Chapter 19WIN - Coisas

8/10/2019 Chapter 19WIN - Coisas

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CHAPTER

19

Communication Systems

his chapter introduces the foundations of electrical communication systems,

emphasizing basic analog communications ideas. An overview of digital

communications concepts is provided in the last section.

The subject of electrical communications is one that touches everyones life:

telephones, TV and radio have been a part of our lives for many decades; today,

new means of communications are becoming as pervasive as the traditional ones.

Computer networks, satellite weather systems, personal communication systems (pagers,

cellular phones, etc.) are becoming essential parts of our everyday lives. The aim of this

chapter is to present the basic mathematics of spectrum analysis, which are the

foundations of all communication systems, and the basic operation of amplitude- andfrequency-modulation systems. The explanation of these concepts is supplemented bythe use of computer-aided tools. In addition, the chapter also includes an overview ofdifferent types of commonly used communication systems.

Upon completing the chapter, you should:

a. Be familiar with the most common types of communication systems in blockdiagram form.

b. Be capable of performing spectral analysis of simple signals using analytical andcomputer-aided tools.

c. Understand the principles of AM modulation and demodulation, and perform basiccalculations and numerical computations on AM signals.

d. Understand the principles of FM modulation and demodulation, and perform basiccalculations and numerical computations on FM signals.

T


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19.1 INTRODUCTION TO COMMUNICATION SYSTEMS

The modern era of communications began with the telegraph and the Morse

code (http://www.soton.ac.uk/~scp93ch/refer/alphabet.html#punctuation), and rapidly

moved towards radio and television. Table 19.1 summarizes some of the major dates in

the history of communication systems.

Table 19.1 A Brief History of CommunicationsDate Event

1838 Samuel F. B. Morse demonstrates telegraph

1876 Alexander Graham Bell patents the telephone

1897 Marconi patents a complete wireless telegraph system

1906 Lee DeForest invents the triode amplifier

1915 Bell System completes a transcontinental telephone line

1918 B.H. Armstrong perfects the superheterodyne receiver

1937 Alec Reeves conceives pulse code modulation

1938 Television broadcasting begins

WW II Radar and microwave systems are developed

1948 The transistor is invented; Claude Shannon publishes

Mathematical Theory of Communications

1956 First transoceanic telephone cable1960 First communications satellite, Telstar I, is launched

1962-66 High speed digital communications

1965 Mariner IV transmits pictures from Mars to Earth

1970 Color TV

1970 Commercial relay satellite telecommunications

1975 Intercontinental computer communication networks

Information, modulation and carriers

The purpose of communication systems is to communicate information; the fourmost common sources of information are:speech (orsound), videoand data. Regardless

of the source, the information that is transmitted and received in a communication system

consists of a signal, encoding the information in some appropriate fashion. Figure 19.1

depicts the general layout of a communication system: an input transducer (e.g., amicrophone) converts the input message into a message signal (e.g., a time varying

voltage) that is transmitted over a channel, and converted by a receiver into an output

signal. An output transducer(e.g., a loudspeaker) converts the received signal into an

output message (e.g.: sound). The transmitter performs a very important function oncommunication signals by encoding the signals in some fashion making use of a carrier

signal. The information is contained in a so-called modulating signal that modulatesa

carrier signal. For example, in FM radio the modulating signal consists of speech and

music, and the carrier is a sinusoidal wave of pre-determined frequency, much higherthan the modulating signal frequency. Table 19.2 summarized the frequency band

allocationand typical applications in each frequency band. More detailed information

can be found at http://www.ntia.doc.gov/osmhome/allochrt.pdf.

There are two principal reasons for the use of a very broad spectrum of carrier

frequencies. The first is that allowing for a broad spectrum permits many simultaneoususers to broadcast information at different frequencies without interference among

different transmissions; the second is that depending on the frequency of the carrier, the

electromagnetic waves that are transmitted have different propagation characteristics.Thus, different carrier frequencies are better suited for propagating over long distances

than others. Table 19.2 summarizes the frequency spectrum allocations used today.


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Input

transducer

Transmitter Channel Receiver Output

Transducer

Carrier

Input message Message signal Transmitted signal Received signal Output signal Output message

Figure 19.1 Block diagram of a communication system

Table 19.2 Frequency bands

Frequency

Band

Name medium Applications

3-30 kHz Very Low Frequency(VLF)

Wire pairs Long-range navigation, sonar.

30-300 kHz Low Frequency (LF) Wire pairs Navigational aids, radio beacons.

300-3000

kHz

Medium Frequency

(MF)

Coaxial

Cable

Maritime radio, direction

finding, Coast Guard,

commercial AM radio.

3-30 MHz High Frequency (HF) Coaxial

Cable

Search and rescue, aircraft

communications with ships,

telegraph, telephone andfacsimile.

30-300

MHz

Very High Frequency

(VHF)

Coaxial

Cable

VHF television channels, FM

radio, private aircraft, air traffic

control, taxi cabs, police.

0.3-3 GHz Ultra High Frequency(UHF)

CoaxialCable

Waveguide

UHF television channels,surveillance radar, satellite

communications.

3-30 GHz Super High

Frequency (SHF)

Waveguide Satellite communications,

airborne radar, approach radar,weather radar, land mobile.

30-300 GHz Extremely High

Frequency (EHF)

Waveguide Railroad service, radar landing

systems, experimental.

> 300 GHz Optical frequencies Opticalfiber

Wideband data, experimental.

Classification of communication systems

Communication systems can be classified into two basic families, based on the

nature of the message signal: analog communication systems and digital

communication systems. In this chapter we shall primarily focus on analog

communications, although it should be remarked that digital communications are taking

an increasingly prominent role even in the most common applications1. Another

classification may be made based on the type of transmission: light wave vs. radio

frequency, or RF transmission, as is explained in the next section. A third classification

is that of carrier vs. direct baseband transmission system. This latter classification is

based on whether the signal of interest is directly transmitted (e.g., as in the case of thetelegraph), or whether the signal modulatesa carrier wave, as in the case of AM and FM

radio transmission.

1An example of this phenomenon is the changeover from analog to digital systems in

cellular telephony. Both systems coexist at the present time, but it is reasonable to

forecast that in a few years allpersonal communication systemswill be digital.


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Communication channels

The modulated transmitted signal can reach the receiver in a number of ways.

In some cases, communication systems are hard wired. Examples of this configuration

are local area computer networks, local telephone systems and local cable TV networks.

Depending on the frequency range, the transmitted signal can be carried by twisted wirepair, coaxial cable, waveguides, or optical fiber. However, in most communications

systems, after the signal had been carried over a wire or cable, it is eventually broadcastover air by an antenna, to be received by a similar antenna elsewhere. Figure 19.2 depicts

some typical communication system components.

Figure 19.2 Communication system components; clockwise from top left: coaxial

cables; RF connectors; detail of coaxial cable; RF cabling components; monopoleantenna; optical fiber bundle.

The range of transmission can be significant consider that signals can be

received from the far reaches of the solar system via radio astronomy. The most

common means of transmission of communication signals is via the broadcast of radio

frequency waves over the air. To understand the different types of wave propagation,

we need to briefly explain the geometry of the earths atmosphere. With reference to

Figure 19.3, the atmosphere is composed of layers, of which the troposphere and theionosphere are the most important for radio wave transmission. The troposphere (up to

about 20 km above sea level) is where the earths air is contained; air density,

temperature and humidity decrease with increasing altitude. The propagation of radio

waves in air depends on various properties of the medium. The speed of propagation of

electromagnetic waves and the refractive index of the medium (causing the deflection ofthe wave) increase with altitude; as a consequence, radio waves tend to bend back

towards the earth as they propagate through the troposphere.

The ionosphere is so called because of the ionization of the small amounts of air

present at these altitudes (50 to 600 km); electromagnetic waves reaching the ionosphere

may propagate through it with some losses (attenuation), or may be reflected down to

earth, depending on the frequency of the transmissions. In general, frequencies above 30MHz will propagate through the ionosphere, and are therefore suitable for space

communications (see Web reference to radio astronomy above).

To achieve long range communications over the earth, use is made of so-called

sky waves. These are waves that are reflected by the ionosphere, and permit reaching

points beyond the horizon. The frequencies used for these waves are below 30 MHz topermit reflection from the ionosphere. Short-wave radio makes use of the sky wave.

Troposheric waves can also propagate beyond the horizon, but instead of being

reflected, as in the case of sky waves, they bend around the earth because of diffusion

(scattering). Direct waves are used in line-of-sight transmission , where the transmitter

and receiver are in the line of sight of one another. The earths curvature is the primary

limitation to the distance of such transmissions; however, due to reflections from the


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ground, and to ground and surface waves, this transmission can achieve greater distances

than one would calculate simply based on the earths curvature and the height of the

antennas.

Coaxial cablesare very commonly used for the transmission of radio-frequencywaves over short to medium distances, typically in the frequency range between afraction of a MHz to hundreds of MHz. Coaxial cable consists of a copper core,

surrounded by an insulating layer, in turn surrounded by a conductive (ground) layer andby an external protective sheath. Today the most common example of the use of coaxial

cables is the distribution of cable television signals from the receiving station to

individual homes.

An increasingly common type of communication systems is based on light wave

transmission. Light is also electromagnetic radiation, but at much higher frequencies

than radio waves. The main drawback in the use of light as a carrier is that it needs to be

enclosed in a guide to travel over significant distances; optical fibers are used to achieve

such transmission. An optical fiber consists of a hair-thin strand of glass, the core,

surrounded by a protective layer, the cladding. Snells law of optics ensures that, if lightenters the fiber at a sufficiently low angle of incidence, the transmission benefits fromtotal internal reflection, confining the light signal to the core with minimal losses. High-speed computer communications networks are increasingly making use of optical fibers.

Direct wave transmission with repeater

Ionosphere

Sky wave

Earth

Figure 19.3 Propagation of radio frequency waves

19.2 SPECTRAL ANALYSIS

Signal spectra.You know from Chapters 4 and 6 that signals can be represented both in time-

domain and in frequency-domain form. The phasor notation introduced in Chapter 4 is

the starting point of the frequency domain representation, or spectral representationofsignals: a phasor describes a sinusoidal signals amplitude and phase as a function of

frequency. The spectrumof a signal consists of the frequency domain representation of

the voltage signal. For example, the signal x (t) = A1 cos( 1t+ 1 ) only contains a singlesinusoidal frequency, 1, and its spectrum therefore consists of a pair spectral linesat

the frequency1. Figures 19.4(a) and (b)-(c) depict the representation of a sinusoidalsignal in the time domainand in thefrequency domain. Note that to completely represent

the frequency domain signal one needs to consider both magnitude and phase, as wasdiscussed in Chapter 4. Note also that the spectrum of the signal exists at both positiveand negative frequencies

2; this is a mathematical consequence of the definition of the

Fourier transform, as will be shown soon.

2 Although negative frequencies have no physical significance, the mathematical form of theFourier Series and Transform requires that we consider the spectrum of the signal at both positiveand negative frequencies.


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-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

time, s

x(t),V

(a)

-200 -150 -100 -50 0 50 100 150 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

-200 -150 -100 -50 0 50 100 150 200

-3

-2

-1

0

1

2

3

Figure 19.4Time-domain (a) and frequency-domain ((b) magnitude, (c) phase)

representation of a sinusoidal voltage (amplitude: 1 V-peak; phase: 0 rad)

Example 19.1 Sinusoidal signal spectrum

Problem:

Generate the spectrum of a signal consisting of the addition of two unity-amplitude sine

waves for different frequencies and phases. Plot the time-domain sum and the spectrum

of the sum.

Solut ion:

Known quantities:

Sine wave amplitude, frequency and phase.

Find:

Plot the time domain sum of the signals and the frequency domain spectrum.

Schematics, diagrams, circuits and given data.

1= 300 rad/s; 2= 500 rad/s;1= 0 rad; 2= /4 rad/s;

Assumptions:

None.


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Periodic signals: Fourier series

Periodic signals (see Chapter 4 for an introduction to signal waveforms) can be

represented by the infinite summation of sinusoidal signals, as was explained in the Box

Fourier Series in Chapter 6. This section explores the idea of the Fourier seriesrepresentation of a periodic signal more formally.

Let the signalx(t)be periodic with period T,that is,

x (t) = x (t+T) = x (t+nT) n = 0,1,2, 3... (19.1)An example of a periodic signal is shown in Figure 19.6.

T 2T 3T t

x(t)

Figure 19.6 A periodic signal

The signal x(t) can be expressed by means of an infinite summation of sinusoidal

components according to one of the following three (equivalent) representations:

In each of these expressions, the period, T, is related to the fundamental frequency of the

signal, 0, by:

0 = 2f0 =2

Trad/s (19.6)

It is straight forward to show that (19.4) is equivalent to (19.3), by expanding (19.4):

c n sin n2

Tt n

=c n cosn( )sin n

2

Tt

+ c n sin n( )cos n

2

Tt

where an =c n sin n( )and bn =cn cosn( ); one can verify that the equality holds if

an2 +bn

2 =cn andb

n

an

= tann( ). (19.7)

x (t) =a0 + an cosn2

Tt

n =1

+ bn sin

n2

Tt

n=1

Sine-cosine (quadrature) representation (19.3)

x (t) =c 0 + c n sin n2

Tt+ n

n=1

or

x (t) =c 0 + c n cos n2

Tt n

n=1

Magnitude and phase representation (19.4)

x (t) = n ejn

2

Tt

n =

Complex representation (19.5


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Similarly, one can show that (19.4) is equivalent to (19.3) if

an2 +bn

2 =cn andb

n

an

= tan n( ) (19.8)

Figure 19.7 is a graphical representation of the equivalence of the {an, bn} and {cn, n}forms of the Fourier series. Equations (19.7-8) permit easy conversion between the two

forms of the Fourier series.

ncb = c sin( )

n n n

a = c cos ( )n n n

n

Figure 19.7 Quadrature and magnitude-phase representation of sinusoidal signals

In each of the above representations, 0 =

2f0 =

2

T is called the

fundamental frequency(in units of radians per second), and the frequencies 2o, 3o,

4o,etc. are called its harmonics.

The computation of the {an,bn}or {cn, n}coefficients for the periodic

functionx(t)is based on the following formulas:

a0 =1

Tx (t)dt

0

T

(19.9)

an =2

Tx (t)cos n

2

Tt

dt

0

T

n = 1, 2,3,... (19.10)

bn =2

Tx (t) sin n

2

Tt

dt

0

T

n = 1, 2,3, ... (19.11)

Note that in the above equations it does not matter where the integration starts, provided

that it is carried out over one entire period. The cnand n coefficients can be derived

from the an and bn coefficients as shown earlier. The coefficients of the complex

representation are computed using the following expression:

n =1

Tx (t)e

jn2

Tt

dt

0

T

n = 0, 1, 2,... (19.12)


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0

T/2

T

0

A

-A

t

Figure 19.9 Sawtooth signal

Assumptions:

The function repeats periodically.

Analysis:

The function of Figure 19.9 is an odd function, since x (t) = x (t) . Thus weonly need compute the bn (sine) coefficients. First, we determine an expression forx(t):

x (t) = A 1 2t

T

then we evaluate the integral of equation 19.11:

bn =2

Tx (t) sin n

2

Tt

dt

0

T

=2

TA 1

2t

T

sin n

2

Tt

dt

0

T

=2

TA sin n

2

Tt

dt

0

T

+2

TA

2t

T

sin n

2

Tt

dt

0

T

= 0

4A

T2 tsin n

2

T t

dt0

T

=

4A

T2

1

n2 2

T

2 sin n

2

T t

t

n2

T

cos n

2

T t

0

T

= 4A

T2

T2

n2

42

sin 2n( ) T2

n2cos 2 n( )

=

2A

n, n = 1,2, 3, ...

Comments:

Note that the final expression for the Fourier series of the sawtooth waveform is actually

quite simple, indicating that the amplitude of the spectral components of the decreases inproportion to the harmonic number, n.


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Example 19.3 Fourier series of pulse train

Problem:

Compute the complete Fourier series of the periodic pulse train shown in Figure 19.10.

Solut ion:

Known quantities: Amplitude, period and functional form of the signal.

Find: bnand cncoefficients as a function of n

Schematics, diagrams, circuits and given data.: =

T= 0. 2

A

/2 t0 T

x(t)

/2 /2

Figure 19.10 Periodic pulse train

Assumptions:

The function repeats periodically.

Analysis:

The function of Figure 19.10 is neither even nor odd. Since there is nothing to

be gained by using the quadrature (odd-even) representation, we shall use the complex

form. The expression forx(t)is very simple in this case:

x (t) = A for 2

< t 2

x (t) = 0 for 2


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n = 1n sin n( ) =1

sinc n( )

Figure 19.11(a) depicts the pulse train corresponding to the numerical values given

above, and Figure 19.11(b) its Fourier series coefficients up to n= 1000. The envelopeof

the discrete-frequency coefficients is the sinc function defined above.

Figure 19.11 Time-domain representation andFourier Series spectrum of periodic pulsetrain with = 0.2.

Comments:

Note that in the complex form of the Fourier series, the coefficients range from negativeinfinity to positive infinity.

Focus on Computer-Aided Tools

A Matlab file used to generate the figures shown in this example may be found on the

book website, http://www.mhhe.com/engcs/electrical/rizzoni.


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Non-periodic signals, Fourier transform

Practical communication signals have both periodic and non-periodic

components. Typically, the carrier waveform is periodic (usually a sine wave), while the

modulating signal, consisting of speech, music, video or data, is non-periodic. The

analysis of non-periodic signals uses a mathematical tool different from (but related to)the Fourier Series: the Fourier transform. The Fourier transform, also named after the

French mathematician Fourier (see box Fourier Series in Chapter 6), is an integraltransform, so called because it is mathematically represented by an integral, and because

it performs a transformationbetween two domains: the time domainand the frequencyor spectral domain.

The Fourier transform of a function x (t) is the function X( ) defined by theintegral

Conversely, if the function X( ) is known, the inverse Fourier transform isdefined by:

The pair x (t) and X( ) represent a Fourier transform pair, a relationship

usually denoted by x (t) X( ) . Table 19.3 provides a useful summary of commonFourier transform pairs.

Before proceeding with the Fourier transform, it will be useful to define a

function called the unit impulseor delta function. This function plays a very important

role in Fourier analysis. The unit impulse function, (t), can be defined as the derivativeof the unit step function introduced in Chapter 6 (equation 6.52):

The unit impulse function has three important properties: 1) it has area equal to

one; 2) it has infinite amplitude; and 3) it has zero duration, that is, its occurrence is

concentrated at one instant in time. Clearly, such a signal is a mathematical abstraction,

since it is impossible to physically generate a signal that has zero duration and infinite

amplitude. Figure 19.12 shows how one can think of the delta function as the limit of a

sequence of rectangular pulses that are increasingly narrow and tall, such that the product

of height (1/) and width () is always equal to 1: the delta function can be thought of asthe limit of this sequence as approaches zero.

The delta function has one further property that is of interest in signal analysis:

x (t) tt0( )dt

= x t0( ) (19.16)

that is, the delta function samples the function x(t)at the time of the occurrence of the

impulse.

A number of properties of the Fourier transform can assist in the computation of

Fourier transforms. The most important properties are summarized in Table 19.3. Some

Fourier transform pairs are listed in Table 19.4.

(t) = du (t)

dt or u(t) = ( t' )dt'

t

Delta or unit impulse function (19.15)

X( ) = x ( t)e jtdt

Fourier transform (19.13)

x (t) =1

2 X( )ej

t

d

Inverse Fourier transform (19.14)

t

(t-t )0

t0

1

Figure 19.12 Deltafunction as the limit of a

sequence of rectangular

pulses of unit area


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Table 19.3 Properties of Fourier transforms

Property Signal Fourier transform

Time shift x (tt0 ) e jt0X( )Frequency shift e

jt0x (t) X( 0 )

Complex conjugate x * (t) X* ( )Time reflection x (t) X( )

Scaling x (at) 1

aX

a

Convolution x (t) y (t) X( ) Y( )Multiplication x (t) y (t) 1

2 X( ) Y( )

Differentiation d

dtx( t)

jX( )

Integrationx (t)

t

dt1

j X( ) + X(0) ( )

Table 19.4 Fourier transform pairs

x(t) X()

1 (t) 1

2 1 2 ( )

3 u( t)

(unit step)2 ( ) +

1

j

4 tu (t)

(unit ramp)2

d ( )

d

1

2

5 et

u(t) > 0 1 + j

6 tet

u( t) > 0 1

+ j( )2

7 ej 0t 2 ( 0 )8 cos0 t( ) ( 0 ) + ( + 0 )[ ]9 sin 0t( ) j ( 0 ) ( + 0 )[ ]10 cos0 t( )u( t)

2 ( 0 ) + ( + 0 )[ ]+ j

02 2

11 sin 0t( )u( t) j2

( 0 ) ( +0 )[ ]+

0

02 2

The importance of the Fourier transform is that it allows us to view signals inthe frequency or spectral domain. As you shall see shortly, the spectral representation of

signals is much more convenient and effective in representing communications signals,

among other reasons because it permits defining important concepts such as bandwidth

and spectrum allocation. You already know that sinusoidal signals are represented by

single frequencies in the spectrum, from the Fourier series discussion in the preceding

subsection. In the following three examples, we compute the Fourier transform of a well-

know signal, a sinusoid; of a signal we have not yet analyzed: a single rectangular pulse;and of a signal that occurs very frequently in communication systems: a sine wave burst,

or RF pulse. These examples will help you develop a feel for the spectral content of

different types of signals.


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Example 19.4 Fourier Transform of sine wave

Problem:

Compute the Fourier transform of an arbitrary sinusoidal signal.

Solut ion:

Known quantities: Functional form of the signal,x(t).

Find: X()

Schematics, diagrams, circuits and given data. Figure 19.13.

Assumptions: None

Analysis:

The signal shown in Figure 19.13, is defined as

)sin()( 0ttx =

The Fourier transform for the signal is calculated using the complex representation

n = 1T x (t)ejn 2T tdt

T2

T

2

= 1T

sin( 0t)ejn 2

T t

dt

T2

T

2

= 1T2j

ej

2T t e j

2 T t

[ ]e

jn 2T tdt

T2

T2

n = 12j1

T e

j2 T ( 1n) t ej

2T ( n 1)t[ ]dt

T2

T

2

Using the relation from Table 19.3, for the defined Fourier transform pairs, the above

integral can be evaluated as:

n =1

2j 2 0( ) 2 + 0( )[ ]n =j 0( ) + 0( )[ ]

Thus, the Fourier transform of an arbitrary sinusoid consists of a pair of delta functions inthe frequency domain, as shown In Figure 19.4.

Comments:

We can extend the result of this example to an arbitrary periodic signal , since we know

that any periodic signal can be represented by the sum of an infinite number of sinusoidal

functions. the Fourier transform,X()of a periodic signal, x(t) is a train of impulsesoccurring at the harmonically related frequencies and for which the area of the impulse at

Figure 19.13 Sinusoidal

signal of frequency 0.

Figure 19.14 Fourier

transform of sinusoid


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the nth harmonic frequency, n0 is 2 times the nth Fourier series coefficient an. The

Fourier series coefficients for this sinusoid signal are

a1 = 12j ,

a1 = 12j ,

ak = 0, |k | 1

Hence, the Fourier transform coefficients are given by,

a1 =

j

,

a1 = j

,

ak

= 0 |k | 1

Example 19.5 Fourier Transform of rectangular pulse signal

Problem:

Compute the Fourier transform of a square pulse signal.

Solut ion:


Find:X()

Schematics, diagrams, circuits and given data. Figure 19.15.

Assumptions:None.

Analysis:

Consider the rectangular pulse, x(t) of duration Tand unity amplitude shown in Figure

19.15. We define this pulse mathematically as follows:

x (t) =1 for |t | T2

0 for |t|> T2

The Fourier transform ofx(t)is

X( ) = e jtdtT2

T2

= ejt

j[ ]T2T2

= 2e

jT2 e

jT2

2 j

= 2 sin( T

2)

X( ) =Tsinc (T2

) where , sinc (x ) = sin(x )x

X(f) =Tsinc( fT)

Figure 19.15

Rectangular pulse.

Figure 19.16 FourTransform of squa

pulse (magnitude onl


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A plot of the spectrum is shown in Figure 19.16. The figure illustrates the characteristics

of the sinc function, with zero crossings at integer multiples of /T rad/s, and peakamplitude of 2T.

Comments:

Single and repetitive square bursts occur commonly in communication systems. The

analysis completed in this example will be useful in the following sections.



book website, http://www.mhhe.com/engcs/electrical/rizzoni.

Example 19.6 Fourier Transform of sine burst (RF pulse)

Problem:

Compute the Fourier transform of the sine wave burst shown in Figure 19.17.

Solut ion:


Find:X()

Schematics, diagrams, circuits and given data: Figure 19.17.

Assumptions:None.

Analysis:

The pulse signal x(t) shown in Figure 19.17 (a) consists of a sinusoidal wave of unit

amplitude and frequency fc, for a duration t = -T/2 to t = T/2.The signal x(t) can bedefined mathematically as follows:

x (t) = A cos( 2fc t) for |t| T2

0 for |t| > T2

The Fourier transform ofx(t)is

X( ) = x ( t)e jtdtT2

T

2

= A2 ejc t + e j c t( )

T2

T

2

ejt

dt

= A2

ej(c )t

j(c )+ e

j(+c ) t

j( +c )[ ]T2T

2

= A e

j(c )T

2

2j( c ) e

j( +c )T

2

2j( + c ) + e

j(c )T

2

2j( c ) + e

j( +c )T

2

2j( +c )

= A sin f +fc( )T2 f+ fc( )

+ sin f fc( )T2 f fc( )

X( ) = AT2

sinc f + fc( )T+ sinc f fc( )T[ ]Note that we could have used the frequency shifting property of the Fourier transform to

obtain the above result without carrying out the integration explicitly. The magnitude

spectrum of the RF pulse is shown in Figure 19.18. It clearly illustrates the frequency-

shifting propertyof the Fourier transform.

Figure 19.17Radio-frequency

(RF) burst.

Figure 19.18Magnitude spectrum

of RF burst


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19

Comments:

This signal is specifically referred to as aRF pulsewhen the frequency,fcof the sinusoid

wave falls in the radio frequency range.



book website, http://www.mhhe.com/engcs/electrical/rizzoni

Bandwidth

The bandwidthof a signal is the range of frequencies comprising the spectrum

of the signal. Bandwidth is a very important concept in communication systems, as the

allocation of radio frequency spectrum for different communication systems permits the

transmission of signal within a certain specified bandwidth. For example, standard FM

radio allows a bandwidth of 200 kHz for each radio station. The most common definition

of bandwidth is that of 3-dB bandwidth, also called half-power bandwidth. The 3-dB

bandwidth of a signal is defined as the frequency range between points where the signal

level is 3 dB below its maximum pass-band value. This informal definition is illustrated

in Figure 19.19, where an arbitrary voltage signal is shown to have a spectrum V(f), with

center frequency f0 and 3-dB bandwidth 2B.You will recall from the definitions given in Chapter 6 that the 3-dB point in a

frequency plot is the frequency where the amplitude has dropped to a value equal to

1 / 2 , or 0.707, times the maximum value. Since signal power is proportional to the

square of the voltage, the 3-dB bandwidth is also called the half-power bandwidth. Thus,

half of the signal power is contained in the frequency band f0 B to f0 +B ; we call 2Bthe bandwidth of the signal. Please observe that this informal definition assumes that the

signal spectrum as a band-pass shape. This is usually the case for most, if not all,

communication signals.

How much bandwidth does a signal require? This depends on twofacts: i) the bandwidth of the signal itself; and ii) the type of modulation. We shall

revisit the concept of bandwidth when we explore amplitude- and frequency-modulationsystems.

f0

f B0 0f +B

Vmax

0.707Vmax

V(f)

f

Figure 19.19 Definition of 3-dB (half-power) bandwidth


20/46

20

Example 19.7 Bandwidth of commercial AM (or TV, or FM) signals

Problem:

Analyze the bandwidth of the signal from a commercial AM station and determine how

many stations can be assigned frequencies over the frequency band assigned tocommercial AM.

Solut ion:

Schematics, diagrams, circuits and given data:

see http://www.ntia.doc.gov/osmhome/allochrt.pdf

Analysis:

As can be seen in the diagram displayed in the above website, the AM band frequency

allocation goes from 535 to 1,605 kHz. Each channel is allocated a bandwidth ofapproximately 10 kHz (we shall see in the next section what this calculation includes).

Thus, the total number of stations that can operate in the same region is approximately

equal to:

N =1605 535

10= 107

Each AM station can operate with a total bandwidth of 10 kHz. As we shall see in the

next section, this actually corresponds to a signal bandwidth of only 5 kHz.

Comments:

You are probably aware of the fact that the FCC licenses many more than 107 AM

stations in the USA. This is possible because AM broadcast has a limited range, and two

stations can be assigned the same frequency if there are located sufficiently far apart.

You may also have noticed that at night it is possible to receive AM radio signals frommuch greater distances (for example, in Ohio one can tune in stations from as far as New

York City and New Orleans late at night). This is a consequence of the change in

ionization density on the ionosphere during the night, permitting reflection of radio

waves over a longer range. The FCC regulates not only the frequency allocation, but also

the power allocated to a given station; a station may be required to switch to a lower-

power transmitter during certain times of the day.

Check Your Understanding

1. Repeat the analysis of Example 19.2 using the complex representation of the Fourierseries (equation 19.5)

2. Repeat the analysis of Example 19.3 using the quadrature representation of theFourier series (equation 19.3).

3. Compute all the coefficients of the Fourier series expansion for the signalx(t)= 1.5

cos(100t).4. Sketch the Fourier transform of a square pulse with unity amplitude and with a

duration of 10 s.

5. The spectrum of a signal can be described by the function X( ) =

2 + 2. Let

= 103, and calculate the 3-dB bandwidth of the signal. What is the center frequency

of the signal?


21/46

21

19.3 AMPLITUDE MODULATION AND DEMODULATION

The concept of amplitude modulation (AM) was introduced in Chapter 4

(Focus on measurements: capacitive displacement transducer), where it was shown that

the signal produced by a capacitive microphone (displacement transducer) inserted in a

Wheatstone bridge circuit, modulated the amplitude of a sinusoidal excitation signal. In

that example, the pressure changes sensed by the microphone constituted the modulation,while the sinusoidal excitation provided a carrier. In Chapter 8 (Focus on

measurements: peak detector for capacitive displacement transducer), it was shown that

a diode circuit was capable of demodulating the AM signal, and of recovering the desired

information (pressure changes corresponding to acoustic waves, that is, sound). In thissection, the same basic principles introduced in the above mentioned examples will be

discussed more formally, as they apply to AM communication systems.

The most common manifestation of amplitude modulation in communication

systems is commercial AM radio, or standard AM. The Federal Communications

Commission(FCC), a body that regulates the usage and allocation of the radio frequency

spectrum in the U.S.A., has assigned the frequency band between 540 and 1600 kHz to

commercial AM radio transmission. Each station can occupy a bandwidth of 10 kHz

centered around its carrier. As we shall see, this corresponds to an effective signal

bandwidth of 5 kHz sufficient for good reproduction of speech, and acceptablereproduction of music.

Basic principle of AM

AM signals are generated by modulating the amplitude of a carrier signal. Let

the carrier signal be a sinusoid at frequency c :

c (t) = A c cosct( ) Carrier signal (19.17)and for illustration purposes - let the modulation also be a single tone (sinusoid), at a

frequency m


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22

Example 19.8 Single-tone amplitude modulation

Problem:

Analyze the spectrum of a single-tone modulation signal based on the WOSU AM 820radio station. Use both analytical and computational tools.

Solut ion:

Known quantities: Carrier frequency; modulation index.

Find: Derive expressions for and plot time-and frequency domain waveforms of the AMsignal.


fc= 0.82 MHz; = 0.5.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 10-4

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time, s

m(t),V)

Message signal

(a)0 1 2

x10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time, s

c(t),V)

Carriersignal

(b)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x10-4

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(c)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 104

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-fm

fm

(d)-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x106

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-fc

fc

(e)

7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 9 9.2

x105

0

0.2

0.4

0.6

0.8

1

(f)

Figure 19.20 Time- and frequency-domain waveforms of single-tone AM signal

Assumptions: Assume unity amplitude for the carrier, Ac = 1, and a modulatingfrequencyfm= 10 kHz

Analysis:

Define the modulating signal m(t)and the carrier signal c(t)asm (t) = A m cos( 2fm t)

c (t) = A c cos( 2fc t)


23/46

23

These waveforms are plotted in Figures 19.20(a) and (b), respectively. The spectra of

these signals are plotted in figures 19.20(d and (e).

The AM waves(t)is given by

s( t) = Ac 1 + cos( 2fm t)[ ]cos( 2fc t)and is plotted in Figure 19.20(c). Using the Fourier transform pairs given in Table 19.3

(Pair 8), the Fourier transform ofs(t)can be expressed as the sum of three delta functions,

centered at the carrier and at the sum (upper sideband) and difference (lower sideband)frequencies. Note also that the spectrum is repeated for negative frequencies, as

explained earlier.

S(f) = A c2

(f fc ) + (f +fc )[ ]+ Ac4 (f fc fm ) + (f + fc + fm )[ ] + Ac

4 (f fc + fm ) + (f +fc fm )[ ]

Thus, the spectrum of an AM wave for the special case of sinusoidal modulation, consists

of delta function at fc,fc fm, and fcfm, as shown in Fig. 19.20(f).

Comments:

If you like, you may experiment with the value of the modulation index and see its effect

on the AM wave. It is recommended that the modulation index, ,benearly equal to 1,but not greater. If the modulation index is greater than 1 for any t, the carrier wave

becomes over-modulated, resulting in carrier phase reversals whenever the function

1 +m( t)( )crosses zero.


A Matlab file used to generate the figures shown in this example may be found on thebook website, http://www.mhhe.com/engcs/electrical/rizzoni. You may use this file to

experiment with changes in the modulation index.

The single-tone modulation example is very valuable to understand the basic properties

of an AM signal. In the next two examples we progress to the double-tone modulation,and then to a general, non-periodic modulating signal to explore the waveforms and

spectrum of more realistic AM signals.


24/46

24

Example 19.9 Double-Tone modulation

Problem:

Plot the frequency spectrum of a carrier signal with unity amplitude and frequencyfc= 1MHz which is amplitude modulated with a modulating signal m(t) consisting of two

sinusoidal frequencies.

Solut ion:

Known quantities: Carrier frequency, and amplitude; modulating signal

Find: Modulation index and frequency spectrum of the AM wave with the defined carrierand modulating signal.

Schematics, diagrams, circuits and given data:

fc= 1 MHz;Ac= 1 V; m(t)= 0.5cos(210000t) + 0.4cos(280000t).

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x10-4

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time, s

m(t),V)

Messagesignal

(a)

-18 -16 -14 -12 -10 -8 -6 -4 -2

x10-6

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time, s

c(t),V)

Carriersignal

(b)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x10-4

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(c)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 104

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-fm1

fm1

(d)

-1 -0.5 0 0.5 1

x106

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-fc

fc

(e)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

x106

0

0.2

0.4

0.6

0.8

1

(f)

Figure 19.21Time- and frequency-domain waveforms of double-tone AM signal

Assumptions:None.

Analysis:

The modulation index for the signal is defined as

= A

mA

c= max(

m (t)) min(m (t))2V

c= 0.9 + 0.8626

2= 0.8813


25/46

25

The spectrum of the AM wave in this case consists of delta functions at fc,fcfm1,fcfm2, -fcfm1, and -fc fm2, wherefm1,fm2, are the frequencies contained in the modulatingsignal. This is seen in Figure 19.21, where all time- and frequency domain waveforms are

plotted..

Comments:The frequency spectrum of the AM wave is just a shifted version of the originalmodulating signal with the shift in frequency equal to the carrier frequency. The portion

of the spectrum of an AM wave lying above the carrier frequency fc is the upper side-

band, whereas the symmetric portion belowfc is called the lower side-band.



book website, http://www.mhhe.com/engcs/electrical/rizzoni. You may use this file toexperiment with changes in the modulation index or in the signal frequencies.

Example 19.10 Non-periodic amplitude modulation

Problem:

Plot the frequency spectrum of a carrier signal with unity amplitude and frequency fc=

0.1 MHz which is amplitude modulated with a non-periodic modulating signal m(t)

having a defined shape.

Solut ion:

Known quantities: Carrier frequency, and amplitude; Modulating wave m(t)defined fora certain interval of time.

Find: Frequency spectrum of the AM wave.


m (t) =sin( 2f1t) + sin( 2f2t) + sin( 2f3t) + sin( 2f4t) +u( t) for t T

0 otherwise

u(t) = 1 for tTc (t) = A c cos( 2fc t)

fc= 0.1 MHz;Ac= 1;f1= 1kHz,f2= 2kHz,f3= 3kHz,f4= 4kHz.


26/46

26

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 10-3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(a)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 10-3

-6

-4

-2

0

2

4

6Amlitude modulated signal, s(t)

Time, s

s(t),V

(b)

-1.5 -1 -0.5 0 0.5 1 1.5

x 104

0

0.2

0.4

0.6

0.8

1

Frequency, Hz

Normalized spectrum of modulating signal, m(t)

|M(f)|,

V

(c)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

x 105

0

0.2

0.4

0.6

0.8

1

Hz

|S(f)|,

V

Normalized spectrum of modulated signal, s(t)

fc

(d)

Figure 19.22Time- and frequency-domain waveforms of non-periodic AM signal

Assumptions:None.

Analysis:

The signal waveform and the frequency spectrum of the modulating signal and of the AMwave are shown in Figures 19.22(a-d). The spectrum of the AM wave is a shifted version

of the modulating signal spectrum around the carrier frequency.

Comments:

If Bmax is the bandwidth of the modulating signal, (the highest frequency in the

modulating signal), the bandwidth of the AM wave is defined as, twice the highest

frequency in the modulating signal, i.e.,B = 2Bmax.



book website, http://www.mhhe.com/engcs/electrical/rizzoni. You will find this fle

useful in exploring more general AM signals.


27/46

27

AM demodulation; integrated circuit receivers

Demodulation is the process of recovering the modulating signal from a

received modulated signal. With reference to Figure 19.1, one can think of the

transmitter in an AM signal as the device that imposes the modulation on a carrier, while

the receiver extracts the modulating signal from a received AM signal. To understand thebasic principle of modulation and demodulation, we observe that amplitude modulation

consists in effect of multiplying the carrier signal times the modulating signal. Thisprocess is often called mixing, and a mixer is the device that implements this function,

that is, multiplication.Consider the AM signal of Equation (19.20), and multiply it by a second signal

at the same frequency as the carrier signal:

s( t) c( t) = Ac 1 +A

m

Ac

cosm t( )

cosc t( )

cosc t( ) (19.23)

The resulting squared term, cos2 ct( )can be expanded to yield:

s( t) c( t) =A c 1 +Am

Accosm t( )

cos

2ct( )=A c

21 +

A m

Accosm t( )

1 + cos 2c t( )( )

= Ac2

+m (t) + Ac2

+m ( t)

cos 2

ct

( )

(19.24)

You see that the result of this mixing operation consists of two terms: a constant

plus the modulation signal what we desire to recover and an amplitude modulated

term at a frequency equal to twice the carrier frequency. Note that the modulation signal

is back to baseband, that is lowfrequencies (for example, 0-5 kHz for speech and music),

and that it is therefore easy to recover the modulating signal by low-pass filtering the

output of the mixer. Figure 19.23 depicts a block diagram of a conceptual AM

demodulator as well as the spectra of the AM signal before and after mixing.The process of demodulating AM signals is carried out today by means of

integrated circuit receivers. Additional information may be found at the followinginternet sites:

http://www.national.com/pf/LM/LM1863.html

http://www.national.com/pf/LM/LM1868.html


19.6 Use the Matlab files that accompany Examples 19.8 and 19.9 to plot only the

positive spectrum of the single- and double-tone AM signals. Determine the

bandwidth of the AM signal in each case.

19.7 Determine the bandwidth of the modulating signal in Figure 19.22(c); what is

the bandwidth of the AM signal? Is this consistent with commercial FMpractice? Would the FCC allow a commercial station to broadcast such a

signal?


28/46


29/46

29

The instantaneous frequency of the FM signal varies linearly with the modulation, that is,

the carrier frequency increases and decreases with the modulating signal, as shown in

equation 19.25:

f( t) = fc +kfAm cos 2fm t( )= fc + f cos 2fm t( ) (19.25)

In the above expression we have implicitly defined the frequency deviation, f:

f =kfAm frequency deviation (19.26)

The frequency deviation is a very important characteristic of an FM signal; frepresentsthe maximum instantaneous deviation of the FM signal frequency from the carrier

frequency. fis dependent on the amplitudeof the modulating signal, and is independentof the modulating signal frequency. The constant kfdepends on the technique used forgenerating the modulated signal.

The instantaneous phase of the FM signal is calculated using equation 19.23:

i = 2 fi t( )dt = 2fc0

t

t+ff

m

sin 2fm t( ) (19.27)Note that equation 19.27 introduces another important constant: the ratio of the frequency

deviation to the modulating frequency is called the modulation index, :

=ffm

(19.28)

The parameter represents the maximum instantaneous deviation of the phase angle ofthe FM signal from the angle of the carrier. Now we can write the instantaneous angle of

the FM signal as follows.

i = 2fc t + sin 2fmt( ) (19.29)

Finally, the sinusoidally modulated FM signal is defined in equation 19.30:

s FM (t) = Ac cos 2fc t+ sin 2fm t( )[ ] (19.30)

It is now possible to make a formal distinction between the two cases mentioned earlier,narrowband FMand wideband FM, on the basis of the modulation index.

Narrowband FM

Narrowband FM corresponds to FM signals with a small modulation index (i.e.,


30/46

30

sAM (t) = Ac cos ct( )+A

c

2cos c m( )t+

Ac

2cosc + m( )t

(19.22)

A reasonable rule of thumb is that the approximation given in equation 19.34 holds for

< 0.3. Figure 19.25 (a)-(d) depicts the spectra of FM signals with various values of .Note how the bandwidth increases with the value of the modulation index. Only in (a)

and (b) is the signal narrowband FM.

-1 -0.5 0 0.5 1

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

-1 -0.5 0 0.5 1

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

-1 -0.5 0 0.5 1

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c)

-1 -0.5 0 0.5 1

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d)Figure 19.25 Bandwidth increases with modulation index in FM signals:

(a) = 0.1; (b) = 0.3; (c) = 0.6; (d) = 1.

Example 19.11 Narrowband FM

Problem:

Compare the spectrum of a narrowband FM waveform with that of an AM waveform

with the same modulating and carrier frequencies.

Solut ion:

Known quantities: Carrier frequency; modulation frequency; modulation index.

Find: Plot the frequency domain waveforms of the narrowband FM and AM signals.


fc= 1,000 Hz;fm= 100 Hz;Ac= 1 V;Am= 0.2 V; = 0.2; = 0.2.


31/46

31

Assumptions: Assume sinusoidal modulation and unity amplitude for the carrier,Ac= 1.

Analysis:

The files Freqmod.m and Ampmod.mare used to generate two signals; the first signal is

an FM waveform with = 0.2, the second is an AM signal with = 0.2. The resultingspectra are plotted in figures 19.26a and b, respectively.

0 2000 4000 6000 8000 10000 120000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 8000 10000 120000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 19.26Comparison of narrowband FM and AM signal spectra (a) FM, = 0.2; (b)

AM, = 0.2.

Comments: Note how the two amplitude spectra are virtually identical. Equations19.34 and 19.22 predict this result. The only difference between the two spectra is in the

phase angle of the signals.



book website, http://www.mhhe.com/engcs/electrical/rizzoni. You may use this file to

observe the difference in phase between the signals.

Wideband FM

The mathematical representation of wideband FM signals is far more complex

than the approximation of equation 19.34. The nonlinearity of the wideband FM signal is

described using Bessel functions; this analysis is beyond the scope of this chapter, and

the interested reader is referred to any one of a number of excellent textbooks in electrical

communications.

Transmission bandwidth of FM signals

The transmission bandwidth of a frequency-modulated signal is theoretically

infinite; however, practical approximations are possible. In the case of narrowband FM

we have already seen that we have the same transmissions bandwidth as in an AM signal:

B = 2fm . For large values of the modulation index, , the bandwidth of the FM signalcan be experimentally observed to be close to the total frequency excursion, f, or

2f. These observations lead to the well-known Carsons rule, relating the approximatetransmission bandwidth to the frequency deviation and to the modulation index:

Carsons rule (19.35)

Carsons rule suggests that, as becomes larger, the bandwidth approaches 2f, while as

decreases, the bandwidth becomes closer to 2fm.

B = 2 f + 2fm = 2f 1 +1


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32

Example 19.12 Commercial FM broadcast

Problem:

Use Carsons rule to analyze the bandwidth of a commercial FM station.

Solut ion:

Known quantities: Carrier frequency; modulation index.

Find: Approximate signal bandwidth


fc= 90.5 MHz;Ac= 1;Am= 1;fm= 10 kHz; kf= 6,000;= 0.2.

Assumptions: Assume sinusoidal modulation.

Analysis:

For sinusoidal modulation, the frequency deviation is f =kfAm= 6 kHz.Then, Carsons rule predicts a bandwidth

B = 2 f + 2fm = 2f 1 +1

= 1.2 104 1 +

1

0.2

= 72 kHz

Example 19.13

Problem:

Given an FM message signal, find:

a. The bandwidth of the message signal,Bm.b. Bandwidth of the modulated carrier signal,Bc.c. Band of frequencies occupied by the signal,B.

Solut ion:

Known quantities: Modulation frequency, carrier frequency; modulation constant.

Find:Bm;Bc;


v m = 5 cos( 200 t) ;fc= 100fm; kf =1000

2

Hz/V

Assumptions: Assume sinusoidal modulation.

Analysis:

a. The message signal is v m = 5 cos( 200 t) . Hence:


33/46


34/46

34

Example 19.14

Problem:

In the United States the assigned band for FM commercial broadcast is from MHz0.88

to MHz0.108 with 100 possible channels. Find the bandwidth for each channel.

Solut ion:

Known quantities:

Band for FM commercial broadcast, number of channels.

Find: The bandwidth of each channel.

Schematics, diagrams, circuits and given data. See problem statement.

Assumptions:None.

Analysis:The bandwidth for each channel is defined as:

B =Total Band

Number of channels=

108.0 - 88

100= 200 kHz

Comments:

Each commercial FM radio station has a bandwidth allocation of 200 kHz.

FM demodulationDemodulation of an FM signal is accomplished by performing a frequency-to-voltage

conversion, that is, by converting the frequency modulation into a voltage signal. This is

the reverse of the modulation process, and can be realized in a number of ways. We

describe two basic approaches in the following sub-sections.

Frequency-to-voltage conversion

If a pulse of fixed amplitude,A, and fixed duration, , is generated at each zero crossingof the sensor waveform, it can be readily shown that a voltage proportional to the

instantaneous signal frequency may be obtained. Figure 19.28 depicts the functional

form of a frequency-to-voltage converter.

v (t)S zero-crossing

detector

one-shot

multivibrator

ti

vZ

vOS v o

v (t)S

vZ

vOS

t

Ti

A

t ti-1 i

Figure 19.28Frequency to voltage conversion

Ideally frequency to voltage (F-V) conversion could be obtained by computing the

following integral:


35/46

35

vosti 1

ti

(t)dt = At

Ti

= 2Afi (19.36)

yielding a voltage proportional to the frequency ofS(t)during the ithcycle of the carrier

waveform. In practice it is quite difficult to reset the integrator of Figure 19.27 at each

zero crossing, so practical F-V converters employ a low-pass filter in place of the ideal

integrator.

Phase-locked demodulation

Another method for implementing F-V conversion utilizes a phase locked loop(PLL) as

a frequency-to-voltage converter, or FM demodulator. The PLL, can act as an FMdemodulator once it is phase-locked to an input signal waveform. When the PLL is in

lock, any change in the input signal frequency generates an error voltage at the output of

the phase detector, which can be either analog (a mixer), or digital, consisting of a pair of

zero crossing detectors.

The output voltage of the PLL, vo(t), is the voltage which is required in order to maintain

a voltage-controlled oscillator (VCO) running at the same frequency as the input signal,

and changes in the input signal frequency are matched by changes in vo(t). In this sense,

the PLL acts as an F-V converter, with the input-output characteristic shown in Figure

19.29. Note that the PLL can offer only a finite lock range.

phasedetector

vS

low-passfilter

amp

voltagecontrolledoscillator

vo

center frequency

lock range

, rad/s

v o

Figure 19.29Phase-Locked Loop F-V conversion

Integrated circuit receivers

FM demodulation is performed using integrated circuit receivers; additional informationmay be found in the following websites:http://www.national.com/pf/LM/LM1868.html

http://www.national.com/pf/DS/DS8906.html

http://www.national.com/pf/LM/LMX3162.html


19.8 Use the Matlab files that accompany Example 19.11 to compute and plot the

phase angle of the two spectra. Are the phase spectra identical?

19.9 What is the effect of changing the amplitude of the modulating signal on thebandwidth of the FM modulated signal?

19.10 Investigate the effect of changing on the FM modulated signal bandwidth and

analyze the spectrum of the FM signal of Example 19.11 with 5.0= .19.11 Find the carrier frequency for Channel 11 used by WCBE Radio, Columbus OH,per the FCC regulations for the commercial broadcast in the US (use data in

Example 19.14).


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19.5 EXAMPLES OF COMMUNICATION SYSTEMS

The objective of this chapter is to summarize some important applications of

modern communication systems. The overview given in this chapter is certainly not

exhaustive, but will give you a good summary of the technology underlying some of

todays communication systems and of their capabilities.

Global Positioning System (GPS)

The Global Positioning System (GPS), illustrated in Figure 19.30, is rapidly

supplanting older navigation technologies based on radio communications used by the

aircraft and marine industries, such as Loran-C. GPS is based on 24 satellites linked to

ground stations, and effectively replaces with much greater accuracy the century-old

system of navigation based on star position. You can think of the satellites as man-madestars. Differential GPS is capable of position measurements with accuracy of a few

centimeters. In recent years, GPS receivers have been miniaturized, and today amateur

sailors, private pilots and other private users have the ability to purchase hand-held GPS

units. Among the uses of GPS we list navigation systems for cars, boats, planes and

guiding agricultural machining for precision agriculture. The operation of GPS is

explained in five basic steps:

Table 19.5 Basic operation of GPS

1. GPS operates based on triangulation of signals received fromsatellites.

2. Triangulation is performed by the GPS receiver by measuringdistance knowing the travel time of radio signals.

3. Travel time is computed with the aid of accurate timingreferences.

4. The exact position of the satellites in space is used to calculatedistance.

5. Delays experienced by the signals in traveling from the satellite tothe ground stations are corrected.

Triangulation

The technique of triangulation is based on distance (range) measurements fromthe receiver location to the satellite. Four satellites are needed to determine exact

position of any receiver location on earth.

Measuring Distance

To measure distance, GPS computes the time required to receive each satellite signal.

The receiver and satellite both generate a synchronized signal; comparison of the signal

received from the satellite with that in the receiver is used to calculate time of travel.Since the speed of travel of electromagnetic waves is known, distance can be calculated.

Timing Accuracy

Satellites carry atomic clocks on board to provide accurate timing. The clock in a low-cost GPS receiver need not be as accurate, as an additional range (distance) measurement

can be used to remove the timing error in the receiver.

Satellite Positions

The position of the satellites is essential in calculating distance. The orbits of GPS

satellites are known, and any deviations are measured by the Deaprtment of Defense; anyerrors are transmitted to the satellites, which in turn transmit this error information to the

receivers along with their timing signals.

Figure 19.30GPS principle(Courtesy: Trimble

Navigation)


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Correcting Errors

Signals traveling through the ionosphere experience additional delays, which turn into

transmission errors. There are many methods for correcting errors; the technique called

differential GPScan eliminate almost all errors.

Further information

http://www.trimble.com/gps/http://www.garmin.com/

http://www.cnde.iastate.edu/staff/swormley/gps/gps.html

http://www.motorola.com/SPS/MCORE/gps/

Radar

The acronym RADAR stands for RAdio Direction And Ranging. Radar

systems operate by radiating electromagnetic waves (typically at microwave frequencies)

and by detecting the echo returned from reflecting objects (targets). Radar technology

was developed during World War II and played a significant part in the success of the

Allied Forces. While military applications are obvious, today Radar finds widespreadcivilian application in air traffic control and in tracking weather conditions, as well as in

marine navigation (see Figure 19.31 for some examples of radar technology used in the

latter application, and Figure 19.32 for an example of weather radar). In addition todetecting the position of a stationary target, radar is capable of determining the trajectory

of a moving target, thus predicting its future location. This function is, for example, very

useful in weather radar where one wishes to predict weather conditions. The principle on

which radar operation is based is that of the doppler shift, a concept with which you are

probably already intuitively familiar (think of the sound of a train whistle as the train

moves by a stationary observer). The doppler shift permits distinguishing a moving

target from a stationary one, thus allowing the radar system to discern the echo of astationary target from that of a moving target.

Further informationhttp://cirrus.sprl.umich.edu/wxnet/radsat.html

http://southport.jpl.nasa.gov/education/classroom/

http://www-cmpo.mit.edu/Radar_Lab/FAQ.html

http://www.skywarn.ampr.org/radartut.htm

Figure 19.31 Radarantenna (top) and

displays. (Courtesy

ProNav)


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Computer Networks

The distinction between communication and computer networks is increasingly

blurred today, as computers are used more and more commonly. In this brief subsection

we describe some of the basic ideas behind todays computer network technology.

Networks are divided into two basic categories: local area networks and wide areanetworks. Generally speaking, a local area network ties computers and other

communication systems within the same building, while wide area networks interconnectbuildings, cities and countries.

Figure 19.34 depicts the three basic network architectures: bus, ring and star.The bus architecture is characterized by a common bus shared by all elements of the

network; all data traffic is present on the bus, and can be read at each node. As an

example of a very common communication standard based on a bus architecture is

Ethernet, the computer communication protocol commonly used for high-speed data

transmission (see Chapter 15). When a computer wishes to transmit on the bus, it first

listensto determine whether the bus is available. If the bus is available, the computer can

transmit. Should a collision occur, a protocol is followed to repeat transmission. The

standard that rules this type of transmission is called carrier-sense multipleaccess/collision detection (CSMA/CD). The bus architecture is effective whenever

nodes share in the communications approximately equally. If one or a few nodes had a

disproportionate share of the traffic, then this architecture would not be very effective.

Bus architecture Ring architecture Star

architecture

Figure 19.34 Communication System architectures

The ring architecture(see Figure 19.32) is characterized by each node having

two connections: inbound and outbound. All network traffic must pass through eachnode. To determine whether a data is to be received by a specific node, a tokenis passedaround the ring. When a node needs to transmit, it captures the token and sends data out

in its place. When the message is received by the intended receiver, the message is

marked read and is sent back around the ring; when the message reaches the original

sender, the latter removes the message from the ring and replaces it with the token. Notethat a node cannot transmit until it receives a token. This method ensures that no one

node can dominate communications around the ring. Among the drawbacks of this

architecture is the fact that if a node is disconnected, the entire ring fails to operate.

The star architecture(see Figure 19.34) connects each node to a central hub

connection. Note that a star architecture can be used to implement both bus and ring

networks. The hub can serve the purpose of monitoring and enforcing various

restrictions.

Local Area Networks

Local area networks (LANs) are physically connected networks of computers in

relative close proximity. Office and school networks are the most common examples.

Connections are made through either copper wire (twisted pair or coaxial), or optical

fiber. Copper wire in the form of unshielded twisted pair (UTP) is capable of data rates

up to 100 Mbit/s (Mbps), and is most commonly used in LANs because of its low cost.

Tabled 19.6 summarizes the performance of different UTP categories.

Table 19.6 Twisted pair copper cable applications


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Category Application Maximum data rate

1 Telephone 1 Mbps

2 Token ring 4 Mbps 4 Mbps

3 10BaseT Ethernet 10 Mbps

4 Token ring 16 Mbps 20 Mbps

5 100BaseT Ethernet 100 Mbps

Coaxial cable is also used for LANs, especially Ethernet3, and comes in thick

and thin forms. Fiber optics are becoming increasingly common, through FDDI (fiber

distributed data interface) backbone networks, which carry the bulk of the traffic in a

large local area network such as corporate networks and university campuses. FDDI can

support a 100 Mbps rate using a token ring architecture.

Wide area networks

Outside of the local area, networking needs are usually handled by a

telecommunications provider, for example a telephone or cable TV company. Services

that are provided by telecommunications providers are based on bandwidth needs, and

include traditional copper lines (DS-n service), Integrated Services Digital Network

(ISDN), also based on copper lines, and optical fiber service (OC-n). Table 19.7

summarizes the data rates achievable through OC-n. Note that the fastest fiber opticservice is nearly 10 Gbps!

Table 19.7Optical fiber service data rates

OC-n Data rate (Mbps)

OC-1 51.84

OC-3 155.52

OC-12 622.08

OC-48 2488.32

OC-192 9953.28

Wireless Networks and Personal Communication Systems

Wireless mobile networks are becoming increasingly common in everyday

applications. With the advent of digital cellular telephony, the performance and cost ofmobile network services have improved to the point that a wireless communication

system may be preferable to a land-line-based one. An interesting statistic is that in a few

European countries the number of wireless telephone lines has recently exceeded that

of fixed telephone installation. The widespread availability of such mobilecommunication capabilities has led to the concept of personal communication systems

(PCS), capable of providing voice and data services in just about any setting.

Mobile wireless networks use a radio frequency (RF) carrier, and usually adopt

a cellular architecture. In such an architecture, a large number of antennas is used; each

antenna is at the center of a cell, as shown in Figure 19.35. A stationary user would thenreceive the signals through the antenna corresponding to the cell where s/he is located. If

the user is in motion, a handoff procedure is used to transfer the communication from

one cell to an adjacent cell. The new generation of digital cellular telephony is based on

the concept of microcells, with antennas spaced very closely. This technology is veryeffective in urban areas, but has limited coverage in rural areas.

A new generation of satellite-based mobile telephones has recently beenlaunched (see e.g.: Motorola Iridium); this technology permits communications to and

from anywhere on the globe, but its cost is very high, making it suited only for

specialized applications (e.g.: explorations in remote areas).

3See brief description of Ethernet communication protocol in Section 15.7.

Antenna

Cell

coverage area

Figure 19.35Coverage pattern in

cellular systemNavi ation


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Further information

http://www.sprintpcs.com/index.html

http://www.mot-sps.com/solutions/isdn/isdn-tutorial.html

http://www.mot.com/SPS/WIRELESS/information/idenoverview.html

http://www.mot.com/SPS/WIRELESS/information/gsmoverview.html

http://www.mot.com/SPS/WIRELESS/information/gsmoverview.html

http://www.mot-sps.com/cgi-bin/get?/books/apnotes/pdf/an1575*.pdfhttp://mot-sps.com/support/technical/tutorials/dspbasics.html

The Internet

The Internet is a very large network that is based on sending information packetsusing the TCP/IP protocol. Today we take the Internet for granted, but it took some 30

years for the Internet to reachits present form. The earliest precursor of the Internet, theARPANet, was originally established as a project of the Defense Advanced Projects

Research Agency (DARPA) in 1962. In the 1980s the ARPANet was replaced by the

NSFNet, administered by the National Science Foundation; NSFNet became todays

Internet in the 1990s, and the system is operated by private as well as by non-profit and

corporate Internet Service Providers. Access to the Internet is today almost ubiquitous; in

many countries a significant percentage of private homes and most businesses make use

of the Internet as a means of communications on a daily basis.


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HOMEWORK PROBLEMS

Section 19.2 Fourier series and transform

19.1 Compute the Fourier series coefficients for a periodic square wave of unit

amplitude, time period , and duty cycle as shown in Figure P19.1 and definedmathematically as.

x (t) =1for |t| 1for |t|


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Figure P19.6 Double exponential signal

a. Compute the Fourier transform of the signal. (Hint: Can use the linearityproperty of Fourier transform)

b. Plot the frequency spectrum of the signal using Matlab for 8=a .19.7 Evaluate the Fourier transform of the damped sinusoidal wave shown in Figure

P19.7 and having the functional form:

)()2cos(exp(-at))( tutftx c=where )(tu is the unit step function.

Figure P19.7 Damped Sinusoidal signal

19.8 An ideal sampling function consists of an infinite sequence of uniformly, spaced

delta functions and is mathematically defined as:

T0( t) = ( tmT0 )

m =

Figure P19.8 Dirac delta function with period T0.

a. Compute the Fourier transform of ideal sampling function shown in Figure P19.8.


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