antenas2

14.1 System Aspects of Antennas

FIGURE 14.1 Photograph of various millimeter wave antennas. Clockwise from top: a high-gain

38 GHz re ector antenna with radome, a prime-focus parabolic antenna, a corru-

gated conical horn antenna, a 38 GHz planar microstrip array, a pyramidal horn

antenna with a Gunn diode module, and a multibeam re ector antenna.

A transmitting antenna can be viewed as a device that converts a guided electromag-

netic wave on a transmission line into a plane wave propagating in free space. Thus, one

side of an antenna appears as an electrical circuit element, while the other side provides

an interface with a propagating plane wave. Antennas are inherently bidirectional, in that

they can be used for both transmit and receive functions. Figure 14.2 illustrates the basic

FIGURE 14.2 Basic operation of transmitting and receiving antennas.

Chapter 14: Introduction to Microwave Systems

operation of transmitting and receiving antennas. The transmitter can be modeled as a

Thevenin source consisting of a voltage generator and series impedance, delivering a power

Pt to the transmitting antenna. A transmitting antenna radiates a spherical wave that, at

large distances, approximates a plane wave over a localized area. A receiving antenna in-

tercepts a portion of an incident plane wave, and delivers a receive power Pr to the receiver

load impedance.

A wide variety of antennas have been developed for different applications, as summa-

rized in the following categories:

r Wire antennas include dipoles, monopoles, loops, sleeve dipoles, Yagi�Uda arrays,

and related structures. Wire antennas generally have low gains, and are most often

used at lower frequencies (HF to UHF). They have the advantages of light weight,

low cost, and simple design.r Aperture antennas include open-ended waveguides, rectangular or circular horns,

re ectors, lenses, and re ectarrays. Aperture antennas are most commonly used at

microwave and millimeter wave frequencies, and have moderate to high gains.r Printed antennas include printed slots, printed dipoles, and microstrip patch an-

tennas. These antennas can be made with photolithographic methods, with both

radiating elements and associated feed circuitry fabricated on dielectric substrates.

Printed antennas are most often used at microwave and millimeter wave frequencies,

and can be easily arrayed for high gain.r Array antennas consist of a regular arrangement of antenna elements with a feed

network. Pattern characteristics such as beam pointing angle and sidelobe levels can

be controlled by adjusting the amplitude and phase excitation of the array elements.

An important type of array antenna is the phased array, in which variable-phase

shifters are used to electronically scan the main beam of the antenna.

While we do not require detailed solutions to Maxwell�s equations for our purposes, we do

need to be familiar with the far-zone electromagnetic !elds radiated by an antenna. Con-

sider an antenna located at the origin of a spherical coordinate system. At large distances,

where the localized near-zone !elds are negligible, the radiated electric !eld of an arbitrary

antenna can be expressed as

E(r, θ, φ) =[

θFθ (θ, φ) + φFφ(θ, φ)]e− jk0r

rV/m, (14.1)

where E is the electric �eld vector, θ and φ are unit vectors in the spherical coordinate

system, r is the radial distance from the origin, and k0 = 2π/λ is the free-space propaga-

tion constant, with wavelength λ = c/ f . Also de�ned in (14.1) are the pattern functions,

Fθ (θ, φ) and Fφ(θ, φ). The interpretation of (14.1) is that this electric �eld propagates in

the radial direction with a phase variation of e− jk0r and an amplitude variation with dis-

tance of 1/r . The electric �eld may be polarized in either the θ or φ direction, but not in the

radial direction, since this is a TEM wave. The magnetic �elds associated with the electric

�eld of (14.1) can be found from (1.76) as

Hφ =Eθ

η0, (14.2a)

Hθ =−Eφ

η0, (14.2b)


where η0 = 377 Ä, the wave impedance of free-space. Note that the magnetic �eld vector

is also polarized only in the transverse directions. The Poynting vector for this wave is

given by (1.90) as

S = E × H∗ W/m2, (14.3)

and the time-average Poynting vector is

Savg =1

2Re {S} =

1

2Re {E × H∗}W/m2. (14.4)

We mentioned earlier that at large distances the near �elds of an antenna are negli-

gible, and that the radiated electric �eld can be written as in (14.1). We can give a more

precise meaning to this concept by de�ning the far- eld distance as the distance where the

spherical wave front radiated by an antenna becomes a close approximation to the ideal

planar phase front of a plane wave. This approximation applies over the radiating aperture

of the antenna, and so it depends on the maximum dimension of the antenna. If we call this

maximum dimension D, then the far-�eld distance is de�ned as

Rff =2D2

λm. (14.5)

This result is derived from the condition that the actual spherical wave front radiated by

the antenna departs less than π/8 = 22.5◦ from a true plane wave front over the maximum

extent of the antenna. For electrically small antennas, such as short dipoles and small loops,

this result may give a far-�eld distance that is too small; in this case, a minimum value of

Rff = 2λ should be used.

EXAMPLE 14.1 FAR-FIELD DISTANCE OF AN ANTENNA

A parabolic re!ector antenna used for reception with the direct broadcast sys-

tem (DBS) is 18 inches in diameter and operates at 12.4 GHz. Find the far-�eld

distance for this antenna.

Solution

The operating wavelength at 12.4 GHz is

λ =c

f=

3× 108

12.4× 109= 2.42 cm.

The far-�eld distance is found from (14.5), after converting 18 inches to 0.457 m:

Rff =2D2

λ=2(0.457)2

0.0242= 17.3 m.

The actual distance from a DBS satellite to Earth is about 36,000 km, so it is safe

to say that the receive antenna is in the far-�eld of the transmitting antenna. ■

Next, de�ne the radiation intensity of the radiated electromagnetic �eld as

U (θ, φ) = r2|Savg| =r2

2Re

{

Eθ θ × H∗φ φ + Eφ φ × H∗

θ θ}

=r2

2η0

[

|Eθ |2 + |Eφ |2

]

=1

2η0

[

|Fθ |2 + |Fφ |2

]

W, (14.6)


where (14.1), (14.2), and (14.4) were used. The units of the radiation intensity are watts,

or watts per unit solid angle, since the radial dependence has been removed. The radiation

intensity gives the variation in radiated power versus position around the antenna. We can

�nd the total power radiated by the antenna by integrating the Poynting vector over the

surface of a sphere of radius r that encloses the antenna. This is equivalent to integrating

the radiation intensity over a unit sphere:

Prad =

2π∫

φ=0

π∫

θ=0

Savg · rr2 sin θdθdφ =

2π∫

φ=0

π∫

θ=0

U (θ, φ) sin θdθdφ. (14.7)

The radiation pattern of an antenna is a plot of the magnitude of the far-zone �eld strength

versus position around the antenna, at a �xed distance from the antenna. Thus the radiation

pattern can be plotted from the pattern function Fθ (θ, φ) or Fφ(θ, φ), versus either the

angle θ (for an elevation plane pattern) or the angle φ (for an azimuthal plane pattern).

The choice of plotting either Fθ or Fφ is dependent on the polarization of the antenna.

A typical antenna pattern is shown in Figure 14.3. This pattern is plotted in polar form,

versus the elevation angle, θ , for a small horn antenna oriented in the vertical direction. The

plot shows the relative variation of the radiated power of the antenna in dB, normalized to

the maximum value. Since the pattern functions are proportional to voltage, the radial scale

of the plot is computed as 20 log |F(θ, φ)|; alternatively, the plot could be computed in

terms of the radiation intensity as 10 log |U (θ, φ)|. The pattern may exhibit several distinct

lobes, with different maxima in different directions. The lobe having the maximum value

is called the main beam, while those lobes at lower levels are called sidelobes. The pattern

of Figure 14.3 has one main beam at θ = 0 and several sidelobes, the largest of which are

located at about θ = ±16◦. The level of these sidelobes is 13 dB below the level of the

main beam. Radiation patterns may also be plotted in rectangular form; this is especially

useful for antennas having a narrow main beam.

-30 -20 -10 0

FIGURE 14.3 The E-plane radiation pattern of a small horn antenna. The pattern is normalized

to 0 dB at the beam maximum, with 10 dB per radial division.


A fundamental property of an antenna is its ability to focus power in a given direction,

to the exclusion of other directions. Thus an antenna with a broad main beam can transmit

(or receive) power over a wide angular region, while an antenna having a narrow main

beam will transmit (or receive) power over a small angular region. One measure of this

focusing effect is the 3 dB beamwidth of the antenna, de�ned as the angular width of the

main beam at which the power level has dropped 3 dB from its maximum value (its half-

power points). The 3 dB beamwidth of the pattern of Figure 14.3 is about 10◦. Antennas

having a constant pattern in the azimuthal plane are called omnidirectional, and are useful

for applications such as broadcasting or for hand-held wireless devices, where it is desired

to transmit or receive equally in all directions. Patterns that have relatively narrow main

beams in both planes are known as pencil beam antennas, and are useful in applications

such as radar and point-to-point radio links.

Another measure of the focusing ability of an antenna is the directivity, de�ned as the

ratio of the maximum radiation intensity in the main beam to the average radiation intensity

over all space:

D =Umax

Uavg=4πUmax

Prad=

4πUmaxπ

∫

θ=0

2π∫

φ=0

U (θ, φ) sin θdθdφ

, (14.8)

where (14.7) has been used for the radiated power. Directivity is a dimensionless ratio of

power, and is usually expressed in dB as D(dB) = 10 log(D).

An antenna that radiates equally in all directions is called an isotropic antenna. Apply-

ing the integral identity thatπ

∫

θ=0

2π∫

φ=0

sin θdθdφ = 4π

to the denominator of (14.8) for U (θ, φ) = 1 shows that the directivity of an isotropic ele-

ment is D = 1, or 0 dB. Since the minimum directivity of any antenna is unity, directivity

is sometimes stated as relative to the directivity of an isotropic radiator, and written as dBi.

Typical directivities for some common antennas are 2.2 dB for a wire dipole, 7.0 dB for a

microstrip patch antenna, 23 dB for a waveguide horn antenna, and 35 dB for a parabolic

re!ector antenna.

Beamwidth and directivity are both measures of the focusing ability of an antenna:

an antenna pattern with a narrow main beam will have a high directivity, while a pattern

with a wide beam will have a lower directivity. We might therefore expect a direct relation

between beamwidth and directivity, but in fact there is not an exact relationship between

these two quantities. This is because beamwidth is only dependent on the size and shape

of the main beam, whereas directivity involves integration of the entire radiation pattern.

Thus it is possible for many different antenna patterns to have the same beamwidth but

quite different directivities due to differences in sidelobes or the presence of more than one

main beam. With this quali�cation in mind, however, it is possible to develop approximate

relations between beamwidth and directivity that apply with reasonable accuracy to a large

number of practical antennas. One such approximation that works well for antennas with

pencil beam patterns is the following:

D ∼=32,400

θ1θ2, (14.9)

where θ1 and θ2 are the beamwidths in two orthogonal planes of the main beam, in degrees.

This approximation does not work well for omnidirectional patterns because there is a

well-de�ned main beam in only one plane for such patterns.


EXAMPLE 14.2 PATTERN CHARACTERISTICS OF A DIPOLE ANTENNA

The far-zone electric �eld radiated by an electrically small wire dipole antenna

oriented on the z-axis is given by

Eθ (r, θ, φ) = V0 sin θe− jk0r

rV/m,

Eφ(r, θ, φ) = 0.

Find the main beam position of the dipole antenna, its beamwidth, and its direc-

tivity.

Solution

The radiation intensity for the above far-�eld is

U (θ, φ) = C sin2 θ,

where the constant C = V 20 /2η0. The radiation pattern is seen to be independent

of the azimuth angle φ, and so is omnidirectional in the azimuthal plane. The

pattern has a �donut� shape, with nulls at θ = 0 and θ = 180◦ (along the z-axis),

and a beam maximum at θ = 90◦ (the horizontal plane). The angles where the

radiation intensity has dropped by 3 dB are given by the solutions to

sin2 θ = 0.5;

thus the 3 dB, or half-power, beamwidth is 135◦ − 45◦ = 90◦.

The directivity is calculated using (14.8). The denominator of this expression

is

π∫

θ=0

2π∫

φ=0

U (θ, φ) sin θdθdφ = 2πC

π∫

θ=0

sin3 θdθ = 2πC

(

4

3

)

=8πC

3,

where the required integral identity is listed in Appendix D. Since Umax = C , the

directivity reduces to

D =3

2= 1.76 dB. ■

Resistive losses, due to nonperfect metals and dielectric materials, exist in all practical

antennas. Such losses result in a difference between the power delivered to the input of an

antenna and the power radiated by that antenna. As with many other electrical components,

we can de�ne the radiation ef ciency of an antenna as the ratio of the desired output power

to the supplied input power:

ηrad =Prad

Pin=

Pin − Ploss

Pin= 1−

Ploss

Pin, (14.10)

where Prad is the power radiated by the antenna, Pin is the power supplied to the input of

the antenna, and Ploss is the power lost in the antenna. Note that there are other factors that

can contribute to the effective loss of transmit power, such as impedance mismatch at the


input to the antenna, or polarization mismatch with the receive antenna. However, these

losses are external to the antenna and could be eliminated by the proper use of matching

networks, or the proper choice and positioning of the receive antenna. Therefore losses

of this type are usually not attributed to the antenna itself, as are dissipative losses due to

metal conductivity or dielectric loss within the antenna.

Recall that antenna directivity is a function only of the shape of the radiation pattern

(the radiated �elds) of an antenna, and is not affected by losses in the antenna itself. To

account for the fact that an antenna having a radiation ef�ciency less than unity will not

radiate all of its input power, we de�ne antenna gain as the product of directivity and

ef�ciency:

G = ηradD. (14.11)

Thus, gain is always less than or equal to directivity. Gain can also be computed directly,

by replacing Prad in the denominator of (14.8) with Pin, since by the de�nition of radiation

ef�ciency in (14.10) we have Prad = ηradPin. Gain is usually expressed in dB, as G(dB) =

10 log(G). Sometimes the effect of impedance mismatch loss is included in the gain of an

antenna; this is referred to as the realized gain [1].

Many types of antennas can be classi�ed as aperture antennas, meaning that the antenna

has a well-de�ned aperture area from which radiation occurs. Examples include re!ector

antennas, horn antennas, lens antennas, and array antennas. For such antennas, it can be

shown that the maximum directivity that can be obtained from an electrically large aperture

of area A is given as

Dmax =4π A

λ2. (14.12)

For example, a rectangular horn antenna having an aperture 2λ × 3λ has a maximum direc-

tivity of 24π , or about 19 dB. In practice, there are several factors that can serve to reduce

the directivity of an antenna from its maximum possible value, such as nonideal amplitude

or phase characteristics of the aperture �eld, aperture blockage, or, in the case of re!ector

antennas, spillover of the feed pattern. For this reason, we de�ne an aperture ef ciency as

the ratio of the actual directivity of an aperture antenna to the maximum directivity given

by (14.12). Then we can write the directivity of an aperture antenna as

D = ηap4π A

λ2. (14.13)

Aperture ef�ciency is always less than or equal to unity.

The above de�nitions of antenna directivity, ef�ciency, and gain were stated in terms

of a transmitting antennas, but they apply to receiving antennas as well. For a receiving

antenna it is also of interest to determine the received power for a given incident plane wave

�eld. This is the converse problem of �nding the power density radiated by a transmitting

antenna, as given in (14.4). Determining received power is important for the derivation of

the Friis radio system link equation, to be discussed in the following section. We expect that

received power will be proportional to the power density, or Poynting vector, of the incident

wave. Since the Poynting vector has dimensions of W/m2, and the received power, Pr , has

dimensions of W, the proportionality constant must have units of area. Thus we write

Pr = AeSavg, (14.14)


where Ae is de�ned as the effective aperture area of the receive antenna. The effective

aperture area has dimensions of m2, and can be interpreted as the �capture area� of a

receive antenna, intercepting part of the incident power density radiated toward the receive

antenna. The quantity Pr in (14.14) is the power available at the terminals of the receive

antenna, as delivered to a conjugately matched load.

The maximum effective aperture area of an antenna can be shown to be related to the

directivity of the antenna as [1, 2]

Ae =Dλ2

4π, (14.15)

where λ is the operating wavelength of the antenna. For electrically large aperture antennas

the effective aperture area is often close to the actual physical aperture area. However,

for many other types of antennas, such as dipoles and loops, there is no simple relation

between the physical cross-sectional area of the antenna and its effective aperture area.

The maximum effective aperture area as de�ned above does not include the effect of losses

in the antenna, which can be accounted for by replacing D in (14.15) with G, the gain, of

the antenna.

We have seen how noise power is generated by lossy components and active devices, but

noise can also be delivered to the input of a receiver by the antenna. Antenna noise power

may be received from the external environment, or generated internally as thermal noise

due to losses in the antenna itself. While noise produced within a receiver is controllable

to some extent (by judicious design and component selection), the noise received from

the environment by a receiving antenna is generally not controllable, and may exceed the

noise level of the receiver itself. Thus it is important to characterize the noise power deliv-

ered to a receiver by its antenna.

Consider the three situations shown in Figure 14.4. In Figure 14.4a we have the simple

case of a resistor at temperature T , producing an available output noise power

No = kTB, (14.16)

where B is the system bandwidth and k is Boltzmann�s constant. In Figure 14.4b we have

an antenna enclosed by an anechoic chamber at temperature T . The anechoic chamber

appears as a perfectly absorbing enclosure, and is in thermal equilibrium with the an-

tenna. Thus the terminals of the antenna are indistinguishable from the resistor terminals of

Figure 14.4a (assuming an impedance-matched antenna), and therefore it produces the

FIGURE 14.4 Illustrating the concept of background temperature. (a) A resistor at temperature T .

(b) An antenna in an anechoic chamber at temperature T . (c) An antenna viewing

a uniform sky background at temperature T .


FIGURE 14.5 Natural and man-made sources of background noise.

same output noise power as the resistor of Figure 14.4a. Figure 14.4c shows the same an-

tenna directed at the sky. If the main beam of the antenna is narrow enough so that it sees

a uniform region at physical temperature T , then the antenna again appears as a resistor at

temperature T and produces the output noise power given in (14.16). This is true regard-

less of the radiation ef�ciency of the antenna, as long as the physical temperature of the

antenna is also T .

In actuality an antenna typically sees a much more complex environment than the

cases depicted in Figure 14.4. A general scenario of both naturally occurring and man-

made noise sources is shown in Figure 14.5, where we see that an antenna with a relatively

broad main beam may pick up noise power from a variety of origins. In addition, noise

may be received through the sidelobes of the antenna pattern or via re!ections from the

ground or other large objects. As in Chapter 10, where the noise power from an arbi-

trary white noise source was represented as an equivalent noise temperature, we de�ne the

background noise temperature, TB , as the equivalent temperature of a resistor required to

produce the same noise power as the actual environment seen by the antenna. Some typ-

ical background noise temperatures that are relevant at low microwave frequencies are as

follows:

r Sky (toward zenith) 3�5 Kr Sky (toward horizon) 50�100 Kr Ground 290�300 K

The overhead sky background temperature of 3�5 K is the cosmic background radiation

believed to be a remnant of the big bang at the creation of the universe. This would be

the noise temperature seen by an antenna with a narrow beam and high radiation ef�-

ciency pointed overhead, away from �hot� sources such as the Sun or stellar radio objects.

The background noise temperature increases as the antenna is pointed toward the horizon

because of the greater thickness of the atmosphere, so that the antenna sees an effective

background closer to that of the anechoic chamber of Figure 14.4b. Pointing the antenna

toward the ground further increases the effective loss, and hence the noise temperature.

Figure 14.6 gives a more complete picture of the background noise temperature, show-

ing the variation of TB versus frequency and for several elevation angles [3]. Note that the

noise temperature shown in the graph follows the trends listed above, in that it is lowest

for the overhead sky (θ = 90◦), and greatest for angles near the horizon (θ = 0◦). Also

note the sharp peaks in noise temperature that occur at 22 and 60 GHz. The �rst is due to

the resonance of molecular water, while the second is caused by resonance of molecular


FIGURE 14.6 Background noise temperature of sky versus frequency. θ is elevation angle mea-

sured from the horizon. Data are for sea level, with surface temperature of 15◦ C

and surface water vapor density of 7.5 gm/m3.

oxygen. Both of these resonances lead to increased atmospheric loss and hence increased

noise temperature. The loss is great enough at 60 GHz that a high-gain antenna pointing

through the atmosphere effectively appears as a matched load at 290 K. While loss in gen-

eral is undesirable, these particular resonances can be useful for remote sensing applica-

tions, or for using the inherent attenuation of the atmosphere to limit propagation distances

for radio communications over small distances.

When the antenna beamwidth is broad enough that different parts of the antenna pat-

tern see different background temperatures, the effective brightness temperature seen by

the antenna can be found by weighting the spatial distribution of background temperature

by the pattern function of the antenna. Mathematically we can write the brightness temper-

ature Tb seen by the antenna as

Tb =

2π∫

φ=0

π∫

θ=0

TB(θ, φ)D(θ, φ) sin θdθdφ

2π∫

φ=0

π∫

θ=0

D(θ, φ) sin θdθdφ

, (14.17)

where TB(θ, φ) is the distribution of the background temperature, and D(θ, φ) is the di-

rectivity (or the power pattern function) of the antenna. Antenna brightness temperature

is referenced at the terminals of the antenna. Observe that when TB is a constant, (14.17)

reduces to Tb = TB , which is essentially the case of a uniform background temperature

shown in Figure 14.3b or 14.4c. Also note that this de�nition of antenna brightness tem-

perature does not involve the gain or ef�ciency of the antenna, and so does not include

thermal noise due to losses in the antenna.


If a receiving antenna has dissipative loss, so that its radiation ef�ciency ηrad is less than

unity, the power available at the terminals of the antenna is reduced by the factor ηradfrom that intercepted by the antenna (the de�nition of radiation ef�ciency is the ratio of

output to input power). This reduction applies to received noise power, as well as received

signal power, so the noise temperature of the antenna will be reduced from the brightness

temperature given in (14.17) by the factor ηrad. In addition, thermal noise will be generated

internally by resistive losses in the antenna, and this will increase the noise temperature of

the antenna. In terms of noise power, a lossy antenna can be modeled as a lossless antenna

and an attenuator having a power loss factor of L = 1/ηrad. Then, using (10.15) for the

equivalent noise temperature of an attenuator, we can �nd the resulting noise temperature

seen at the antenna terminals as

TA =Tb

L+

(L − 1)

LTp = ηradTb + (1− ηrad)Tp. (14.18)

The equivalent temperature TA is called the antenna noise temperature, and is a combi-

nation of the external brightness temperature seen by the antenna and the thermal noise

generated by the antenna. As with other equivalent noise temperatures, the proper interpre-

tation of TA is that a matched load at this temperature will produce the same available noise

power as does the antenna. Note that this temperature is referenced at the output terminals

of the antenna; since an antenna is not a two-port circuit element, it does not make sense

to refer the equivalent noise temperature to its �input.�

Observe that (14.18) reduces to TA = Tb for a lossless antenna with ηrad = 1. If the

radiation ef�ciency is zero, meaning that the antenna appears as a matched load and does

not see any external background noise, then (14.18) reduces to TA = Tp, due to the thermal

noise generated by the losses. If an antenna is pointed toward a known background tem-

perature different than T0, then (14.18) can be used to determine its radiation ef�ciency.

EXAMPLE 14.3 ANTENNA NOISE TEMPERATURE

A high-gain antenna has the idealized hemispherical elevation plane pattern shown

in Figure 14.7, and is rotationally symmetric in the azimuth plane. If the antenna

is facing a region having a background temperature TB approximated as given in

Figure 14.7, �nd the antenna noise temperature. Assume the radiation ef�ciency

of the antenna is 100%.

Solution

Since ηrad = 1, (14.18) reduces to TA = Tb. The brightness temperature can be

computed from (14.17), after normalizing the directivity to a maximum value

FIGURE 14.7 Idealized antenna pattern and background noise temperature for Example 14.3.


of unity:

Tb =

2π∫

φ=0

π∫

θ=0

TB(θ, φ)D(θ, φ) sin θdθdφ

2π∫

φ=0

π∫

θ=0

D(θ, φ) sin θdθdφ

=

1◦∫

θ=0

10 sin θdθ+30◦∫

θ=1◦0.1 sin θdθ+

90◦∫

θ=30◦sin θdθ

1◦∫

θ=0

sin θdθ +90◦∫

θ=1◦0.01 sin θdθ

=−10 cos θ |1

◦

0 − 0.1 cos θ |30◦

1◦ − cos θ |90◦

30◦

−cosθ |1◦

0 − 0.01 cos θ |90◦

1◦

=0.00152+ 0.0134+ 0.866

0.0102= 86.4 K.

In this example most of the noise power is collected through the sidelobe region

of the antenna. ■

The more general problem of a receiver connected through a lossy transmission line to

an antenna viewing a background noise temperature distribution TB can be represented by

the system shown in Figure 14.8. The antenna is assumed to have a radiation ef�ciency ηrad,

and the connecting transmission line has a power loss factor of L ≥ 1, with both at physical

temperature Tp. We also include the effect of an impedance mismatch between the antenna

and the transmission line, represented by the re!ection coef�cient Ŵ. The equivalent noise

temperature seen at the output terminals of the transmission line consists of three contri-

butions: noise power from the antenna due to internal noise and the background brightness

temperature, noise power generated from the lossy line in the forward direction, and noise

power generated by the lossy line in the backward direction and re!ected from the antenna

mismatch toward the receiver. The noise due to the antenna is given by (14.18), but re-

duced by the loss factor of the line, 1/L , and the re!ection mismatch factor, (1− |Ŵ|2).

The forward noise power from the lossy line is given by (10.15), after reduction by the loss

factor, 1/L . The contribution from the lossy line re!ected from the mismatched antenna

is given by (10.15), after reduction by the power re!ection coef�cient, |Ŵ|2, and the loss

factor, 1/L2 (since the reference point for the back-directed noise power from the lossy

line given by (10.15) is at the output terminals of the line). Thus the overall system noise

temperature seen at the input to the receiver is given by

TS =TA

L(1− |Ŵ|2) + (L − 1)

Tp

L+ (L − 1)

Tp

L2|Ŵ|2

=(1− |Ŵ|2)

L[ηradTb + (1− ηrad)Tp] +

(L − 1)

L

(

1+|Ŵ|2

L

)

Tp. (14.19)

BackgroundtemperatureTB ( , !)

AntennaTP, "rad

Receiver

TA TS

�

Lossy"lineTP, �

FIGURE 14.8 A receiving antenna connected to a receiver through a lossy transmission line. An

impedance mismatch exists between the antenna and the line.

14.2 Wireless Communications

Observe that for a lossless line (L = 1) the effect of an antenna mismatch is to reduce the

system noise temperature by the factor (1− |Ŵ|2). Of course, the received signal power

will be reduced by the same amount. Also note that for the case of a matched antenna

(Ŵ = 0), (14.19) reduces to

TS =1

L[ηradTb + (1− ηrad)Tp] +

L − 1

LTp, (14.20)

as expected for a cascade of two noisy components.

Finally, it is important to realize the difference between radiation ef�ciency and aper-

ture ef�ciency, and their effects on antenna noise temperature. While radiation ef�ciency

accounts for resistive losses, and thus involves the generation of thermal noise, aperture ef-

�ciency does not. Aperture ef�ciency applies to the loss of directivity in aperture antennas,

such as re!ectors, lenses, or horns, due to feed spillover or suboptimum aperture excitation

(e.g., a nonuniform amplitude or phase distribution), and by itself does not lead to any ad-

ditional effect on noise temperature that would not be included through the pattern of the

antenna.

The antenna noise temperature de�ned above is a useful �gure of merit for a receive

antenna because it characterizes the total noise power delivered by the antenna to the input

of a receiver. Another useful �gure of merit for receive antennas is theG/T ratio, de�ned as

G/T (dB) = 10 logG

TAdB/K, (14.21)

where G is the gain of the antenna, and TA is the antenna noise temperature. This quantity

is important because, as we will see in Section 14.2, the signal-to-noise ratio (SNR) at the

input to a receiver is proportional to G/TA. The ratio G/T can often be maximized by in-

creasing the gain of the antenna, since this increases the numerator and usually minimizes

reception of noise from hot sources at low elevation angles. Of course, higher gain requires

a larger and more expensive antenna, and high gain may not be desirable for applications

requiring omnidirectional coverage (e.g., cellular telephones or mobile data networks), so

often a compromise must be made. Finally, note that the dimensions given in (14.21) for

10 log(G/T ) are not actually decibels per degree kelvin, but this is the nomenclature that

is commonly used for this quantity.

Wireless communications involves the transfer of information between two points without

direct connection. While this may be accomplished using sound, infrared, optical, or radio

frequency energy, most modern wireless systems rely on RF or microwave signals, usually

in the UHF to millimeter wave frequency range. Because of spectrum crowding and the

need for higher data rates, the trend is to higher frequencies, so the majority of wireless

systems today operate at frequencies ranging from about 800 MHz to a few gigahertz. RF

and microwave signals offer wide bandwidths, and have the added advantage of being able

to penetrate fog, dust, foliage, and even buildings and vehicles to some extent. Historically,

wireless communication using RF energy has its foundations in the theoretical work of

Maxwell, followed by the experimental veri�cation of electromagnetic wave propagation

by Hertz, and the practical development of radio techniques and systems by Tesla, Marconi,

and others in the early part of the 20th century. Today, wireless systems include broadcast

radio and television, cellular telephone and networking systems, direct broadcast satellite

(DBS) television service, wireless local area networks (WLANs), paging systems, Global

Positioning System (GPS) service, and radio frequency identi�cation (RFID) systems [4].


These systems are beginning to provide, for the �rst time in history, worldwide connectivity

for voice, video, and network communications.

One way to categorize wireless systems is according to the nature and placement of

the users. In a point-to-point radio system a single transmitter communicates with a single

receiver. Such systems generally use high-gain antennas in �xed positions to maximize

received power and minimize interference with other radios that may be operating nearby

in the same frequency range. Point-to-point radios are typically used for satellite commu-

nications, dedicated data communications by utility companies, and backhaul connection

of cellular base stations to a central switching of�ce. Point-to-multipoint systems connect

a central station to a large number of possible receivers. The most common examples are

commercial AM and FM radio and broadcast television, where a central transmitter uses

an antenna with a broad azimuthal beam to reach many listeners and viewers. Multipoint-

to-multipoint systems allow simultaneous communication between individual users (who

may not be in �xed locations). Such systems generally do not connect two users directly,

but instead rely on a grid of base stations to provide the desired interconnections between

users. Cellular telephone systems and some types of WLANs are examples of this type of

application.

Another way to characterize wireless systems is in terms of the directionality of com-

munication. In a simplex system, communication occurs only in one direction�from the

transmitter to the receiver. Examples of simplex systems include broadcast radio, televi-

sion, and paging systems. In a half-duplex system, communication may occur in two direc-

tions, but not simultaneously. Early mobile radios and citizens band radio are examples of

duplex systems, and generally rely on a �push-to-talk� function so that a single channel can

be used for both transmitting and receiving at different times. Full-duplex systems allow si-

multaneous two-way transmission and reception. Examples include cellular telephone and

point-to-point radio systems. Full-duplex transmission clearly requires a duplexing tech-

nique to avoid interference between transmitted and received signals. This can be done by

using separate frequency bands for transmit and receive (frequency division duplexing), or

by allowing users to transmit and receive only in certain prede�ned time intervals (time

division duplexing).

While most wireless systems are ground based, it is also possible to use satellite sys-

tems for voice, video, and data communications [5]. Satellites offer the possibility of

communication with a large number of users over wide areas, perhaps including the en-

tire planet. Satellites in a geosynchronous earth orbit (GEO) are positioned approximately

36,000 km above Earth, and have a 24-hour orbital period. When a GEO satellite is posi-

tioned above the equator, it becomes geostationary, and will remain in a �xed position rel-

ative to Earth. Such satellites are useful for point-to-point radio links between widely sep-

arated stations, and are commonly used for television and data communications through-

out the world. At one time transcontinental telephone service relied on such satellites,

but undersea �ber optics cables have largely replaced satellites for transoceanic connec-

tions as being more economical and avoiding the annoying delay caused by the very long

round-trip path between the satellite and Earth. Another drawback of GEO satellites is

that their high altitude greatly reduces the received signal strength, making it dif�cult

for two-way communication with handheld transceivers. Low Earth orbit (LEO) satel-

lites orbit much closer to Earth, typically in the range of 500�2000 km. The shorter path

length may allow line-of-sight communication between LEO satellites and hand-held ra-

dios, but satellites in LEO orbits are visible from a given point on the ground for only

a short time, typically between a few minutes and about 20 minutes. Effective coverage

therefore requires a large number of LEO satellites in different orbital planes. The ill-fated

Iridium system is probably the best-known example of a LEO satellite communications

system.


Gt Gr

PrPt

R

FIGURE 14.9 A basic radio system.

A general radio system link is shown in Figure 14.9, where the transmit power is Pt , the

transmit antenna gain is G t , the receive antenna gain is Gr , and the received power (de-

livered to a matched load) is Pr . The transmit and receive antennas are separated by the

distance R.

From (14.6)�(14.7), the power density radiated by an isotropic antenna (D =

1 = 0 dB) at a distance R is given by

Savg =Pt

4πR2W/m2. (14.22)

This result re ects the fact that we must be able to recover all of the radiated power by

integrating over a sphere of radius R surrounding the antenna; since the power is distributed

isotropically, and the area of a sphere is 4πR2, (14.22) follows. If the transmit antenna has

a directivity greater than 0 dB, we can !nd the radiated power density by multiplying by

the directivity, since directivity is de!ned as the ratio of the actual radiation intensity to the

equivalent isotropic radiation intensity. In addition, if the transmit antenna has losses, we

can include the radiation ef!ciency factor, which has the effect of converting directivity to

gain. Thus, the general expression for the power density radiated by an arbitrary transmit

antenna is

Savg =G t Pt

4πR2W/m2. (14.23)

If this power density is incident on the receive antenna, we can use the concept of effective

aperture area, as de!ned in (14.14), to !nd the received power:

Pr = AeSavg =G t Pt Ae

4πR2W.

Next, (14.15) can be used to relate the effective area to the directivity of the receive antenna.

Again, the possibility of losses in the receive antenna can be accounted for by using the

gain (rather than the directivity) of the receive antenna. Then the !nal result for the received

power is

Pr =G tGrλ

2

(4πR)2Pt W. (14.24)

This result is known as the Friis radio link formula, and it addresses the fundamental

question of how much power is received by a radio antenna. In practice, the value given

by (14.24) should be interpreted as the maximum possible received power, as there are a

number of factors that can serve to reduce the received power in an actual radio system.

These include impedance mismatch at either antenna, polarization mismatch between the

antennas, propagation effects leading to attenuation or depolarization, and multipath effects

that may cause partial cancellation of the received !eld.


Observe in (14.24) that the received power decreases as 1/R2 as the separation be-

tween transmitter and receiver increases. This dependence is a result of conservation of

energy. While it may seem to be prohibitively large for large distances, in fact the space

decay of 1/R2 is usually much better than the exponential decrease in power due to losses

in a wired communications link. This is because the attenuation of power on a transmission

line varies as e−2αz (where α is the attenuation constant of the line), and at large distances

the exponential function decreases faster than an algebraic dependence like 1/R2. Thus

for long-distance communications, radio links will perform better than wired links. This

conclusion applies to any type of transmission line, including coaxial lines, waveguides,

and even �ber optic lines. (It may not apply, however, if the communications link is land

or sea based, so that repeaters can be inserted along the link to recover lost signal power.)

As can be seen from the Friis formula, received power is proportional to the prod-

uct PtG t . These two factors�the transmit power and transmit antenna gain�characterize

the transmitter, and in the main beam of the antenna the product PtG t can be interpreted

equivalently as the power radiated by an isotropic antenna with input power PtG t . Thus,

this product is de�ned as the effective isotropic radiated power (EIRP):

EIRP = PtG t W. (14.25)

For a given frequency, range, and receiver antenna gain, the received power is propor-

tional to the EIRP of the transmitter and can only be increased by increasing the EIRP. This

can be done by increasing the transmit power, or the transmit antenna gain, or both.

The various terms in the Friis formula of (14.24) are often tabulated separately in a link

budget, where each of the factors can be individually considered in terms of its net effect

on the received power. Additional loss factors, such as line losses or impedance mismatch

at the antennas, atmospheric attenuation (see Section 14.5), and polarization mismatch can

also be added to the link budget. One of the terms in a link budget is the path loss, account-

ing for the free-space reduction in signal strength with distance between the transmitter

and receiver. From (14.24), path loss is de�ned (in dB) as

L0(dB) = 20 log

(

4πR

λ

)

> 0. (14.26)

Note that path loss depends on wavelength (frequency), which serves to provide a normal-

ization for the units of distance.

With the above de�nition of path loss, we can write the remaining terms of the Friis

formula as shown in the following link budget:

Transmit power Pt

Transmit antenna line loss (−)L t

Transmit antenna gain Gt

Path loss (free-space) (−)L0

Atmospheric attenuation (−)L A

Receive antenna gain Gr

Receive antenna line loss (−)Lr

Receive power Pr

We have also included loss terms for atmospheric attenuation and line attenuation. Assum-

ing that all of the above quantities are expressed in dB (or dBm, in the case of Pt ), we can


write the receive power as

Pr (dBm) = Pt − L t + G t − L0 − L A + Gr − Lr . (14.27)

If the transmit and/or receive antenna is not impedance matched to the transmitter/

receiver (or to their connecting lines), impedance mismatch will reduce the received power

by the factor (1− |Ŵ|2), where Ŵ is the appropriate re ection coef!cient. The resulting

impedance mismatch loss,

L imp(dB) = −10 log(1− |Ŵ|2) ≥ 0, (14.28)

can be included in the link budget to account for the reduction in received power.

Another possible entry in the link budget relates to the polarization matching of the

transmit and receive antennas, as maximum power transmission between transmitter and

receiver requires both antennas to be polarized in the same manner. If a transmit antenna is

vertically polarized, for example, maximum power will only be delivered to a vertically po-

larized receiving antenna, while zero power would be delivered to a horizontally polarized

receive antenna, and half the available power would be delivered to a circularly polarized

antenna. Determination of the polarization loss factor is explained in references [1], [2],

and [4].

In practical communications systems it is usually desired to have the received power

level greater than the threshold level required for the minimum acceptable quality of service

(usually expressed as the minimum carrier-to-noise ratio (CNR), or minimum SNR). This

design allowance for received power is referred to as the link margin, and can be expressed

as the difference between the design value of received power and the minimum threshold

value of receive power:

Link margin (dB) = LM = Pr − Pr(min) > 0, (14.29)

where all quantities are in dB. Link margin should be a positive number; typical values may

range from 3 to 20 dB. Having a reasonable link margin provides a level of robustness to the

system to account for variables such as signal fading due to weather, movement of a mobile

user, multipath propagation problems, and other unpredictable effects that can degrade

system performance and quality of service. Link margin that is used to account for fading

effects is sometimes referred to as fade margin. Satellite links operating at frequencies

above 10 GHz, for example, often require fade margins of 20 dB or more to account for

attenuation during heavy rain.

As seen from (14.29) and the link budget, link margin for a given communication

system can be improved by increasing the received power (by increasing transmit power or

antenna gains), or by reducing the minimum threshold power (by improving the design of

the receiver, changing the modulation method, or by other means). Increasing link margin

therefore usually involves an increase in cost and complexity, so excessive increases in link

margin are usually avoided.

EXAMPLE 14.4 LINK ANALYSIS OF DBS TELEVISION SYSTEM

The direct broadcast system in North America operates at 12.2�12.7 GHz, with a

transmit carrier power of 120 W, a transmit antenna gain of 34 dB, an IF band-

width of 20 MHz, and a worst-case slant angle (30◦) distance from the geostation-

ary satellite to Earth of 39,000 km. The 18-inch receiving dish antenna has a gain

of 33.5 dB and sees an average background brightness temperature of Tb = 50 K,

with a receiver low-noise block (LNB) having a noise !gure of 0.7 dB. The re-

quired minimum CNR is 15 dB. The overall system is shown in Figure 14.10.


FIGURE 14.10 Diagram of the DBS system for Example 14.4.

Find (a) the link budget for the received carrier power at the antenna terminals,

(b) G/T for the receive antenna and LNB system, (c) the CNR at the output of the

LNB, and (d) the link margin of the system.

Solution

Wewill take the operating frequency to be 12.45 GHz, so the wavelength is 0.0241

m. From (14.26) the path loss is

L0 = 20 log

(

4πR

λ

)

= 20 log

(

(4π) (39× 106)

0.0241

)

= 206.2 dB

(a) The link budget for the received power is

Pt = 120 W = 50.8 dBm

G t = 34.0 dB

L0 = (−)206.2 dB

Gr = 33.5 dB

Pr = −87.9 dBm = 1.63× 10−12 W.

(b) To �nd G/T we �rst �nd the noise temperature of the antenna and LNB cas-

cade, referenced at the input of the LNB:

Te = TA + TLNB = Tb + (F − 1)T0 = 50+ (1.175− 1)(290) = 100.8 K.

Then G/T for the antenna and LNB is

G/T (dB) = 10 log2239

100.8= 13.5 dB/K.

(c) The CNR at the output of the LNB is

CNR =PrGLNB

kTeBGLNB=

1.63× 10−12

(1.38× 10−23)(100.8)(20× 106)= 58.6 = 17.7 dB.

Note that GLNB, the gain of the LNB module, cancels in the ratio for the output

CNR.

(d) If the minimum required CNR is 15 dB, the system link margin is 2.7 dB. ■

The receiver is usually the most critical component of a wireless system, having the over-

all purpose of reliably recovering the desired signal from a wide spectrum of transmit-

ting sources, interference, and noise. In this section we will describe some of the critical

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