Communication electronics by louis e frenzel

2-1, you can see that the modulating signal uses the peak value of DEGREE VERSUS RADIAN MEASURE OF ANGLES You may recall that a sine wave reaches 70.7 percent of its maximum value at a

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This chapter will help you to:

1 Recognize an AM signal in the time

domain (oscilloscope display), the fre-

quency domain (spectral display), or in

trigonometric equation form

2 Calculate the percentage of modulation of

an AM signal given waveform measure-

ments

ne of the principal techniques used in

47 electronic communications is modn-

REY lation Modulation is the process of having the information to be transmitted alter

a higher frequency signal for the purpose of

transmitting the information somewhere in

2 electromagnetic spectrum via radio, wire,

or fiberoptic cable Without modulation,

electronic communications would not exist

as we know it today Communications elec-

tronics is largely the study of various modu-

2-1 Amplitude Modulation

Principles Information signals such as voice, video, or bi-

nary data are sometimes transmitted directly

from one point to another over some commu-

nications medium For example, voice signals

are transmitted by way of wires in the tele-

phone system Coaxial cables carry video sig-

nals between two points, and twisted-pair ca- ~

ble is often used to carry binary data from one

point to another in a computer network

However, when transmission distance Ave far,

cables are sometimes imipractical Ia such

Amplitude Modulation and Single-Sideband Modulation

and Single- “Sideband Modulation

3 Calculate the upper and lower sidebands

of an AM signal given the carrier and ˆ modulating signal frequencies

4 Calculate the sideband power in an AM

waye given the carrier power and the per-

centage of modulation

5 Define the terms DSB and SSB and state

the benefits of SSB over an AM signal

jation techniques and of the modulator and demodulator circuits that make modulation possible The three principal types of electronic communications are amplitude modu-

lation (AM), frequency modulation (FM),

and phase modulation (PM) The oldest and

simplest form of modulation is AM In this

chapter we will cover AM along with a

derivation known as single-sideband modula-

tion Chapter 3 will cover amplitude modulator and dk gauatr circuits

cases, ratio communications is used Te carry

out reliable long-distance radio communica-

tion, a high-frequency signal must be used It

is simply irapractical to convert the informa-

tion signal directly to electromagnetic radia-

tion Excessively long antennas and interfer-

would result if information signals were transmitted directly ;

ence between signals

For this ‘reason, it is desirable to translate the

information signal to a point higher in the elec-

tromagnetic frequency spectrum It is the

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sing wave carrier with the simple expression

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* Retype the Web address Most errors will be cleared up by

.cor/MIMS/SG/howto/acrolisÐ, first try mot.com and then

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others will not

Modulation is the process of modifying the char-

acteristic of one signal in accordance with some

characteristic of another signal In most cases, the

information signal, be it voice, video, binary data,

or some other information, is normally used to

modify a higher-frequency signal known as the carrier Tne information signal is usually called

the modulating signal, and the higher-frequency

signal which is being modulated is called the carrier or modulated wave The carrier is usually a sine wave, while the information signal can be of any shape, permitting both analog and digital sig-

nals to be transmitted In most cases, the carrier

frequency is considerably higher than the highest

information frequency to be transmitted

Amplitude Modulation

with Sine Waves

In AM, the information signal varies the ampli-

tude of the carrier sine wave In other words, the

instantaneous value of the carer amplitude changes in accordance with the amplitude and

frequency variations of the modulating signal

Figure 2-1 shows a single-frequency sine wave

modulating a higher-frequency carrier signal

Note that the camer frequency remains constant during the modulation process but that its am-

plitude varies in accordance with the modulat-

ing signal An increase in the modulating signal

amplitude canses the amplitude of the carrier to

increase Both the positive and negative peaks

of the carrier wave vary with the modulating

signal An increase or decrease in the amplitude

of the modulating signal causes a corresponding

increase or decrease in both the positive and

negative peaks of the carrier amplitude

If you interconnect the positive and negative peaks of the carrier waveform with an

imaginary line (shown dashed in Fig 2-1), then

you re-create the exact shape of the modulat-

ing information signal This imaginary line on the carrier waveform is known as the envelope,

and it is the same as the modulating signal

Because complex waveforms like that

shown in Fig 2-1 are difficult to draw, they

are usually simplified by representing the high-

frequency carrier wave as simply many equally

spaced vertical lines whose amplitudes vary in accordance with a modulating signal Figure 2-2 shows a sine wave tone modulating a

higher-frequency carrier We will use this method of representation throughout this book

The signals illustrated in Figs 2-1 and 2-2

show the variation of the carrier signal with re-

spect to time Such signals are said to be in the time domain Time-domain signals are the ac-

tual variation of voltage over time They are

what you would see displayed on the screen of

an oscilloscope In this section we show the

time-domain signals created by the various

wet

Fig 2-2 A simplified method of representing

an AM high-frequency sine wave

types of modulation Later you will see that

modulated signals can also be expressed in tne

frequency domain

ve = Vo sin Wafer

In this expression, v; represents the instantaneous value of the sine wave voltage at some specific time in the cycle The Vc represents

the peak value of the sine wave as measured between zero and the maximum amplitude of

either the positive- or negative-going alterna-

tions See Fig 2-1 The term fe is the frequency

of the carrier sine wave Finally, ¢ represents

some particular point in time during the ac

cycle

In the same way, a sine wave modulating sig-

nal can also be expressed with a similar formula:

Vn = Vin Sin 2atfint the frequency of the modulating

signal

were fa =

Referring back.to Fig 2-1, you can see that

the modulating signal uses the peak value of

DEGREE VERSUS RADIAN MEASURE OF ANGLES

You may recall that a sine wave reaches 70.7 percent of its maximum value at a phase an-

gle of 45° In general, the instantaneous value

of a sine wave can be found by

v= Vn &X sind

Where v = the instantaneous value

Vn = the maximum value

@ = the phase angle

Example: A 1-MHz sine wave has a peak ar

} maximum value of 18 V What is its instanta-

neous value at a phase angle of 45°?

sin 45° = 0.707

:y= 18 Vx 0707 = 187 Ý

- in communications, ‘the phase drigle may be’

ted i in an equivalent Way using the frequency

of the ‘signal and some time of interest This is

a5 radia measure, For éxample, the in

Vin X sin(2aft) ST the’ frequency’ “of the signal "

he time ot interest

Amplitude Modulation and Single-Sideband Modulation Chapter2 s%

lue ofa 'signal can be found with

A 1-MHz signal has a period of 1 ps One pe-

tiod equals one cycle with 360° A phase

angle of 45° corresponds to 45/360 or Vg cycle, which is a time of ips divided by

8(0.125 5)

Example: Use the second equation for finding

the instantaneous value of 4 1-MHz sine signal

with a peak value of 18 volts (V) at a time of

0.125 ps (Your calculator must be in the

angular measure, and the second is based

on radian Measure Keep in mind that 1 rad -

` = 57, 3 in our r example, 45” A557 3= 9 785

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varies above and below the peak carrier am-

plitude That is, the zero reference fine of the

modulating signal coincides with the peak

value of the unmodulated carrier Because of

this, the relative amplitudes of the carrier and

modulating signal are important In general, the amplitude of the modulating signal should

be less than the amplitude of the carrier If the

iplication amplitude of the modulating signal is greater

than the amplitude of the carrier, distortion will -

occur, Distortion causes incorrect information

to be transmitted is important in AM that the peak value of the modulating signal be less than the peak value of the carrier *

Using the mathematical expressions for the

carrier and the modulating signal, we can cre-

ate a new mathematical expression for the com- plete modulated wave First, keep in mind that the peak value of ghe carrier is the reference

point for the modulating signal The modulating signal value adds to or subtracts from the peak value of the carrier This instantaneous

value of either the top or bottom voltage enve-

~./ Jope can be computed by the simple expression

All this expression says is that the instanta-

neous value of the modulating signal alge-

braically adds to the peak value of the carer

As you can see, the value of vy is really the en-

velope of the carrier wave For that reason, we

can write the instantaneous value of the com- plete modulated wave v2 as

v2 = Vị sin 2t

In this expression, the peak value of carrier

wave V, from the first equation given is replaced

by rr Now, substituting the previously derived

:apression jor vy and expanding, we get

là

fit (Sin Bolt)

Ve Fig 2-3 Amplitude modulator showing input

‘ and output signals

v2 = (Vo + Vin Sin 27ifnt) Sin 2aefet

Carrier + modulation X carrier

This expression consists of two parts: the first part is simply the carrier waveform, and the second part is the carrier waveform multi-

plied by the modulating signal waveforin It is

this second part of the expression that is char-

acteristic of AM A circuit must be able to pro-

duce mathematical multiplication of analog”

signals in order for AM to occur :

The circuit used for producing AM is called a

modulator Its two inputs, the carrier and the mod-

ulating signal, and the resulting output are shown

in Fig 2-3 Amplitude modulators compute the

product of the carrier and modulating signals

Amplitude Modulation with Digital Signals

Digital, usually binary, signals may also be

usedto amplitude modulate a carrier Figure

2-4 shows a binary signal modulating a sine

_ waye carrier In Fig 2-4(a), the binary 1 level

: Digital data transmissions ;

: Computer modems (used in combinatio

"with phase modulation QAM) :

| 6 NIST time signals

i lh—'

Fig 2-4 A

ing (ASK) (b) On-off keying (Ok),

f

mpktude modulation of a carrier with

@ binary signal (a) Amplitude shift key-

_ signal To improve the speed of digital trans-

mission in computer modems, 4-, 8-, 16- and 32-level digital Signals are com

Amplitude modulation is usu

Answer the following questions

I Modulation causes the information signal t

e be — 10a higher frequency

Or more efficient transmission

2 During modulation, the in

nai —

formation sig-

— the amplitude of a high-

frequency Signal called the -

3 The circuit used to produce

| Fae: maxim carrier amplitude and the and " - ma

y Ö level produces ạ lower-valu¢ carrier 4 In AM, the instantaneous

Amplitude modulation in whith

Ve switched between two different carn

_is known as amplitude shift keying (ASK)

A Special form of ASK is one in

cartier is sim

2-4(b), The binary | Jey

“and the binary 0 level

‘is called on-off keying (00K)

32-5 Multilevel digital AM,

Amplitude Modulation and Single-Si

the carrier is

ier levels

which the

ply switched on or off, See Fig

el turns the carrier on, turns the carrier off This

al signals have more than two

of

the carrier varies in accor

information signal

5 True or false The carrier frequency is

usually lower than the modulating fre-

6, The outline Of the peaks Gf the carrier

signal is called the

the same shape as the

The carrier frequency re-

Mains constant during AM,

10 An amplitude nicdulator performs the

mathematical operation of

a Addition c Multiplication,

b Subtraction — đ Division, _ 11, AM with binary signals is called

12, AM using the presence and absence of a Carrier is called

2-2 Modulation index and

Percentage of Modulation

In order for proper AM to occur, the modulating

Signal voltage Vm must be less than the cartier voltage Ye, Therefore, the relationship between

the amplitudes of the modulating signal and car-

ner is important This relationship is expressed

in terms of a ratio known as the modulation

deband Modulation Chapter

Quadrature

ampitude meadulation

(QAM)

Amplitude shift keying (ASK)

On-off keying

(OOK)

r2 GER ag

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S Stray end distributed capacitances

and inductances can greatly alter the

tt operation and performance of a circuit

visit the follow-

ing Internet site:

The modulation index should be a number between 0 and 1 If the amplitude of the mod-

ulating voltage is higher than the carrier volt-

age, m will be greater than 1 This will cause severe distortion of the modulated waveform, This is illustrated in Fig 2-6 Here a sine wave

- information signal modulates a sine wave car-

rier, but the modulating voltage is much greater

than the carrier voltage This condition is called

overmodulation As you can see, the waveform

is flattened near the zero line The received’

signal will produce an output waveform in the shape of the envelope, which in this case is a

sine wave whose negative peaks have been clipped off By keeping the amplitude of the’

modulating signal Jess than the carrier amplitude, no distortion will occur The ideal condi-

Fig 2-6 Distortion of the envelope caused by

overmodulation where the modulating signal amplitude Vm is greater than

the carrier signal V;

tion for AM is where Vn = Ve orm = I, since

this will produce the greatest output at the receiver with no distortion

The modulation index can be determined by measuring the actual values of the modulation

voltage and the carrer voltage and computing

the ratio However, it is more common to com-

pute the modulation index from measurements

taken on the composite modulated wave itself

Whenever the AM signal is displayed on an

oscilloscope, the modulation index can be com-

puted from Vmax and Vmin as shown in Fig 2.7

The peak value of the modulating signal Vin

is one-half the difference of the peak and trough values and is computed with the expression

of the modulated wave The Vmax is one-half

the peak-to-peak value of the AM signal or

Vmax(p-n/2 Subtracting Vmin from Vmax produces the peak-to-peak value of the modulat-

ing signal One-half of that, of course, is sim-

ply the peak value

The peak value.of the carrier signal Vz is the

average of the Vmax and Vin values and is

computed with the expression

The values for Vmax and Vmin can be read

directly: from an oscilloscope screen and

plugged into the formula to compute the mod-

For example, suppose that the Vmax value

read from the graticule on the oscilloscope screen is 4.6 divisions and Vmin is 0.7 divisions, The modulation index is then

Whenever the tncdulation index is multiplied

by 100, tue degree of modulation is expressed

as a cercentage The amount or depth of AM is more commonly expressed as percent modula-

tion rather than as a fractional value In the example above, the percent of modulation is

100% * m or 73.6 percent The maximum

amount of modulation without signal distor-

tion, of course, is 100 percent where V; and Vin

are equal At this time, Vinin = 0 and Vmax = 2Vm, where Vin is the peak.value of the modu-

lating signal

In practice, it is desirable to operate with as

close to 100 percent modulation as possible, In

this way, the maximum information signal amplitude is transmitted More information signal power is transmitted, ‘thereby producing a

“stronger, more intelligible signal When the

~ modulating signal amplitude varies randomly

over a wide range, it is impossible to maintain

100 percent modulation A voice signal, for ex-

ample, changes amplitude as a person speaks,

Only the peaks of the signal produce 100 per-

a Vm should be greater than Vo

b V should be greater than Vi:

c Vm should be equal to or less than Ve

d Ve must always equal Va -

14 Which of the following is not another

name for modulation index?

a Modulation reciprocal ˆ

'b Modulation factor

c Degree of modulation

d Modulation coefficient

15 The degree or depth of modulation ex-

pressed as a percentage is computed us-

ing the expression

16 The modulation index is the ratio of the

peak voltage of the _ to the

17 An AM wave displayed on an oscillo-

scope has values of Vinax = 3.8 and Vinia

= 1.5 as read from the graticule The per-

centage of modulation is _ per-

cent,

Amplitude Modulation and Single-Sideband Modulation Chanter >

Percentage of modulation

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Sidebands

Side freyuencies

18 The ideal percentage of modulation for

maximum amplitude of information trans-

mussion is percent

19 To achieve 85 percent modulation of a

carcier Of Ve = 40 volts (V), a modulating

signal of Vin = — is needed

20 The peak-to-peak vaiue of an AM signal

is 30 V The peak-to-peak value of the

modulating signal is 12 V The percentage

of modulation is percent

21 In Fig 2-4(a), the carrier maximum value

is 600 mV, and the carrier minimum is

300 mV The percentage of modulation is

2-3 Sidebands and the

Frequency Domain Whenever a carrier is modulaied by an infor-

mation signal, new signals at different fre-

quencies are generated as part of the process / These new frequencies are called side frequencies, or sidebands and occur in the frequency ,;

spectrum directly above and directly below the carer frequency

Sidebands

If the modulating signal is a single-frequency

sine wave, the resulting new signals produced

by modulation are called side frequencies If the modulating signal contains multiple frequencies such as voice, video, or digital sig-

nals, the result is a range of multiple side fre-

quencies These are referred to as sidebands

The sidebands occur at frequencies that are the sum and difference of the carrier and mod-

ulating frequencies, Assuming a cartier fre-

quency of f and a modulating frequency of fin,

the upper sideband fuss and Jower “sideband

Fisp are computed as follows:

fuss = fo + fn

fuss = fe — fm

The existence of these additional new sig-

nals that result from the process of modulation can also be proven mathematically This can be done by starting with the equation for an AM

signal v2 described previously

v2 = Ve tin 2ã? + (Vm sin 2/„f)(sìn 2/2!)

There is a uigonometric identity that says

tnat the product of two sine waves is

Cà Cellular phor 4s transmitiing

above 3 W, suci as some portable and *

mobile cellular phones, should be held ạt

least 12 in away from a pacemaker, or

they could interfere with its function

Even some high-power stereo speakers

could be dangerous to pacemaker users,

because they contain large magnets

eee =s=m

`

cos (Á — P)

sin A sin B= ———* 2 cos (A + B) 2

By substituting this identity into the expres-

| sion for our modulated wave, the instantaneous

, amplitude of the signal becomes

AS you can see, lạc econd and third terms

of this expression cgntain the sum fo + fin and

duiverence fo — fn ar the carrier and modulat-

iag signal frequencies The first element in the expression is simply the carrier wave to which

is added the difference frequency and the sum frequency

By algebraically adding the carrier and the

two sideband signals together, the standard AM

waveform described earlier is obtained This is

illustrated in Fig, 2-8 This is solid proof that

an AM wave contains not only the carrier but also the sideband frequencies Observing an

AM signal on an oscilloscope, you can see the

amplitude variations of the carrier with respect

to ume, This is called a time-domain display

It gives no indication of the existence of the

sidebands, although the modulation process

does indeed produce them

The Frequency Domain

Another method of showing the sideband signals is to plot the carrier and sideband ampli-

tudes with respect to frequency This is illus-

trated in Fig 2-9 Here the horizontal axis

represents frequency, and the vertical axis rep-

Tesents the amplitudes of the signals A plot of

a Producas this sum

wa Envelope is the originat

- modulating signa!

Fig 2-8 The algebraic sum of the carrier and sideband signals is the AM signal

Signal amplitude versus frequency is referred

\ to as a frequency-domain display A test in-

Atrument known as a spectrum analyzer will

iy Y display the:frequenéy ‘domain of a signal

Xx | Whenever the modulating signal is more

complex than-a single sine wave tone, multiple

\upper and lower side frequencies will be pro-

duced For example, a voice signal consists of

many different-frequency sine wave compo-

nents mixed together Recall that voice fre:

quencies occur in the 300- to 3000-Hz range

Therefore, voice signals will produce a range

of frequencies above and below the carrier frequency as shown in Fig 2-10 These sidebands take up spectrim space You can compute the

total bandwidth of the AM signal by comput-

ing the maximum and minimum sideband fre-

quencies This is done by finding the sum and

difference of the carrier frequency and maxi-

mum modulating frequency, 3000 Hz, or 3 kHz,

for voice transmission If the carrier frequency

is 2.8 MHz, or 2800 kHz, then the maximum

and minimum sideband frequencies are

fuss = 2800 + 3 = 2803 kHz fiss = 2800 — 3 = 2797 kHz

The total bandwidth (BW), then, is simply

the difference between the upper and lower

s2 Chapter 2 9 & pati Ge Muduletion and Single-Sideband Modulation , Tố ¬ ‘

Amplitude Modulation and Single-Sideband Modulation Chaptar2 8% 29

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24 The total bandwidth of the signal in the Ps = Pisa = Pusg = — Pe (m?)

with respect to time is called a lation where the modulation fac- Whether or not you are

-domain display The test in- torm = I the powerineach side- Working toward a four-year strument used to present such adisplay is band is one-fourth, or 25 percent, degree, look for companies

the nà of the carrier power, Since there — that are advertising posi-

i 26 A signal whose amplitude is displayed are two sidebands, their power

tions for engineers These

with respect to frequency is called a ~domain display The test in- together represents 50 percent of €Ompanies are sure to need the carrier power For example, — technical support staff with

Sttument used to present such a display is if the carrier power is 100 watts Jess than a four-year degree,

-ASit turns out, the bandwidth of the AM sig-

~ nal is simply twice the highest frequency in the

modulating signal With a voice signal whose maximum frequency is 3 kHz, the total hand-

width would simply be twice this, or 6 kHz

broad spectrum of sidebands is produced

According to the Fourier theory, complex signals such as square waves, triangular waves,

sawtooth waves or distorted sine waves are sim-

ply made up of a fundamental sine wave and nu- mere - harmonic signals at different amplitudes

TT’ sssic example is that of a square wave

whi s made up of a fundamental sine wave

and ull odd harmonics A modulating square

wave will produce sidebands at frequencies of

the fundamental square wave as well as at the third, fifth, seventh, etc., harmonics The result-

ing frequency-domain plot would appear like

that shown in Fig 2-11 (a) Pulses generate ex-

tremely wide bandwidth signals In order for the square wave to be transmitted and received with-

out distortion or degradation, all the sidebands

must be passed by the antennas and the trans-

mitting and receiving circuits

' Figure 2-11(b) shows the relationship be-

tween the time and frequency domain presen-

tations of the modulating square war, The time domain shows the individual sine wave

harmonics that, when added together, produce the square wave The frequency domain shows

nal modulated by a square v

Relationship between time o quency domain displays of the modulating signal

the signal amplitudes of the harmonics that

modulate the carrier and produce sidebands

TEST

Answer the following questions,

22 New signals generated by the madulaiion

process that appear diréctly above and be-

low the carrier frequency are called

23 An AM radio station operating at 630

kHz is permitted to broadcast audio frequencies up to $ KHz The upper and

lower side frequencies are

fuss = KHz fisp= _ si Hz

27 Complex modulating signals containing

many frequencies produce multiple

thus occupying more Spec-

28 The AM signal that occupies the greätest

bandwidth is the one modulated by a

a 1-KHz sine wave

b 5-kHZ sine wave

c 1-KHz square wave

d 5-kHz square wave:

29 The composite AM signal can be re-

created by algebraically adding which

three signals? ¬

30 True or False, Digital modulating signals

typically produce an AM signal that has a-

greater bandwidth than an AM signal pro-

duced by an analog modulating signal

2-4 Amplitude Modulation

Power Distribution

To communicate by radio, the AM signal is amplified by a-power amplifier and fed to the

antenna with’a characteristic impedance R The

total transmitted power divides itself between

the carrier and the upper and lower sidebands

The total transmitted power Pr is simply the sum of the cartier power Pe, and the power in ~

the two sidebands Pusp, and Pysp ‘This is ex-

j Pressed by this simple equation:

vy Pr= Pc + Pisa + Pusn

Sideband Powers

The power in the sidebands depends upon the value of the modulation index The greater the percentage of modulation, the higher the sideband power Of course, maximum power appears in the sidebands when the carrier is 100

percent modulated.’The power in each side-

band Py is given by the expression

: m- Ma

`4

will be transmitted When modulation

Sidebands are produced It is easy to conclude,

(W), then at 100 percent modulation, 50 W will appear in the side-

bands, 25 W in each The total transmitted

power then is the sum of the cartier and sideband powers or 150 W

"AS you can see, the carrier power tepre- Sents two-thirds of the total transmitted

power assuming 100 percent modulation

With a carrier power of 100 W and a total power of 150 W, the ‘carrier power percent-

“age can be computed,

The carrier itself conveys no inforniation.~

The carrier can be transmitted and received, but unless modulation occurs, no infornfation

CUTS,”

therefore, that all the transmitted information

‘is contained within the sidebands Only one- third of the total transmitted power is allotted

to the sidebands, while the temaining two-

thirds of it is literally wasted on the carrier

Obviously, although it is quite effective and still widely used, AM is a very inefficient

At lower percentages of modulation, the

power in the sidebands is even less You can

‘compute the amount of power in a sideband

With the previously given expression Assume

" a Carrier power of 500 Wand a modulation

of 70 percent The power in each sideband then is

ì «g% Chapter2 Amplitude Modulation and Single -Sideband Modulation

TH

Amplitude Modulat on and Single-Sideband Modulation Chapter2 <f 31

Trang 7

cars rented in Miami come with cellular phones, GPS

ion, and a panic button that contacts police about

One way to calculate the total AM power is

“to use the formula

, Pp = PL + m2)

where Pp = unmodulated carrier power

mt = modulation index '

For example, if the carrier power is 1200 W

‘and the percentage of modulation is 90 pcr-

cent, the total power is

= 1200( + 0.99/2) =

if you subtract the carrier power, this will

leave the power in both sidebands

Pr = Pe + Pisa + Puss

Prsg + Puss = Pr~ Pe

= 1686 — 1200 = 486 W

"Since the sideband powers are equal, the

power in each sideband is 486/2 = 243, W,

In practice, 100 percent modulation is difficult to ‘maintain The reason for this is that

typical information signals, such as voice and video, do not have constant amplitudes

The complex voice-and video signals vary over a wide amplitude and frequency range,

so 100 percent modulation only occurs on

1200( 1.405) = 1686 W

, reason, the average sideband power is con-

siderably less than the ideal.50 percent produced by full 100 percent modulation With

less sideband power transmitted, the received

the peaks of the modulating signal For this,

signal is weaker and communication is less reliahle

Despite its inefficiency, AM is sill widely

used because it is simple and effective It is

used in AM radio broadcasting, CB radio, TV

aircraft communications and

broadcastin

computer modems

that there are three basic way’s

to calculate the power dissi-

pated in a load These are: +

Simply select the formula for which you,

have the values of current, voltage, or resis-

tance Jn an AM radio transmitting station, R is

the load resistance which is an antenna, To 4 „ transmitter the antenna looks like a resistance :

Although an antenna is not actually a physical '

resistor, it does appear to be one This resis-

tanee is referred to as the characteristic resistance of the antenna You will lear more about

it ina Jater chapter

;

về:

Power Calculations ¡

Acommon way to determine modulated power i

is to measure antenna current, Current in an

antenna can be measured because accurate

radio-frequency current meters are available

For example, if youd know that the unmodu- |

lated carrier produces a current of 2.5 Ain an:

antenna with a characteristic resistance of 73:

2, the power is:

P= PR= (2.5) (73) = 6.25(73) = 456.25 W

If the carrier is modulated, the antenna cur-

rent will be higher because of the additional

power in the sidebands, The total antenna current /7 is

To determine the total power monitor the

total modulated aateara current and anake the +

r8

calculation abows, coven tne dutengd PS istance,

2 tne modulated and the

v HT” dL,EẺRP2 VUEEeR!š, VOU cần

Hf vou maauire pet

Thun hat

svinnpute tas Peres `

ing this formeta: ae SỰ code ion by us-

Assume that you measured the unmodulated

carrier current and found it to be 1.8 A With

modulation, the total current was 2 A The per-

centage of modulation is:

31 The total sideband power is what percent-

age of the carrier power for 100 percent modulation?

a 25 percent

b 50 percent c- 100 percent _d 150 percent -

32 Information in an AM signal is conveyed

34 The total transmitted power is the sum of

the and “powers

35 A S-kW carrier with 60 percent modula-

tion produces _ kW in each

36 In an AM signal with a carrier of 18 W

and a modulation percentage of 75 per-

cent, the total power in the sidebands is

W,

37 An AM signal with a cartier of 1 kW has

100 W in each sideband The percentage

of modulation is percent

38 An AM transmitter has a carrier power of

200 W The percentase of modulation is

60 percent The total signal power is

_W

39 The total AM signal power is 2800 W

The carrier power is 2000 W The power

in one sideband is _ _ W The perceniage of modulation is

40, The uninodulated cartier current in an an-

tenna is 1.5 A When the carrier is modu-

lated by 95 percent the total antenna cur-

‘AM is an inefficient and, therefore, wasteful method of communications Two-thirds of the

transmitted power appears in the carrier which itself conveys no information The real information is contained within the sidebands One

way lo overcome this problem is simply to

suppress the carrier Since the carrier does not

provide any useful information, there is no reason why it has to be transmitted By Suppress:

ing the carrier, the resulting signal is simply

the upper and lower sidebands Such a signal

“2

is referred to as a dowble-sideband suppressed| Double-

carrier (DSSC or DSB) signal The benefit, of | sideband

course, is that no power is wasted on the car- suppressed

, rier and that the power saved can be put into carrier

the sidebands Double-sideband suppressed

carrier modulation is simply a special case of

Double and Single Sidebands

Amplitude modulation generates two sets of _sidebands, cach containing the same information The information is redundant in an AM or

a DSB signal Therefore, all the information | can be conveyed in just one sideband ⁄⁄

Eliminating one sideband produces a single-

sideband (SSB) signal Eliminating the carrier

and one sideband produces a more efficient_-

_ AM signal

Atypical DSB signal is shown in Fig 2-12

This signal is simply the algebraic sum of the

two sinusoidal sidebands This is the signal

produced when a carrier is modulated by a Single-tone sine wave information signal

During the modulation process, the carrier is

suppressed, but the two sideband signals re- main, Even though the carrier is suppressed,

& Chapter2 Amplitude Modulation and Single-Sideband Modulation

Trang 8

Carver frequency sie wave

line phase transition ;

shown Note that the envelope of this wave-

form is not the same as the modulating signe)

as itis in a pure AM signal with carrier

A frequency-domain display of a DSB sig-

é ~ nal js given in Fig 2-13 Note that the spec-

trum space occupied by a DSB signal is the

Single-sideband vị same as that for a conventional AM signal

suppressed carrier (" DPouble-sideband suppressed carrier signals

are generated by a circuit called a balanced

x ,| modulator The purpose of the balanced mod-

J ulator is to produce the sum and difference fre-

quencies but to cancel or balance out the car-

ier, You will learn more about balanced

modulators in Chap 3

| ˆ When DSB AM is used, considerable power

| is saved by eliminating the carrier This power

Ự can be put into the sidebands for stronger sig-

nals over longer distances Although a DSB

AM signal is relatively easy to generate, DSB

Balanced

modulator

Suppressed carrier

@ For SSB transmissions, it does not

matter whether the upper or lower side-

band is used, since the information is

carrier and modulating signals, the information

must be contained in both of them As it tums out, there is absolutely no reason to transmit both sidebands in order to convey the infor-

mation One sideband may be suppressed The :

remaining sideband is called a single-sideband

suppressed carrier (SSSC ar SSB) signal |

The SSB sicnal offers four major benefits :

First, the spectrum space occupied by the SSB : signal is only half that of AM and DSB signals © Ị

‘This greatly conserves spectrum space and al- | ị

Jows more signals to be transmitted in the same

frequency range It also means that there should

be less interference between signals

The second benefit is that all the power pre-

viously devoted to the carrier and other side-

band can be channeled into the signal side-

" band, thereby producing a stronger signal that should carry farther and be more reliably received at greater distances

The third benefit is that there is less noise _on the signal Noise gets added to all signals in the communications medium or in the receiver

itself Noise is a random voltage made up of an

almost infinite number of frequencies If the

signal bandwidth is restricted and the receiver

circuits are made with a narrower bandwidth,

a great deal of the noise is fihered out Since

the SSB signal has less bandwidth than an AM

or a DSB signal, logically there will be less |

noise on it This is a major advantage in weak \

signal long-distance communications

The fourth advantage of SSB signals is that they experience less fading than an AM signal

Fading means that a signal alternately increases

and decreases in Strength as itis picked up by

\ 1

ị '

i

~/

' fading does not occur

the receiver It occurs because the carrier and)

sidebands may reach the receiver shifted in|

tuume end phase with respect to one another, }

The cartier and sidebands, whict: are on sepa- rate frequencies, are atfected by the ionosphere

in different ways, The lonosphere bends the

cartier and sideband signals hack to earth at

slightly different angles so that sometimes they!

reach the receiver in such a way that they can-|

ce] out one another rather than adding up to th

desired AM wave The result is fading Wit

SSB there is only one sideband, so this kind o

An SSB signal has some unusual character

istics First when no information or modulat

ing signal is present, no RF signal is transit

ied In a standard AM transmitter, the carrier is still transmitted even though it may not be

modulated This is the condition that might oc-

cur during a voice pause on an AM broadcast

station But since there is no cartier transmit-

ted in an SSB system, no signals are present if

the information signal is zero Sidebands are

generated only during the modulation process, such as when someone speaks into a microy

Figure 2-14 shows the frequency- and time- domain displays of an SSB signal produced

when a steady 2-kHz sine wave tone modu-

lates a 14.3-MHz carrier Amplitude modulation would produce sidebands of 14.298 and

14.302 MHz In SSB, only one sideband is used Figure 2-14(a) shows that only the upper sideband is generated The RF signal is simply

a constant-power [4.302-MHz sine wave A

time-domain display of the SSB signal appears

Of course, most information signals trans-

mitted by SSB are not pure sine waves.-A more

common modulation signal is voice with its varying frequency and amplitude content The

voice signal will create a complex RE SSB signal which varies in frequency and amplitude over the narrow spectrum defined by the voice

Ptolemy was an astronomer in ancient Exy,

also the name of a digital signa! processin

software product can aid in design, in the generation of

assembly code, and in making prototypes of new digital sig

naling processors Perhaps you will use a program like Ptolemy at your job site °

EE PRT EAGER EME SELON GES COT RTE ER,

Upper side frequencies:

14,300,000 + 200 = 14,300.200 Hz 14,300.000 + 4000 = 14,304,000 Hz

14.300.000 — 200 = 14,299,500 Hz 14.300.000 ~ 4000 = 14,296,000 Hz

The upper sideband extends from 14.300,200

to 14,304.000 Hz and occupies a bandwidth of

Fig 2-14 An $58 signa! produced by a 2-kHz

sine wave modulating a 14.2-Mu2

sine wave Carrier

Amplitude Modulation and Single-Sidehaad Modulation

Trang 9

velope ¢

PEP) is the maximum power produced on voice am-

The lower sideband extends fram 14,296,000

to 14,299,000 Hz and occupies a bandwidth of

BW = 14,299,800 14,296,000 = 3800 Hz

The DSB signal consumes a total bandwidth of

4000 X 2 = 8000 Hz (8 kHz) If we transmit

only one sideband, the bandwidth is 3800 Hz

The SSB signal may be cither the upper side-

band (USB) or the lower sideband (LSB) In

practice, an SSB transmitter generates both sidebands and a switch is used to select zither

the USB or the LSB for transmission

When the voice or other modulating signal

is zero, no SSB signal is produced An SSB RF

signal is produced only when modulation oc:

curs In AM, with no modulating signal; the

carmier would stil be transmitted This explains

why SSB is so much more efficient than AM,

In conventional AM, the transmitter power is

distributed among the carrier and two side-

bands If we assume a carrier power of 100 W

and 100 percent modulation, each sideband will

contain 25 W of power The total transmitted

power will be’ 150 W The communication ef-

fectiveness of this conventional AM transmitter

is established by the combined power in the

sidebands, or 50 W in this example

_ An SSB transmitter sends no carrier, so the

carrier power is zero Such a transmitter will have the same communication effectiveness as

a conventional AM unit running much more

power, A 50-W SSB transmitter will equal the

performance of an AM transmitter running a total of 150 W, since they both show 50 W of

total sideband power The power advantage of

SSB over AM is 3:1

In SSB, the transmitter output is expressed

in terms of peak envelope power (PEP) This

- plitude peaks The PEP is computed by the fa-

miliar expression

pa R

where P = output power

V = root mean square (rms) output

“across a 50-ohm ((2) load The rms voltage is

0.707 times the peak value The peak value is

l Chapter 2

one-halt the peak-to-peak voltage In this ex-

ample, the rms voltage is

the PEP input power is simply the dircct-

current (dc) input power of the transmitter fi- -

nal amplifier stage at the instant of the voice envelope peak It is the final amplifier stage de

supply voltage multiplied by the maximum amplifier current thầt occurs at the peak or

-PLP = Vohinax

where Vs = the amplifier supply voltage

Imax = the current peak

A 300-V supply with a peak current of 0.6

ampere (A) produces a PEP of

It is important to point out that the PEP oc-

curs only occasionally

Voice amplitude peaks are produced only

when very Joud sounds are generated during certain speech patterns or when some word or

sound is emphasized During normal speech

‘levels, the input and output power levels are

_ much less than the PEP level, The average

power is typicaffy only one-fourth to one-

third of | the REP value with typical human

PEP

With a PEP of 180 W the average power is

only 45 to 60 W Typical SSB transmitters are designed to handle only the average power

level on a continuous basis not the PEP

The transmitted sideband will, of course, change in frequency and amplitude as a complex voice signal is applied This sideband will

occupy thé same bandwidth as one sideband in

a fully modulated AM signal with carrier,

Incidentally, it does not matter whether the

upper or lower sideband is used since the in-

formation is contained in either A filter is typically used to remove the unwanted sideband

_ =

058 and SSB Applications

Both DSB and SSB technique’ are widely used

in communications Pure SSB signals are used

in telephone systems as well as in two-way ra-

dio Two-way SSB cominunicaHloas is used in

the military, in CB radio and by hobbyists

known as radio amateurs

The DSB signals are used in FM and TV broadcasting to transmit two-channel stereo

signals They are also used in some types of

phase-shift keying which is used for transmit-

Ung binary data

An unusual form of AM is that used in television broadcasting A T¥ signal consists

of the picture (video) signal and the audio

signal which have different carrier frequencies

The audio carrier is frequency-modulated, bui the video information amplitude- modulates the picture carrier The picture carrier is transmitted but one sideband is partially suppressed

Video information typically contains fre-

quencies as hygdy as 4.2 MHz A fully amplitude-

modulated te vision signal would then occupy

2(4.2) = et MHz This is an excessive amount <f tundwidth that is wasteful of spec-

ve because not all of it is required to transmit a TV signal To reduce the

relis

bandw: dth to the 6-MHz maximum allowed °

by the FCC or TY signals, a portion’ of the

lower sideband of the TV signal is suppressed

leaving only a small vestige of the lower side-

band Such an arrangement is know as a ves-

tigial sideband signal It is illustrated ï in Fig

2-15 Video signals above 0.75 MHz (750 kHz)

are suppressed in the loser sideband, and all video frequencies are transmitted in the upper sideband

TEST Answer the following questions

41 An AM signal without a carrier is called

a(n) signal

42 True or false Two sidebands must be

transmitted to retain all the information

43 The acronym SSB means

a Single sideband with carrier

b Single sideband with Suppressed carricr

c Double sideband with no carrier

d Double sideband with carrier

A major benefit of DSB and SSB is

a Higher power can be put into the side-

band(s)

b Greater power consumption

d Double the sideband power

List four benefits of SSB over AM and

True or false In SSB, no signal is trans-

mitted unless the information signal is Vestigial

The type of AM signal that is used in TV

trans-

An SSB signal produces a peak-to-peak

voltage of 720 V on voice peaks across a 75-2 antenna The PEP output is

WwW,

An SSB transmitter has a 150-V supply

Voice peaks produce a current of 2.3 A

The PEP input is WwW

The average output power of an SSB

transmitter rated at 12 W PEP is in the

lO _ range

Trang 10

1 Modulation is the process of having the infor-

mation to be cornmunicated modify a higher-

requency signal called a camer

2 Annnuds ha dulation (AM) is the oldest and

simplest form of modulation

3 In AM, the amplitude of the carrier is changed

in accordance with the amplitude and frequency

or the characteristics of the modulating signal

The carrier frequency remains constant

4, The amplitude variation of the cartier peaks has

the shape of the modulating signal and is referred to as the envelope

5 Á time- dornain display shows amplitude versus

time variation of AM and other signals

_ 6 Amplitude modulation is produced by a circuit

called a modulator which has two inputs and an output

7 The modulator performs ¢ a mathematical multi-

plication of the cartier and information signals

The output is their MAI product

8 The carrier in an AM sigifal is a sine wave that

may be modulated by either analog or digital

information signals

9, Amplitude modulation of a carrier by a binary

signal produces amplitude shift keying (ASK)

10, The ratio of the peak voltage value of the mod-

ulating signal Vn to the peak value of the car-

rier V- is caligck the modulation index m (m=

V,/Vc) It is also referred to as the moduiation

coefficient or factor and the degree of modula-

tion

11 The ideal value for m is 1 Typically m is ng

than 1 The condition where i is greater than

should be avoided as it introduces serious distortion of the modulating signal This is called

12 When the modulation index is multiplied by

100, it is called nen sec of modulation

13 The percentage ofsfnodulation can be computed

from AM waveforms displayed on an oscilloscope by using the expression

1C0(Vinax — Vmin)

38 #2 Chapter 2

where Vinax is the maximum peak cartier ampli-

tude and Vmin is the minimum peak carrier am-

Hude

14 The tà signals generated by the modulation

process are called sidebands and cccur at frequencies above and below the carrier fre-

uency

15 The upper fUSB and lower fisp sideband

frequencies are the sum and difference of the carrier frequency fe and the modulating

frequency fm and are computed with the,

‘expressions

* fuss = fe + fn

fuse = fe ~ fn

16 A dis y of signal amplitudes with respect to

freque ney ig called a frequency- -domain display

17 An AM signal:can be viewed as the carrier sig-

nal added to the sideband signals produced by

18 The total transmitted power in an AM signal is

the sum of the carrier and sideband powers Pr

= Po + Pusp + Pisp) and is distributed among

the canier and sidebands This power distribu-

tion vangs, with the percentage of modulation

The total power is

›; 8n” figher the percentage of modulation, the

“greater the sideband power and the stronger and

more intelligible the transmitted and received

~Agghal - ˆ

20 e its simplicity and effectiveness, AM is

‘a ‘highly inefficient method of modulation

21 In an AM signal, the carrier contains no infor-

* mation Any transmitted information lies solely , itvthe sideband For that reason, the carrier may

bể suppressed and not transmitted

22 An AM signal with suppressed carrier is called ,

a dotBle- sideband (DSB) signal

4

‘

a

23 Since the same transmitted information is con-

tained in both upper and lower sidebands, one

is redundant Full information can be transmit-

ted using only one sideband

24 An AM signal with no carrier and one sideband

is called a single-sideband (SSR) signal The upper and lower sidebands contain the same infor-

mation, and one is not preferred over the other

25 The main advantage of an SSB AM signal over

an AM or DSB signal is that it occupies one-

Both DSB and SSB signals are more efficient

in terms of power usage The power wasted in

the useless carrier is saved, thereby allowing

more power to be put into the sidebands

27 Power in an SSB transmitter is rated in terms

of peak envelope power (PEP), the power that

is produced on voice peaks PEP output is com-

puted using the expression

26

t2

Ep 2 1

PEF 5

where PEP is in watts and Vis the rims vol

across the antenna load impedance R The input is computed using the expression

PEP = Vv, X đnax

where Vs is the de supply voltage of the fin amplifier stage and /nax is the amplifier cur

on voice peaks

28 The average output of an SSB transmitter is

one-fourth to one-third of the PEP yalue

29 DSB AM is not widely used However, SSE

widely used in two-way radio communicatic

30 A special form of amplitude modulation is t

in TV transmission Known as vestigial side

* band, this method filters out a portion of the

lower video sidebands to decrease the overa

bandwidth of the AM picture signal to 6 ME

Chapter Review Questions

Choose the letter that best answers each question, 2-1 Having an information Signal change some,

characteristic of a carrier signal is called

a Multiplexing

b Modulation

c Duplexing

d Linear mixing

2-2 Which of the following is not true about AM?

a The carrier amplitude varies

b The carrier frequency remains constant

c The carrier frequency changes

d The information signal amplitude changes

the carrier amplitude

2-3 The opposite of modulation is

2-5 A modulator circuit performs what mathemati-

cal operation on its two inputs?

a Addition

b Multiplication

c Division

d Square root

2-6 The ratio of the peak modulating signal vc

age to the peak carrier voltage is referred t

a The voltage’ ratio

b Decibels,

c The modulation index

d The mix factor,

2-7 If m is greater than 1, what happens?

a Normal operation:

b Carrier drops to zero

‘ce Carrier frequency shifts,

d Information signal is distorted

2-8 For ideal AM, which of the following is tru

am bm=

€C m<

dđm>]

2-9 The outline of the peaks of a carrier has the

shape of the modulating signal and is called

Trang 11

The new signals produced by modulation are

AAD wave on an oscilloscope

A carrier of SSO-KEZ is modufated by a/3.8-

KHz sine wave The LSB itd USB are respee-

„An AM signal bas a cartier power of 3 W, The

percentage of modulation is SO percent, The

total sideband power is

a 0.8 W

h 166M

2.5 W

d 4Ow,

For 100 percent modulation, what percentige

ẹ power fs in each sideband?

a 25 percent

33.3 percent

c 50 percent

d 100 percent

An AM transmitter has a percentage of modu-

lation of 88 The carrier power is 440 W The

power in one sideband is

_ An AM transmitter antenna current is rmea-

sured with no modulation and found to be 2.6

amperes, With modulation the current rises to

2.9 amperes The percentage of modulation is

ad 3S percent

6 70 pereent,

ce 42 percent

d, SY percent

What is the carrier powerin the problem

above the antenna resistance is 75 Ohms?

a 19S W

b b1 W,

œ S07 ÁN,

d 793W,

bra AM signal the transmitted information is

contuined within the

a Less spectrum space is used

b Simpler equipment is used,

c Less power fs consumed,

d Ahigher modulation percentage

d Depends upon the use

The typical audio modulating frequency range used in radio and telephone communications is

ter2 Amohtude Modulation a: d Sir.gle Sideband Modulation

2-28 Distortion of the modulating signal produces

4(h), the peak carrier value is 7 V

What is th entags of modulation?

1 The output of an SSB transmitter with a 3.85-

-MHz¿ camier and a 1.5-KHz sine wave modulat-

ALR bude a

a 4.3.8485-MHz sine

b a 3.85-MHz sine wave

3.85-, 3.8485-, and 3.8515-MHz sine waves

d 3848.5- and 3851.5-Mriz sing waves,

An SSB transmitter produces a 400-¥ peak-to-

peak signal across a 52-0 antenna load The

d Peak etivelope power

An SSB transmitter has a PEP rating of 1 kW

The average output power ts in the runge of

a, 150 to 450 W

b, 100 to 300 W

c 259 to 333 W

Py aye

Critical Thinking Questions -

1 Explain why an overmodulated AM signal oc-

cupies a lot of bandwidth

2 Would it be possible to transmit one intelli-

gence signal in the upper sideband and a different intclligence signal in tne lower sideband

of an AM or a DSB signal? Explain

3 Explain how a potentiometer could be con-

nected to demonstrate AM

What are the side frequencies produced by a carrier modulated with a signal equal to the

cartier frequency?

During a weak AM signal transmission, will talking louder produce a stronger and more intelligible signal? Explain

An AM communication system consists of 30

channels spaced 5 kHz from one another

a had een RAE APC RA OO MEL

“Name two ways that can be used to prevent

one station from interfering with adjacent channel stations

2-7 An AM signal is restricted to a channel 4.5

kHz wide What is the highest frequency that

can be transmitted without going out of the

channel?

2-8 Could a voice signal (300-3000 Hz) be trans-

2-

mitted without modulation by amplifying the

signal aucio power amplifier and connecting its output to an antenna? Explain What are the

problems with this system?

5, A constant-amplitude signa! of 8.361 MHz is

received It is known that the source of the signal uses AM or SSB Name three possible con- ditions the signal may represent

translated

varies, Carrier modulator, carrier, modulating signa!

Trang 12

camier, sideband

0.45 5.06 63.25 236W

400 W, 89.44 percent

18 DSB false

345 3-, 4-W

htde or no

Chapter Objectives This chapter will help vou to:

1 Explain the operation of low-level diode

modulators and high-level collector mad-

ulators

2 Explain the operation of a diode detector

circuit

3 Stare the function of a balanced modulator

and explain the operation'of diode and IC

balanced modulators

4 Draw a block diagram of a filter-type SSB

generator and name three types of filters

used to eliminate one sideband

ver the years, hundreds of circuits

have been developed to produce AM

ge’ ‘Lhese modulator curculis cause une

“amplitude of the carrier to be varied in accor-

dance with the modulating information signal

There are circuits to produce AM, DSB, and

SSB In this chapter, ycu will examine some

3-1 Amplitude Modulators

There are twu basic ways to produce amplitude

modulation The first is to mudtiply the carrier by

4 gain or attenuation factor that varies with the

modulating signal The second is to linearly mix

or algebraically add the carrier and modulating

signals and then apply the composite signal to a nonlinear device or circuit All amplitude modu-

lators are based upon one of these two methods

Analog Multiplication

You can see how the first method works by referring to the basic AM equation in Chap 2 It is:

Vam = Ve sin Zaft + (Vaz sin 2afath(sin 27)

We know that the modulation index is

m= V/V,

Therefore

, Vin = mV, Substituting this in the equation above and

of the more common and widely used dis-

crete-coinponent and integrated-circuit (IC)

1D HOUU8Ó15, LAs CH ARCL aso sure

ers demodulators, A demodulator is the circuit that recovers the original information signal from the modulated wave Demodutators for

AM, DSB, and SSB will also be discussed

Vậm = Sin 22/⁄/(V¿ + mV_ sin 2 tfiat)

= V¿sỉn 2m#0(1 + m sin Dafa!)

You can see in this equation that AM is ac-

complished by multiplying the carrier by a factor equal tog plus the modulating sine wave

Uf we can create a circuit with a gain or atten-

uation that can be varied in accordance with

thể modulatins signal, AM is produced by pass-

ing the carrier through it Certain types of am- plifiers and voltage dividers (for example, PIN

diodes) can be created to do this Better still,

simply use one of the many available analog

multiplier or modulator integrated circuits

Nonlinear Mixing

With this method, we linearly mix the carrier and

modulating signals Then we use this composite

voltage to vary the currant in a nonlinear device

A nonlinear device is one whose current is pro-

portional to but does not vary linearly with the

applied voltage One nonlinear device is the

Amplitude Modulation Circuits Chapter? $2

Multiplication

4n

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