Chap. 3: Amplitude (Linear) Modulation

Now that we are familiar with the basic signal analysis techniques, we want to move ahead todiscuss operational communication systems.

• Amplitude Modulation (AM) and its variants;

• Angle Modulation with special case - FM;

Message signal: m(t) baseband signal with bandlimited to B Hz. For efficient transmission,m(t) has to be transmitted over a bandpass communication channel.

• This requires a shift of the range of frequencies in m(t) into other frequency rangessuitable for transmission over a communication channel (usually shifting up infrequency);

• At the receiver, this process has to be reversed (down shifting in frequency);

• Why do we need to move to higher frequency band – antenna size, available bandwidth,multiple access, etc. (for example, cellular phone, antenna size - !/4. For f1 = 3k,!1 = c/f1 = 105m, !1/4 = 2.5! 104m; while f2 = 900M , !2 = c/f2 = 1/3m,!2/4 = 8cm.)

1

Definition: The process by which some characteristics of a carrier signal is varied inaccordance with a modulation signal is called MODULATION.

• A common form of the carrier we will use, is a sinusoidal wave. Let

c(t)""carrier wave/signal m(t)""modulating signal

• m(t) modulates c(t) by changing some of its characteristics.

Modulators(t)

c(t): carrier

m(t): message signal

modulating signal modulated waveform (bandpass)

The modulated waveform/signal can be generally expressed as

s(t) = a(t) cos("(t)) (1)

If a(t) changes as m(t), then we have amplitude modulation (AM). If "(t) changes as m(t),then we have angle modulation (PM, FM).

2

In the case of AM,

"(t) = constant phase and frequency = 2#fct+ $0

where we assume that $0 = 0 without loss of generality, and

a(t) # m(t)

3

3.1 DSB-SC

Let us consider the simple and intuitive AM case. Let the message signal to be m(t) with abandwidth of B Hz. The carrier signal is

c(t) = Ac cos(2#fct)

with fc >> B.

fc

fc

4

upper sideband

lower sideband

Figure 1: Illustration of upper/lower side-band.

Then the AM wave s(t) is given by the expression

s(t) = m(t) · c(t) = Acm(t) cos(2#fct)

S(f) =Ac

2[M(f + fc) +M(f " fc)]

is called DSB-SC where the acronym DSB-SC represents “double sideband - suppressedcarrier”.

• Transmission bandwidth = 2B (twice the baseband BW).

• upper sideband (USB): the portion that lies outside of ±fc; lower sideband (LSB): theportion that lies inside of ±fc.

• M(f) doesn’t contain a discrete component of the carrier frequency fc (the modulation

5

process doesn’t introduce a sinusoid at fc).

• This product modulation is named as DSB-SC (double-sideband, suppressed carrier).

• Recovery (demodulation) of m(t) from s(t)

– Modulation: m(t) $ DSB-SC by multiply the carrier

– Demodulation: DSB-SC $ m(t) shift in frequency again

s(t) cos(2#fct) = Acm(t) cos2(2#fct) =Ac

2[1 + cos(2#fct)]m(t)

=Ac

2m(t) +

Ac

2m(t) cos(2#fct)

• Main advantage: extra power not needed for added carrier.

• Main disadvantage: carrier phase and frequency synchronization needed at the receiver.

6

spectrum @ A

fc

carrier carrier

LPF

Demodulatorm(t) A C

Modulator

2fc−2fc

k m(t)

2fmspectrum @ C

Figure 2: Modulation and demodulation of DSB-SC.

7

Generation of DSB-SC Signals

There are many methods that allow us to generate AM signals.

(1) Nonlinear Modulator

Consider a non-linear devicey(t) = ax(t) + bx2(t)

which is used in the following system (diagram p184 textbook)

8

x1(t) = m(t) + cos(%ct) x1(t) = m(t)" cos(%ct)

y1(t) = ax1(t) + bx21(t) y2(t) = ax2(t) + bx2

2(t)

z(t) = y1(t)" y2(t) (2)

= a[m(t) + cos(%ct)] + b[m(t) + cos(%ct)]2

"a[cos(%ct)"m(t)]" b[cos(%ct)"m(t)]2

= am(t) + a cos(%ct) + bm2 + 2bm cos(%ct) + b cos2(%ct)

am(t)" a cos(%ct)" bm2 + 2bm cos(%ct)" b cos2(%ct)

= 2am(t) + 4bm(t) cos(%ct)

where the first term can be filtered out and the second term can be kept by applying abandpass filter (BPF). The BPF is tuned to %c with a bandwidth 2B Hz.

(2) Switching Modulator:

The multiplication operation required for modulation can be replaced by a simpler switchingoperation. Modulated signal can be obtained by multiplying m(t) by any periodic signal $(t)with fundamental frequency fc.

9

Let $(t) be such a periodic waveform, then

$(t) =!

n

Cn cos(n%ct+ "n)

andm(t)$(t) =

!

n

Cnm(t) cos(n%ct+ "n)

10

such that

F [m(t)$(t)] =1

2

!

n

CnM(f)%[&(f"nfc)+&(f+nfc)] =1

2

!

n

Cn[M(f+nfc)+M(f"nfc)]

Thus, the spectrum of the product m(t)$(t) is the spectrum of the modulating waveform,M(f) shifted to ±fc,±2fc,±3fc, · · · and scaled by Cn. If we pass this signal through aBPF of bandwidth 2B Hz and tuned to fc, then we have the desired modulated signalm(t) cos(2#fct).

The above operation is also known as mixing, frequency conversion, or heterodyning.

Let’s apply the square pulse train as the periodic function. The gate function is x(t) = tT/2 .

We have X(f) = T2 sinc(#fT/2), then

Dn =1

TX(nf0) =

T

2Tsinc

"# · n 1

T· T2

#=

1

2sinc

$#

2n%. (3)

Obviously, if n equals even numbers, Dn = 0;D0 = 1/2. We want to express w(t) astrigonometrical form as

w(t) = C0 +!!

n=1

Cn cos(n%ct+ "n)

11

For n = 1,

D1 =1

2sinc

$#

2

%=

1

2· 2#

=1

#(4)

D"1 =1

2sinc

$"#

2

%=

1

2· "2

#= " 1

#(5)

Similarly, D3 = "1/(3#), D5 = 1/(5#), · · · . Therefore, the coefficients of {Cn} are:

Cn =

&'

(D0 = 1

2 n = 0

2Dn = ("1)(n"1)/2 · 2n! n = 1, 3, 5, 7, · · · ,

(6)

The gate function can be written as

w(t) = 12 + 2

!

)cos(%ct)" 1

3 cos(3%ct) +15 cos(5%ct) + · · ·

*.

(7)

12

so

m(t)w(t) =1

2m(t) +

2

#

+m(t) cos(%ct)"

1

3m(t) cos(3%ct) +

1

5m(t) cos(5%ct) + · · ·

,.

After bandpass filter centered at %c, we have DSB-SC signal

s(t) =2

#m(t) cos(%ct) (8)

13

Unsynchronized Demodulation

For demodulation of DSB-SC, the receiver must generate a carrier that is synchronous withthe incoming carrier (the requirement for a local oscillator to generate L02 = fc). Thesedemodulators are synonymously called “synchronous” or “coherent” demodulators.

We may notice potential problems: What happens when LO2 is mismatched to LO1 ? Let theoutput of the LO2 be

l2(t) = cos(2#(fc +!f) + $0)

Then, the demodulated waveform expression is

s(t)l2(t) = Acm(t) cos(2#fct) cos(2#(fc +!f)t+ $0)

=Ac

2m(t) cos(2#!ft+ $0) +

Ac

2m(t) cos(2#(2fc +!f)t+ $0) (9)

We assume that the error/mismatch between LO1 and LO2 is small, i.e., !f << fc.Therefore, the 2nd component in (9) will be centerred about f = 2fc and will be filtered outby the demodulator LPF. Thus, the LPF output equals

y(t) =Ac

2m(t) cos(2#!ft+ $0)

14

Thus, y(t) is oscillating at a slow rate. For example, let m(t) = exp("2t). The demodulatoroutput is an oscillation function as shown in the figure below.

Assume that !f = 0, then

y(t) =Ac

2m(t) cos($0)

It can be written as

y(t) =

&'

(

Ac2 m(t) if $0 = 0 (best)

Ac2 m(t) cos(#/2) = 0 if $0 = #/2 (worse)

Therefore, if we are lucky to have $0 = 0, it corresponds to perfectly synchronization of LO1

and LO2. But is the phase difference is #/2, there is no signal at the output of thedemodulator.

Hence, we may conclude that successful demodulation of DSB-SC signal requires carefuland accurate synchronization of the two LOs.

15

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

ampl

itude

(a) message signal: x(t)=exp(−2t)

0 0.5 1 1.5 2 2.5−1

−0.5

0

0.5

1

ampl

itude

(b) demodulated message with Δf=10 Hz

Figure 3: Synchonization error !f = 10 Hz.

16

Amplitude Modulation (AM) – Large Carrier

• We have seen that the use of suppressed carrier waveforms requires fairly complicatedcircuitry at the receiver in order to acquire and maintain phase synchronization(expensive receivers).

• Especially in applications where we have one (or few) TX and a much larger number ofRXs, it makes economic sense that the RXs are as simple as possible.

• To address this issue we entertain the idea of transmitting a carrier in the same frequencyband as the DSB-SC, which is DSB-LC

• This approach will certainly ruin the DC response of m(t), but many signals do not haveany DC content anyway. Hence we have

DSB-SC : s(t) = m(t) cos(%ct)

DSB-LC : s(t) = m(t) cos(%ct) +Ac cos(%ct) = [Ac +m(t)] cos(%ct)

and

SAM (f) =1

2[M(f + fc) +M(f " fc)] +

Ac

2[&(f + fc) + &(f " fc)]

17

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0

2

4

ampl

itude

(a) message signal

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

−1

0

1

ampl

itude

(b) carrier

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

−2

0

2

4

ampl

itude

(c) modulated signal

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0

2

4

ampl

itude (d) output of envelope detector

seconds

Figure 4: AM Large Carrier.

18

We will refer to the envelope of the signal, where we use definition of the envelope asfollows: given the signal E(t) cos(%ct), E(t) is the envelope if E(t) & 0 for all t. In a moregeneral sense we will use |E(t)| as the envelope of E(t) cos(%ct).

19

• For envelope detection to properly detect m(t), two conditions must be met:

a). fc >> bandwidth of m(t)

b). Ac +m(t) & 0

• For AM signal we will insist that the envelope |E(t)| = E(t) = Ac +m(t), i.e., E(t)

remains positive for all t. This will allow the use of a simple/inexpensive detector (forexample, envelope detector) for demodulation of AM signal.

• If m(t) & 0 for all t, there is no sense to add a carrier signal Ac cos(%ct). We can applyenvelope detection to the signal m(t) cos(%ct). Therefore, we are interested toinvestigate the case that m(t) has negative amplitude for some t.

• Let us define "mp as the minimum value of m(t). Then,

µ = mp/Ac

is known as the modulation index of the AM signal.

• Conversely, if E(t) = Ac +m(t) < 0, then the envelope will no longer be a scaled andshifted version of m(t), and due to the phase reversals as Ac +m(t) crossed thehorizontal axis, we have to use a synchronous detector as in the DSB-SC case,eliminating all the advantages of AM.

20

• The requirement that Ac +m(t) & 0 for all t is then equivalent to

0 ' µ ' 1

This will be a sufficient condition for proper/easy demodulation of AM signals.

• Main advantage: carrier phase and frequency synchronization are not needed at thereceiver.

• Main disadvantage: power needed for large carrier does not convoy message signal.

Example: consider the single-tone modulation case, where m(t) = B cos(%mt) (modulatingsignal is a pure sinusoid or tone).

In this case, mp = B and the modulation index is

µ = B/Ac

Hence B = µAc andm(t) = B cos(%mt) = µAc cos(%mt)

Therefore,s(t) = [Ac +m(t)] cos(%ct) = Ac[1 + µ cos(%mt)] cos(%ct)

21

0 0.05 0.1 0.15 0.2 0.25 0.3−3

−2

−1

0

1

2

3

ampl

itude

0 0.05 0.1 0.15 0.2 0.25 0.3−3

−2

−1

0

1

2

3

ampl

itude

µ=0.5

µ= 1

Figure 5: Tone-modulation.

22

Sideband and Carrier Power

The advantage of envelope detection in AM has its price. In AM, the carrier term does notcarry any information, and hence, the carrier power is wasted. Let’s look at an AM signal

sAM (t) = Ac cos%ct+m(t) cos%ct

The carrier power Pc and sideband power Ps are given as

Pc =A2

c

2Ps =

1

2E[m2(t)]

Ps is the useful power and Pc is the power wasted for convenience. The power efficiency is

' =useful powertotal power

=Ps

Pc + Ps=

E[m2(t)]

A2c + E[m2(t)]

For tone modulation,

m(t) = µAc cos%mt $ E[m2(t)] =(µAc)2

2

Hence

' =µ2

2 + µ20 ' µ ' 1

It can be seen that ' increases monotonically with µ, and 'max = 33% occurs at µ = 1.

23

Generation of AM Signals

We can use the same DSB-SC generation methods except that an additional carriercomponent A cos%c needs to be added.

The gate function w(t) can be expanded by using FS as

w(t) =

+1

2+

2

#

"cos%ct"

1

3cos 3%ct+

1

5cos 5%ct" · · ·

#,

24

vbb!(t) = [Ac cos%ct+m(t)] · w(t) (10)

= [Ac · cos%ct+m(t)]

+1

2+

2

#

"cos%ct"

1

3cos 3%ct+

1

5cos 5%ct" · · ·

#,

=Ac

2cos%ct+

2

#m(t) cos%ct+ other terms

25

Demodulation of AM Signals

Coherent/synchronous detection with a locally generated carrier (as in the case of DSB-SC)is possible, but this approach defeats the whole purpose of AM.

(a) Rectifier Detector

26

These operations can also be described analytically (diagram, p196)

vR(t) = (A+m(t)) cos(%ct)w(t)

= (A+m(t)) cos(%ct)

+1

2+

2

#

"cos(%ct)"

1

3cos(3%ct) + · · ·

#,

=A+m(t)

#+ (· · · ) cos(%ct) + (· · · ) cos(2%ct) + · · ·

At the output of the LPF we haveA+m(t)

#

and after the DC-eliminator, we have output m(t)/#.

27

(b) Envelope detector

The biggest selling point for the AM signals was the simple/inexpensive detection, i.e.,envelope detector (ED). ED is a simple but effective device well suited to the demodulationof narrowband AM signals with µ ' 1. Ideally the output of the ED follows the envelope ofs(t):

Vc

s(t)

28

• s(t) > Vc:

– diode conducts (forward biased)

– C charges quickly to the max value of s(t)

• s(t) < Vc:

– diode does not conduct (reverse biased)

– C discharges over RL until s(t) > Vc

• The output of the ED is then lowpass filtered to eliminate the ripple, followed byblocking out the DC component.

29

Single Sideband Modulation (SSB)

Standard AM and DSB-SC techniques are wasteful of bandwidth because they both requiretransmission bandwidth of 2B Hz, where B is the bandwidth of the baseband modulatingsignal m(t).

In both cases the transmission bandwidth (BT ) is occupied by the upper sideband (USB) andlower sideband (LSB).

Observations

• USB and LSB are uniquely related to each other, as they are symmetric wrt fc.Therefore, to transmit information contained within m(t) we used to transmit only oneside band.

• As far as demodulation is concerned, we can coherently demodulate SSB (as we did theDSB-SC signal) by multiplying SSB with cos(%ct) followed by LPF.

Frequency domain representation of SSB signals

Given the baseband signal m(t) with spectrum M(%), the spectrum of DSB-SC and SSB areshown below (textbook, p174):

30