Lecture 3 DSB AM

LECTURE 3

CONVENTIONAL DOUBLE SIDEBAND

AMPLITUDE MODULATION

A/Prof Dr Zhuquan Zang

Dept of Electrical and Computer Engineering

Curtin University

Perth, Western Australia

CE304-DC603 LECTURE NOTES Lecturer & Tutor: [email protected]

LECTURE 3: CONVENTIONAL AMPLITUDE MODULATION

Generally speaking, DSB-SC AM systems need sophisticated circuitry at the receiver

for the purpose of generating a local carrier of exactly the right frequency and phase for

synchronous demodulation. But such systems are very efficient from the point of view

of power requirements at the transmitter. In point-to-point communications, where

there is one transmitter for each receiver, substantial complexity in the receiver system

can be justified, provided it results in a large enough saving in expensive high-power

transmitting equipment. On the other hand, for a broadcast system with a multitude of

receivers for each transmitter, it is more economical to have one expensive high-power

transmitter and simpler, less expensive receivers. For such applications, a large carrier

signal is transmitted along with the suppressed-carrier modulated signal m(t) cos(ωct),

thus eliminating the need to generate a local carrier signal at the receiver. This is the

so-called AM (amplitude modulation), in which the transmitted signal ϕAM(t) is given

by

ϕAM(t) = m(t) cos(ωct) + A cos(ωct) = [A + m(t)] cos(ωct)

The spectrum of ϕAM(t) is the same as that of m(t) cos(ωct) plus two additional

impulses at ±ωc

ϕAM(t) ↔ 1

2[M(ω + ωc) + M(ω − ωc)] + πA[δ(ω + ωc) + δ(ω − ωc)]

The condition for demodulation by an envelope detector is

A + m(t) > 0 for all t

This is the same as

A ≥ −min{m(t)}

We define the modulation index µ as

µ =−min{m(t)}

A

µ ≤ 1 is the required condition for proper demodulation of AM by an envelope detector.

For µ > 1 (overmodulation), the option of envelope detection is no longer available.

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We then need to use synchronous demodulation. Note tha synchronous demodulation

can be used for any value of µ. The envelope detector, which is considerably simpler

and less expensive than the synthronous detector, can be used only for µ ≤ 1. For

µ ≥ 1, it is possible to extract the required local carrier from the received signal by

using a narrowband filter tuned to ωc (the carrier can be readily extracted by using a

phas-lock loop).

Sideband and carrier power. The advantage of envelope detection in AM has its

price. In AM, the carrier term does not carry any information, and hence, the carrier

power is wasted.

ϕAM(t) = A cosωct︸︷︷︸

carrier

+ m(t) cosωct︸︷︷︸

sidebands

The carrier power Pc is the mean squre value of A cos ωct, which is A2/2. The sideband

power Ps is the power of m(t) cos ωct, which is 0.5 m2(t). Hence,

Pc =A2

2and Ps =

1

2m2(t)

The sideband power is the useful power and the carrier power is the power wasted for

convenience. The total power is the sum of the carrier (wasted) power and the sideband

(useful) power. Hence, η, the power efficiency, is

η =useful power

total power=

Ps

Pc + Ps=

m2(t)

A2 + m2(t)× 100%

For the special case of tone modulation

m(t) = µA cos ωmt and m2(t) =(µA)2

2

Hence,

η =µ2

2 + µ2× 100%

with the condition that 0 ≤ µ ≤ 1. It can be seen that η increases monotonically with

µ, and ηmax occurs at µ = 1, for which ηmax = 33%. Thus, for tone modulation, under

best conditions (µ = 1), only one-third of the transmitted power is used for carrying

message. For practical signals, the efficiency is even worse – on the order of 35% or

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lower –compared to that of DSB-SC case. The best condition implies µ = 1. Smaller

values of µ degrade efficiency further. For this reason volume compression and peak

limiting are commonly used in AM to ensure that full modulation (µ = 1) is maintained

most of the time.

Generation of AM Signals

Nonlinear modulators. Modulation can be achieved by using nonlinear devices. A

semicondductor diode or a transistor is an example of such a device. Nonlinear charac-

teristics such as these may be approximated by a power series:

i = ae + be2

AM signals can be generated by any DSB-SC generator if the modulating signal is

A + m(t) instead of just m(t). AM can be generated in simpler ways, however. For

example, the nonlinear modulator ( see Fig. 1) used for DSB-SC is a balanced modulator.

We can show that to generate AM, we need use only one of the two branches. Assume

Figure 1: Nonlinear DSB-SC modulator

that the nonlinear characteristic between i and e is given by

i = ae + be2

The output of the upper modulator is i1R, given by

i1R = R[a(cos(ωct) + m(t)) + b(cos(ωct) + m(t))2]

= aR cos(ωct) + 2bRm(t) cos(ωct) + aRm(t) + bRm2(t) + bR cos2(ωct)

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When this signal is passed through a bandpass filter tuned to ωc, the last three terms

are suppressed, leaving only the first two terms, which in fact represent an AM signal.

Switching modulators. Fig. 2 shows an AM modulator with a single diode acting as

a switch. The input is c cos(ωct) + m(t) with c � m(t), so that the switching action

of the diode is controlled by c cos(ωct). The diode opens and shorts periodically with

Figure 2: Switching modulator - AM generator

cos(ωct), in effect multiplying the input signal [c cos(ωct)+m(t)] by k(t). The voltage

across terminall bb′, is

vbb′(t) = [c cos(ωct) + m(t)]k(t)

= [c cos(ωct) + m(t)][1

2+

2

π

(

cos(ωct) −1

3cos(3ωct) +

1

5cos(5ωct) + · · ·

)]

=c

2cos(ωct) +

2

πm(t) cos(ωct) + other terms

The bandpass filter tuned to ωc suppresses all the other terms, yielding the desired AM

signal at the output.

Demodulation of AM Signals

The AM signal can be demodulated coherently by a locally generated carrier. Co-

herent, or synthronous, demodulation of AM, however, will defeat the very purpose

of AM and, hence, is rarely used in practice. We shall consider here three noncoher-

ent methods of AM modulation: 1) rectifier detection, 2) envelope detection, and 3)

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square-law detection.

Rectifier detector. If AM is applied to a diode and a resistor circuit (see Fig. 3), the

negative part of the AM wave will be suppressed.

Figure 3: Rectifier detector for AM

The output across the resistor is a rectified version of the AM signal. In essence,

the AM signal is multiplied by k(t). Hence, the rectified output vk is

vk(t) = {[A + m(t)] cos(ωct)}k(t)

= [A + m(t)] cos(ωct)[1

2+

2

π

(

cos(ωct) −1

3cos(3ωct) +

1

5cos(5ωct) + · · ·

)]

=1

π[A + m(t)] + other terms of higher frequencies

When vR(t) is applied to a lowpass filter of cutoff B, the output is [A + m(t)]/π, and

all other terms in vR(t) of frequencies higher than B are suppressed. The dc term A/π

may be blocked by a capacitor to give the desired output m(t)/π. The output cna be

doubled by using a full-wave rectifier.

It is interesting to note that rectifier detection is in effect synchronous detection

performed without using a local carrier. The high carrier content in the received signal

makes this possible.

Envelope detector. In an envelope detector, the output of the detector follows the

envelope of the modulated signal. The circuit show in Fig. 4 functions as an envelope

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detector. On the positive cycle of the input signal, the capacitor C charges up the

Figure 4: Envelope detector for AM

peak voltage of the input signal. As the input signal falls below this peak value, the

diode is cut off, because the capacitor voltage (which is very nearly the peak voltage)

is greater than the input signal voltage, thus causing the diode to open. The capacitor

now discharges through the resistor R at a slow rate. During the next positive cycle,

when the input signnal becomes greater than the capacitor voltage, the diode conducts

again. The capacitor again charges to the peak value of this (new) cycle. The capacitor

discharges slowly during the cut-off, thus changing the capacitor voltage very slightly.

During each positive cycle, the capacitor charges up to the peak voltage of the input

signal and then decays slowly until the next positive cycle. The output voltage thus

follows the envelope of the input. Making RC too large would make it impossible for

the capacitor voltage to follow the envelope (see Fig. 4). A ripple signal of frequency

ωc, however, is caused by capacitor discharge between positive peaks. This ripple is

reduced by increasing the time constand RC so that the capacitor discharges very little

between the positive peaks (RC � 1/ωc). Making RC too large, however, would

make it impossible for the capacitor voltage to follow the envelope. Thus, RC should

be large compared to 1/ωc but should be small compared to 1/2πB, where B is the

highest frequency in m(t). This, incidentally, also requires that ωc � 2πB, a condition

that is necessary for a well-defined envelope.

The envelop-detector output is A + m(t) with a ripple of frequence ωc. The dc

term A can be blocked out by a capacitor or a simple RC highpass filter. The ripple

may be reduced further more another (lowpass) RC filter

Example: For tone modulation (m(t) = Am cos(ωmt)), determine the upper limit on

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RC to ensure that the capacitor voltage follows the envelope.

Solution: Fig. 5 shows the envelope and the voltage across the capacitor. The capacitor

Figure 5: Single tone envelope detection

discharges from the peak value E at some arbitrary instant t = 0. The voltage vc across

the capacitor is given by

vc(t) = Ee−t/RC

Because the time constant is much larger than the interval between the two successive

cycles of the carrier (RC � 1/ωc). the capacitor voltage vc discharges exponentially

for a short time compared to its time constant. Hence, the exponential can be ap-

proximateed by a straight line obtained form the first two terms in Taylor’s series of

Ee−t/RC .

vc(t) ≈ E(

1 − t

RC

)

The slope of the discharge is −E/RC. In order for the capacitor to follow the enve-

lope E(t), the magnitude of the slope of the RC discharge must be greater than the

magnitude of the slope of the envelope E(t). Hence

∣∣∣∣

vc

dt

∣∣∣∣ =

E

RC≥∣∣∣∣∣

dE

dt

∣∣∣∣∣

(1)

But the envelope E(t) of a tone-modulated carrier is

E(t) = A[1 + µ cos(ωmt)]

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dE

dt= −µAωm sin(ωmt)

Hence, Eq. 1 becomes

A(1 + µ cos(ωmt))

RC≥ µAωm sin(ωmt)

for all t, or

RC ≤ A(1 + µ cos(ωmt))

µωm sin(ωmt)

for all t. The worst possible case occurs when the right-hand side is the minimum.

This is found (as usual, by taking the derivative and setting it to zero) to be when

cos(ωm) = −µ. For this case, the right-hand side is√

(1 − µ2)/µωm. Hence,

RC ≤ 1

ωm

(√1 − µ2

µ

)

Square-law detector. An AM signal can be demodulated by squaring it and then

passing the squared signal through a lowpass filter. This can be readily seen from the

fact that

ϕAM(t) = [A + m(t)] cos(ωct)

and

ϕ2

AM(t) =[A2 + 2Am(t) + m2(t)]

2(1 + 2 cos(2ωct)

The lowpass filter output yo(t) is

yo(t) =A2

2

[

1 + 2m(t)

A+

m2(t)

A2

]

Usually, m(t) � A for most of the time. Only when m(t) is near its peak is this

violated. Hence,

yo(t) ≈A2

2+ Am(t)

A blocking capacitor will suppress the dc term, yielding the output Am(t). Note that

square-law detector causes signal distortion. The distortion, however, is negligible when

µ is small. This type of detection can be performed by any nonlinear device that does

not have odd symmetry. Whenever a modulated signal pass through any nonlinearity

of this type, it will create a demodulation component, whether intended or not.

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Comparison of Various AM Systems

The AM system has an advantage over the AM-SC systems (that is, DSB-SC and

SSB-SC) at the receiver. The detectors required for AM are relatively simpler (rectifier

or envelope detectors) than those required for suppressed-carrier systems. For this

reason, all AM broadcast systems AM. In addition, AM signals are easier to generate at

high power levels, as compared to suppressed-carrier signals. The balanced modulators

required in the latter are somewhat difficult to design.

Suppressed-carrier systems have an advantage over AM in that they require less

power to transmit the same information. Under normal conditions, the carrier takes

up to 75 percent (or even more) of the totaltransmitted power. This necessitates

a rather expensive transmitter for AM. For suppressed-carrier systems, however, the

receiver is much more complex and consequently more expensive. For a point-to-point

communication system, where there are only a few receivers for one transmitter,, the

complexity in a receiver is justified, whereas for public broadcast systems, where there

are millions of receivers for each transmitter, AM is the obvious choice.

The AM signal also suffers from the phenomenon of fading, as mentioned earlier.

Fading is strongly frequency dependent; that is, various frequency components suffer

diffenrnt attenuation and nonlinear phase shifts. This is known as selective fading.

The effect of fading is more serious on AM signals than on AM-SC signals, because in

AM the carrier must maintain a certain strength in relation to the sidebands. Because

of selective fading, the carrier may be attenuated to the point where the modulation

index is no longer less than 1. In such a case, the received signal detected by an

envelope detector will show severe distortion. Even if the carrier is not badly attenuation

and the modulation index is still less than 1, selective fading can still badly distort

the AM signal because of unequal attenuation and nonlinear phase shifts of the two

sidebands and the carrier. The effect of selective fading becomes pronounced at higher

frequencies. Therefore, suppressed-carrier systems are prefrred at higher frequencies.

The AM system is generally used for medium-frequency broadcast transmission.

Next we compare the DSBN-SC system with the SSB-SC system. Here we find

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that the balance is mostly in favour of SSB-SC. The following are the advantages of

the SSB-SC over DSB-SC.

1. SSB-SC needs only half the bandwidth needed for DSB-SC. Although this dif-

ference can be balanced out by quadrature multiplexing two DSB-SC signals,

practical difficulties of crosstalk are much more serious in quadrature multiplex-

ing.

2. Frequency and phase errors in the local carrier used for demodulation have more

serious effects in DSB-SC than in SSB-SC, particularly for voice signals.

3. Selective fading disturbs the relationship of the two sidebands in DSB-SC and

causes more serious distortion than in the case of SSB-SC, where only one side-

band exists.

For these reasons, DSB-SC is rarely used in audio communication. Long-haul telephone

systems use SSB-SC multiplexed systems with a pilot carrier. For short-haul systems,

DSB is sometimes used., PCM, however, is gradually replacing both.

SSB compares poorly to DSB in one respect: the generation of high-level SSB

signals is more difficult than that of DSB signals. This disadvantage is overcome in

what is called vestigial sideband transmission.

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Lecture 3 DSB AM

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