external electromagnetic disturbances,such as lightning, and faults and flaws in the communications system.

Impulse noise is generally only a minor annoyance for analog data. For exam-ple, voice transmission may be corrupted by short clicks and crackles with no loss ofintelligibility. However, impulse noise is the primary source of error in digital datatransmission. For example, a sharp spike of energy of 0.01 s duration would notdestroy any voice data but would wash out about 560 bits of data being transmittedat 56 kbps.

The Expression Eb/N0

Chapter 2 introduced the signal-to-noise ratio (SNR). There is a parameter relatedto SNR that is more convenient for determining digital data rates and error ratesand that is the standard quality measure for digital communication system perfor-mance. The parameter is the ratio of signal energy per bit to noise power densityper Hertz, Eb/N0. Consider a signal, digital or analog, that contains binary digitaldata transmitted at a certain bit rate R. Recalling that 1 watt � 1 J/s, the energy perbit in a signal is given by Eb � STb, where S is the signal power and Tb is the timerequired to send one bit. The data rate R is just R � 1/Tb. Thus

Eb

N0�

S>RN0

�S

kTR



or, in decibel notation,

The ratio Eb/N0 is important because the bit error rate for digital data is a (decreas-ing) function of this ratio. Given a value of Eb/N0 needed to achieve a desired errorrate, the parameters in the preceding formula may be selected. Note that as the bitrate R increases, the transmitted signal power, relative to noise, must increase tomaintain the required Eb/N0.

Let us try to grasp this result intuitively by considering again Figure 2.9. Thesignal here is digital, but the reasoning would be the same for an analog signal.In several instances, the noise is sufficient to alter the value of a bit. If the data ratewere doubled, the bits would be more tightly packed together, and the same pas-sage of noise might destroy two bits. Thus, for constant signal and noise strength, anincrease in data rate increases the error rate.

The advantage of Eb/N0 over SNR is that the latter quantity depends on thebandwidth.

aEb

N0b

dB� SdBW � 10 log R � 10 log k � 10 log T

� SdBW � 10 log R � 228.6 dBW � 10 log T

Example. Suppose a signal encoding technique requires that Eb/N0 � 8.4 dB for a bit errorrate of 10�4 (one bit error out of every 10,000). If the effective noise temperature is 290°K(room temperature) and the data rate is 2400 bps, what received signal level is requiredto overcome thermal noise?

We have8.4 � SdBW � 10 log 2400 � 228.6 dBW � 10 log 290

� SdBW � (10)(3.38) � 228.6 � (10)(2.46)S � �161.8 dBW

We can relate Eb/N0 to SNR as follows. We have

The parameter N0 is the noise power density in watts/hertz. Hence, the noise in asignal with bandwidth BT is N � N0BT. Substituting, we have

(5.4)

Another formulation of interest relates to Eb/N0 spectral efficiency. Recall,from Chapter 2, Shannon’s result that the maximum channel capacity, in bits persecond, obeys the equation

C � B log2 (1 � S/N)

Eb

N0�

S

N BT

R

Eb

N0�

S

N0R



where C is the capacity of the channel in bits per second and B is the bandwidth ofthe channel in Hertz. This can be rewritten as

Using Equation (5.4), and equating BT with B and R with C, we have

This is a useful formula that relates the achievable spectral efficiency C/B to Eb/N0.

Eb

N0�

B

C 12C>B � 12

S

N� 2C>B � 1

Example. Suppose we want to find the minimum Eb/N0 required to achieve a spectral effi-ciency of 6 bps/Hz. Then Eb/N0 � (1/6)(26 � 1) � 10.5 � 10.21 dB.

Atmospheric Absorption

An additional loss between the transmitting and receiving antennas is atmosphericabsorption. Water vapor and oxygen contribute most to attenuation. A peak atten-uation occurs in the vicinity of 22 GHz due to water vapor. At frequencies below 15GHz, the attenuation is less. The presence of oxygen results in an absorption peakin the vicinity of 60 GHz but contributes less at frequencies below 30 GHz. Rainand fog (suspended water droplets) cause scattering of radio waves that results inattenuation. This can be a major cause of signal loss. Thus, in areas of significantprecipitation, either path lengths have to be kept short or lower-frequency bandsshould be used.

Multipath

For wireless facilities where there is a relatively free choice of where antennas areto be located, they can be placed so that if there are no nearby interfering obstacles,there is a direct line-of-sight path from transmitter to receiver. This is generally thecase for many satellite facilities and for point-to-point microwave. In other cases,such as mobile telephony, there are obstacles in abundance. The signal can bereflected by such obstacles so that multiple copies of the signal with varying delayscan be received. In fact, in extreme cases, there may be no direct signal. Dependingon the differences in the path lengths of the direct and reflected waves, the com-posite signal can be either larger or smaller than the direct signal. Reinforcementand cancellation of the signal resulting from the signal following multiple paths canbe controlled for communication between fixed, well-sited antennas, and betweensatellites and fixed ground stations. One exception is when the path goes acrosswater, where the wind keeps the reflective surface of the water in motion. Formobile telephony and communication to antennas that are not well sited, multipathconsiderations can be paramount.

Figure 5.9 illustrates in general terms the types of multipath interferencetypical in terrestrial, fixed microwave and in mobile communications. For fixed



(a) Microwave line of sight

(b) Mobile radio

Figure 5.9 Examples of Multipath Interference

microwave, in addition to the direct line of sight, the signal may follow a curved paththrough the atmosphere due to refraction and the signal may also reflect from theground. For mobile communications, structures and topographic features providereflection surfaces.

Refraction

Radio waves are refracted (or bent) when they propagate through the atmosphere.The refraction is caused by changes in the speed of the signal with altitude or byother spatial changes in the atmospheric conditions. Normally, the speed of thesignal increases with altitude, causing radio waves to bend downward. However, onoccasion, weather conditions may lead to variations in speed with height that differsignificantly from the typical variations. This may result in a situation in which onlya fraction or no part of the line-of-sight wave reaches the receiving antenna.

5.4 FADING IN THE MOBILE ENVIRONMENT

Perhaps the most challenging technical problem facing communications systemsengineers is fading in a mobile environment. The term fading refers to the time vari-ation of received signal power caused by changes in the transmission medium or


5.4 / FADING IN THE MOBILE ENVIRONMENT 119

R

R

D

S

Lamppost

Figure 5.10 Sketch of Three Important Propagation Mechanisms: Reflection (R),Scattering (S), Diffraction (D) [ANDE95]

3On the other hand, the reflected signal has a longer path, which creates a phase shift due to delay rela-tive to the unreflected signal. When this delay is equivalent to half a wavelength, the two signals are backin phase.

path(s). In a fixed environment, fading is affected by changes in atmospheric con-ditions, such as rainfall. But in a mobile environment, where one of the two anten-nas is moving relative to the other, the relative location of various obstacles changesover time, creating complex transmission effects.

Multipath Propagation

Three propagation mechanisms, illustrated in Figure 5.10, play a role. Reflectionoccurs when an electromagnetic signal encounters a surface that is large relative tothe wavelength of the signal. For example, suppose a ground-reflected wave nearthe mobile unit is received. Because the ground-reflected wave has a 180° phase shiftafter reflection, the ground wave and the line-of-sight (LOS) wave may tend tocancel, resulting in high signal loss.3 Further, because the mobile antenna is lowerthan most human-made structures in the area, multipath interference occurs. Thesereflected waves may interfere constructively or destructively at the receiver.

Diffraction occurs at the edge of an impenetrable body that is large comparedto the wavelength of the radio wave. When a radio wave encounters such an edge,waves propagate in different directions with the edge as the source. Thus, signalscan be received even when there is no unobstructed LOS from the transmitter.



ReceivedLOS pulse

Receivedmultipath

pulses

Time

Time

Transmittedpulse

Transmittedpulse

ReceivedLOS pulse

Receivedmultipath

pulses

Figure 5.11 Two Pulses in Time-Variant Multipath

If the size of an obstacle is on the order of the wavelength of the signal or less,scattering occurs. An incoming signal is scattered into several weaker outgoingsignals. At typical cellular microwave frequencies, there are numerous objects, suchas lamp posts and traffic signs, that can cause scattering. Thus, scattering effects aredifficult to predict.

These three propagation effects influence system performance in various waysdepending on local conditions and as the mobile unit moves within a cell. If a mobileunit has a clear LOS to the transmitter, then diffraction and scattering are generallyminor effects, although reflection may have a significant impact. If there is no clearLOS, such as in an urban area at street level, then diffraction and scattering are theprimary means of signal reception.

The Effects of Multipath Propagation

As just noted, one unwanted effect of multipath propagation is that multiplecopies of a signal may arrive at different phases. If these phases add destructively,the signal level relative to noise declines, making signal detection at the receivermore difficult.

A second phenomenon, of particular importance for digital transmission, isintersymbol interference (ISI). Consider that we are sending a narrow pulse at agiven frequency across a link between a fixed antenna and a mobile unit. Figure 5.11shows what the channel may deliver to the receiver if the impulse is sent at two dif-ferent times. The upper line shows two pulses at the time of transmission. The lowerline shows the resulting pulses at the receiver. In each case the first received pulse isthe desired LOS signal. The magnitude of that pulse may change because of changesin atmospheric attenuation. Further, as the mobile unit moves farther away from thefixed antenna, the amount of LOS attenuation increases. But in addition to this pri-mary pulse, there may be multiple secondary pulses due to reflection, diffraction,



50 10 15

Position (m)

Am

plit

ude

(dB

m)

20 25 30

�130

�120

�110

�100

�90

�80

Figure 5.12 Typical Slow and Fast Fading in an Urban Mobile Environment

and scattering. Now suppose that this pulse encodes one or more bits of data. In thatcase, one or more delayed copies of a pulse may arrive at the same time as the pri-mary pulse for a subsequent bit. These delayed pulses act as a form of noise to thesubsequent primary pulse, making recovery of the bit information more difficult.

As the mobile antenna moves, the location of various obstacles changes; hencethe number, magnitude, and timing of the secondary pulses change. This makes itdifficult to design signal processing techniques that will filter out multipath effectsso that the intended signal is recovered with fidelity.

Types of Fading

Fading effects in a mobile environment can be classified as either fast or slow. Refer-ring to Figure 5.10, as the mobile unit moves down a street in an urban environment,rapid variations in signal strength occur over distances of about one-half a wave-length. At a frequency of 900 MHz, which is typical for mobile cellular applications,a wavelength is 0.33 m. The rapidly changing waveform in Figure 5.12 shows anexample of the spatial variation of received signal amplitude at 900 MHz in an urbansetting. Note that changes of amplitude can be as much as 20 or 30 dB over a shortdistance. This type of rapidly changing fading phenomenon, known as fast fading,affects not only mobile phones in automobiles, but even a mobile phone user walk-ing down an urban street.

As the mobile user covers distances well in excess of a wavelength, the urbanenvironment changes, as the user passes buildings of different heights, vacant lots,intersections, and so forth. Over these longer distances, there is a change in the aver-age received power level about which the rapid fluctuations occur. This is indicatedby the slowly changing waveform in Figure 5.12 and is referred to as slow fading.

Fading effects can also be classified as flat or selective. Flat fading, or non-selective fading, is that type of fading in which all frequency components of thereceived signal fluctuate in the same proportions simultaneously. Selective fadingaffects unequally the different spectral components of a radio signal. The term selec-



tive fading is usually significant only relative to the bandwidth of the overallcommunications channel. If attenuation occurs over a portion of the bandwidth ofthe signal, the fading is considered to be selective; nonselective fading implies thatthe signal bandwidth of interest is narrower than, and completely covered by, thespectrum affected by the fading.

The Fading Channel

In designing a communications system, the communications engineer needsto estimate the effects of multipath fading and noise on the mobile channel. Thesimplest channel model, from the point of view of analysis, is the additive whiteGaussian noise (AWGN) channel. In this channel, the desired signal is degraded bythermal noise associated with the physical channel itself as well as electronics at thetransmitter and receiver (and any intermediate amplifiers or repeaters). This modelis fairly accurate in some cases, such as space communications and some wire trans-missions, such as coaxial cable. For terrestrial wireless transmission, particularly inthe mobile situation, AWGN is not a good guide for the designer.

Rayleigh fading occurs when there are multiple indirect paths between trans-mitter and receiver and no distinct dominant path, such as an LOS path. This rep-resents a worst-case scenario. Fortunately, Rayleigh fading can be dealt withanalytically, providing insights into performance characteristics that can be used indifficult environments, such as downtown urban settings.

Rician fading best characterizes a situation where there is a direct LOS pathin addition to a number of indirect multipath signals. The Rician model is oftenapplicable in an indoor environment whereas the Rayleigh model characterizes out-door settings. The Rician model also becomes more applicable in smaller cells or inmore open outdoor environments. The channels can be characterized by a parame-ter K, defined as follows:

When K � 0 the channel is Rayleigh (i.e., numerator is zero) and when K � �,the channel is AWGN (i.e., denominator is zero). Figure 5.13, based on [FREE98a]and [SKLA01], shows system performance in the presence of noise. Here bit errorrate is plotted as a function of the ratio Eb/N0. Of course, as that ratio increases,the bit error rate drops. The figure shows that with a reasonably strong signal, rel-ative to noise, an AWGN exhibit provides fairly good performance, as do Ricianchannels with larger values of K, roughly corresponding to microcells or an opencountry environment. The performance would be adequate for a digitized voiceapplication, but for digital data transfer efforts to compensate would be needed.The Rayleigh channel provides relatively poor performance; this is likely to be seenfor flat fading and for slow fading; in these cases, error compensation mechanismsbecome more desirable. Finally, some environments produce fading effects worsethan the so-called worst case of Rayleigh. Examples are fast fading in an urbanenvironment and the fading within the affected band of a selective fading channel.In these cases, no level of Eb/N0 will help achieve the desired performance, and

K �power in the dominant path

power in the scattered paths



(Eb /N0) (dB)0 5 10 15 20 25 30 35

10-4

10�3

10�2

10�1

1

Prob

abili

ty o

f bi

t err

or (

BE

R)

Flat fadingand slow fadingRayleigh limit

Frequency-selective fading orfast fading distortion

Rician fading

K �

16

Rician fading

K �

4

Additive w

hite

gaussian noise

Figure 5.13 Theoretical Bit Error Rate for Various Fading Conditions

compensation mechanisms are mandatory. We turn to a discussion of those mecha-nisms next.

Error Compensation Mechanisms

The efforts to compensate for the errors and distortions introduced by multi-path fading fall into three general categories: forward error correction, adaptiveequalization, and diversity techniques. In the typical mobile wireless environment,techniques from all three categories are combined to combat the error ratesencountered.



Forward Error Correction

Forward error correction is applicable in digital transmission applications:those in which the transmitted signal carries digital data or digitized voice or videodata. The term forward refers to procedures whereby a receiver, using only infor-mation contained in the incoming digital transmission, corrects bit errors in the data.This is in contrast to backward error correction, in which the receiver merelydetects the presence of errors and then sends a request back to the transmitter toretransmit the data in error. Backward error correction is not practical in manywireless applications. For example, in satellite communications, the amount of delayinvolved makes retransmission undesirable. In mobile communications, the errorrates are often so high that there is a high probability that the retransmitted blockof bits will also contain errors. In these applications, forward error correction isrequired. In essence, forward error correction is achieved as follows:

1. The transmitter adds a number of additional, redundant bits to each transmit-ted block of data. These bits form an error-correcting code and are calculatedas a function of the data bits.

2. For each incoming block of bits (data plus error-correcting code), the receivercalculates a new error-correcting code from the incoming data bits. If the cal-culated code matches the incoming code, then the receiver assumes that noerror has occurred in this block of bits.

3. If the incoming and calculated codes do not match, then one or more bits arein error. If the number of bit errors is below a threshold that depends on thelength of the code and the nature of the algorithm, it is possible for the re-ceiver to determine the bit positions in error and correct all errors.

Typically in mobile wireless applications, the ratio of total bits sent to data bitssent is between 2 and 3. This may seem an extravagant amount of overhead, in thatthe capacity of the system is cut to one-half or one-third of its potential, but the mobilewireless environment is so difficult that such levels of redundancy are necessary.

Chapter 8 examines forward error correction techniques in detail.

Adaptive Equalization

Adaptive equalization can be applied to transmissions that carry analog infor-mation (e.g., analog voice or video) or digital information (e.g., digital data, digi-tized voice or video) and is used to combat intersymbol interference. The processof equalization involves some method of gathering the dispersed symbol energyback together into its original time interval. Equalization is a broad topic; tech-niques include the use of so-called lumped analog circuits as well as sophisticateddigital signal processing algorithms. Here we give a flavor of the digital signal pro-cessing approach.

Figure 5.14 illustrates a common approach using a linear equalizer circuit. Inthis specific example, for each output symbol, the input signal is sampled at five uni-formly spaced intervals of time, separated by a delay . These samples are individ-ually weighted by the coefficients Ci and then summed to produce the output. Thecircuit is referred to as adaptive because the coefficients are dynamically adjusted.



τ τ τ τ

×× × × ×

Σ

Algorithm for tapgain adjustment

Unequalizedinput

Equalizedoutput

C�2 C–1 C0 C1 C2

Figure 5.14 Linear Equalizer Circuit [PROA94]

Typically, the coefficients are set using a training sequence, which is a knownsequence of bits. The training sequence is transmitted. The receiver compares thereceived training sequence with the expected training sequence and on the basis ofthe comparison calculates suitable values for the coefficients. Periodically, a newtraining sequence is sent to account for changes in the transmission environment.

For Rayleigh channels, or worse, it may be necessary to include a new trainingsequence with every single block of data. Again, this represents considerable over-head but is justified by the error rates encountered in a mobile wireless environment.

Diversity Techniques

Diversity is based on the fact that individual channels experience independentfading events. We can therefore compensate for error effects by providing multiplelogical channels in some sense between transmitter and receiver and sending part ofthe signal over each channel. This technique does not eliminate errors but it doesreduce the error rate, since we have spread the transmission out to avoid being sub-jected to the highest error rate that might occur. The other techniques (equalization,forward error correction) can then cope with the reduced error rate.

Some diversity techniques involve the physical transmission path and arereferred to as space diversity. For example, multiple nearby antennas may be usedto receive the message, with the signals combined in some fashion to reconstruct themost likely transmitted signal. Another example is the use of collocated multipledirectional antennas, each oriented to a different reception angle with the incomingsignals again combined to reconstitute the transmitted signal.

More commonly, the term diversity refers to frequency diversity or time di-versity techniques. With frequency diversity, the signal is spread out over a larger



Fade giving rise to errorsTime

(a) TDM stream

D A B C

Fade giving rise to errorsTime

(b) Interleaving without TDM

AD B C D A AB C D

A2 A3 A4 A5 A6 A7 A8 A9

A2 A6 A10 A14 A3 A7 A11 A15

Figure 5.15 Interleaving Data Blocks to Spread the Effects of Error Bursts

frequency bandwidth or carried on multiple frequency carriers. The most importantexample of this approach is spread spectrum, which is examined in Chapter 7.

Time diversity techniques aim to spread the data out over time so that a noiseburst affects fewer bits. Time diversity can be quite effective in a region of slowfading. If a mobile unit is moving slowly, it may remain in a region of a high levelof fading for a relatively long interval. The result will be a long burst of errors eventhough the local mean signal level is much higher than the interference. Even pow-erful error correction codes may be unable to cope with an extended error burst. Ifdigital data is transmitted in a time division multiplex (TDM) structure, in whichmultiple users share the same physical channel by the use of time slots (see Figure2.13b), then block interleaving can be used to provide time diversity. Figure 5.15a,


5.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 127

based on one in [JONE93], illustrates the concept. Note that the same number ofbits are still affected by the noise surge, but they are spread out over a number of log-ical channels. If each channel is protected by forward error correction, the error-correcting code may be able to cope with the fewer number of bits that are in errorin a particular logical channel. If TDM is not used, time diversity can still be appliedby viewing the stream of bits from the source as a sequence of blocks and thenshuffling the blocks. In Figure 5.15b, blocks are shuffled in groups of four. Again,the same number of bits is in error, but the error-correcting code is applied to setsof bits that are spread out in time. Even greater diversity is achieved by combiningTDM interleaving with block shuffling.

The tradeoff with time diversity is delay. The greater the degree of inter-leaving and shuffling used, the longer the delay in reconstructing the original bitsequence at the receiver.

5.5 RECOMMENDED READING

[FREE97] provides good coverage of all of the topics in this chapter. A rigorous treatmentof antennas and propagation is found in [BERT00]. [THUR00] provides an exceptionallyclear discussion of antennas.

BERT00 Bertoni, H. Radio Propagation for Modern Wireless Systems. Upper SaddleRiver, NJ: Prentice Hall, 2000.

FREE97 Freeman, R. Radio System Design for Telecommunications. New York: Wiley, 1997.THUR00 Thurwachter, C. Data and Telecommunications: Systems and Applications.

Upper Saddle River, NJ: Prentice Hall, 2000.

5.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

adaptive equalization flat fading optical LOSantenna forward error correction parabolic reflective antennaantenna gain (FEC) radiation patternatmospheric absorption free space loss radio LOSattenuation ground wave propagation reception patternbeam width Hertz antenna reflectioncrosstalk impulse noise refractiondiffraction intermodulation noise scatteringdipole isotropic antenna selective fadingdiversity line of sight (LOS) sky wave propagationfading multipath slow fadingfast fading noise thermal noise



Review Questions1 What two functions are performed by an antenna?2 What is an isotropic antenna?3 What information is available from a radiation pattern?4 What is the advantage of a parabolic reflective antenna?5 What factors determine antenna gain?6 What is the primary cause of signal loss in satellite communications?7 Name and briefly define four types of noise.8 What is refraction?9 What is fading?

10 What is the difference between diffraction and scattering?11 What is the difference between fast and slow fading?12 What is the difference between flat and selective fading?13 Name and briefly define three diversity techniques.

Problems1 For radio transmission in free space, signal power is reduced in proportion to the

square of the distance form the source, whereas in wire transmission, the attenuationis a fixed number of dB per kilometer. The following table is used to show the dBreduction relative to some reference for free space radio and uniform wire. Fill in themissing numbers to complete the table.

2 Find the optimum wavelength and frequency for a half-wave dipole of length 10 m. 3 It turns out that the depth in the ocean to which airborne electromagnetic signals can

be detected grows with the wavelength. Therefore, the military got the idea of usingvery long wavelengths corresponding to about 30 Hz to communicate with submarines

Distance (km) Radio (dB) Wire (dB)

1 �6 �3248

16

throughout the world. If we want to have an antenna that is about one-half wave-length long, how long would that be?

4 The audio power of the human voice is concentrated at about 300 Hz. Antennas ofthe appropriate size for this frequency are impracticably large, so that to send voiceby radio the voice signal must be used to modulate a higher (carrier) frequency forwhich the natural antenna size is smaller.a. What is the length of an antenna one-half wavelength long for sending radio at 300 Hz?b. An alternative is to use a modulation scheme, as described in Chapter 6, for trans-

mitting the voice signal by modulating a carrier frequency, so that the bandwidth ofthe signal is a narrow band centered on the carrier frequency. Suppose we would likea half-wave antenna to have a length of 1 m. What carrier frequency would we use?

5 Stories abound of people who receive radio signals in fillings in their teeth. Supposeyou have one filling that is 2.5 mm (0.0025 m) long that acts as a radio antenna. Thatis, it is equal in length to one-half the wavelength. What frequency do you receive?


6 Section 5.1 stated that if a source of electromagnetic energy is placed at the focus ofthe paraboloid, and if the paraboloid is a reflecting surface, then the wave will bounceback in lines parallel to the axis of the paraboloid. To demonstrate this, consider theparabola y2 � 2px shown in Figure 5.16. Let P(x1, y1) be a point on the parabola, andPF be the line from P to the focus. Construct the line L through P parallel to the x-axis and the line M tangent to the parabola at P. The angle between L and M is , andthe angle between PF and M is �. The angle � is the angle at which a ray from Fstrikes the parabola at P. Because the angle of incidence equals the angle of reflec-tion, the ray reflected from P must be at an angle � to M. Thus, if we can show that� � , we have demonstrated that rays reflected from the parabola starting at F willbe parallel to the x axis.a. First show that tan � (p/y1). Hint: Recall from trigonometry that the slope of a

line is equal to the tangent of the angle the line makes with the positive x direction.Also recall that the slope of the line tangent to a curve at a given point is equal tothe derivative of the curve at that point.

b. Now show that tan � � (p/y1), which demonstrates that � � . Hint: Recall fromtrigonometry that the formula for the tangent of the difference between two angles�1 and �2 is tan(�2 � �1) � (tan �2 � tan �1 ) / (1 � tan �2 � tan �1).

7 For each of the antenna types listed in Table 5.2, what is the effective area and gainat a wavelength of 30 cm? Repeat for a wavelength of 3 mm. Assume that the actualarea for the horn and parabolic antennas is �.

8 It is often more convenient to express distance in km rather than m and frequency inMHz rather than Hz. Rewrite Equation (5.2) using these dimensions.

9 Assume that two antennas are half-wave dipoles and each has a directive gain of 3 dB.If the transmitted power is 1 W and the two antennas are separated by a distance of

5.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 129

y

x

β

α

F(p/2, 0)

L

M

0

P(x1, y1)

Figure 5.16 Parabolic Reflection



10 km, what is the received power? Assume that the antennas are aligned so that thedirective gain numbers are correct and that the frequency used is 100 MHz.

10 Suppose a transmitter produces 50 W of power.a. Express the transmit power in units of dBm and dBW.b. If the transmitter’s power is applied to a unity gain antenna with a 900-MHz carrier

frequency, what is the received power in dBm at a free space distance of 100 m?c. Repeat (b) for a distance of 10 km.d. Repeat (c) but assume a receiver antenna gain of 2.

11 A microwave transmitter has an output of 0.1 W at 2 GHz. Assume that this trans-mitter is used in a microwave communication system where the transmitting andreceiving antennas are parabolas, each 1.2 m in diameter.a. What is the gain of each antenna in decibels?b. Taking into account antenna gain, what is the effective radiated power of the trans-

mitted signal?c. If the receiving antenna is located 24 km from the transmitting antenna over a free

space path, find the available signal power out of the receiving antenna in dBmunits.

12 Section 5.2 states that with no intervening obstacles, the optical line of sight can beexpressed as , where d is the distance between an antenna and the hori-zon in kilometers and h is the antenna height in meters. Using a value for the earth’sradius of 6370 km, derive this equation. Hint: Assume that the antenna is perpendic-ular to the earth’s surface, and note that the line from the top of the antenna to thehorizon forms a tangent to the earth’s surface at the horizon. Draw a picture showingthe antenna, the line of sight, and the earth’s radius to help visualize the problem.

13 Determine the height of an antenna for a TV station that must be able to reach cus-tomers up to 80 km away.

14 What is the thermal noise level of a channel with a bandwidth of 10 kHz carrying 1000watts of power operating at 50°C? Compare the noise level to the operating power.

15 The square wave of Figure 2.5c, with T � 1 ms, is passed through a low-pass filter thatpasses frequencies up to 8 kHz with no attenuation.a. Find the power in the output waveform.b. Assuming that at the filter input there is a thermal noise voltage with N0 �

0.1 �W/Hz, find the output signal to noise ratio in dB.16 If the received signal level for a particular digital system is �151 dBW and the

receiver system effective noise temperature is 1500°K, what is Eb/N0 for a link trans-mitting 2400 bps?

17 Suppose a ray of visible light passes from the atmosphere into water at an angle to thehorizontal of 30°. What is the angle of the ray in the water? Note: At standard atmos-pheric conditions at the earth’s surface, a reasonable value for refractive index is1.0003. A typical value of refractive index for water is 4/3.

d � 3.572h


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