13Continuous-TimeModulation

In this lecture, we begin the discussion of modulation. This is an importantconcept in communication systems and, as we will see in Lecture 15, also pro-vides the basis for converting between continuous-time and discrete-time sig-nals. In its most general sense, modulation means using one signal to vary aparameter of another signal. In communication systems, for example, if achannel is particularly suited to transmission in a certain frequency range, theinformation to be transmitted may be embedded in a carrier signal matched tothe channel. The mechanism by which the information is embedded is modu-lation; that is, the information to be transmitted is used to modulate some pa-rameter of the carrier signal. In sinusoidal frequency modulation, for example,the information is used to modulate the carrier frequency. In sinusoidal ampli-tude modulation, the carrier is sinusoidal at a frequency that the channel canaccommodate, and the information to be transmitted modulates the ampli-tude of this carrier. Furthermore, in communication systems, if many differ-ent signals are to be transmitted over the same channel, a technique referredto asfrequency division multiplexing is often used. In this method each sig-nal is used to modulate a carrier of a different frequency so that in the com-posite signal the information for each of the separate signals occupies non-overlapping frequency bands.

The modulation property for Fourier transforms applies directly to am-plitude modulation, that is, the interpretation in the frequency domain of theresult of multiplying a carrier signal by a modulating signal. From the modula-tion property we know that for amplitude modulation the spectrum of themodulated output is the convolution of the spectra of the carrier and the mod-ulating signal. When the carrier is either a complex exponential or a sinu-soidal signal, the spectrum of the carrier is one or a pair of impulses and theresult of the convolution is then to shift the spectrum of the modulating signalto a center frequency equal to the carrier frequency. Modulation with a singlecomplex exponential and with a sinusoidal signal are closely related.

With a complex exponential carrier, demodulation, i.e., recovery of theoriginal modulating signal, is relatively straightforward, basically involvingmodulating a second time with the complex corjugate signal. With sinusoidal

13-1

Signals and Systems13-2

amplitude modulation, demodulation consists of modulating again with a si-nusoidal carrier followed by lowpass filtering to extract the original signal.This form of demodulation is typically referred to as synchronous demodula-tion since it requires synchronization between the sinusoidal carrier signals inthe modulator and demodulator. However, by adding a constant to the modu-lating signal or equivalently injecting some carrier signal into the modulatedoutput, a simpler form of demodulator can be used. This is referred to asasynchronous demodulation and typically results in a less expensive demod-ulator. However, the fact that a carrier signal is injected into the modulatedsignal represents an inefficiency of power transmission.

Suggested ReadingSection 7.0, Introduction, pages 447-448

Section 7.1, Continuous-Time Sinusoidal Amplitude Modulation, pages 449-459

Section 7.2, Some Applications of Sinusoidal Amplitude Modulation, pages459-464

Section 7.3, Single-Sideband Amplitude Modulation, pages 464-468

Continuous-Time Modulation

13-3

SINUSOIDAL AMPLITUDE MODULATION

SINUSOIDAL FREQUENCY MODULATION

TRANSPARENCY13.1Sinusoidal amplitudeand frequencymodulation with asinusoidal carrier.

TRANSPARENCY13.2Block diagram ofamplitude modulationand some examplesof commonly usedcarrier signals.

A A A t

Signals and Systems13-4

TRANSPARENCY13.3Spectra associatedwith amplitudemodulation with acomplex exponentialcarrier.

xt 1-x(t) CMt 4 ( ff X(w) * CMI~

YGw)

x(t) -

t + OG 43.

2e0*

Iz

I I W

Y~w)

y(t) + 3M

(WC - WM 4 1w. + "M)i

C(w)

21e-ws

t 0 -twot + .)

liwe

X(w)

~WM WM

00s (w~t + 6,)

TRANSPARENCY13.4Representation ofamplitude modulationwith a complexexponential carrier interms of amplitudemodulation with twosinusoidal carrierswith a 90* phasedifference.

xKt) X y(t)

X()

-- M WM

y"w)

y(t)

lY(C) + Y* (-w)j

Re jy(t)

(Y() - Y*(-w)

jIm jy(t)}

Re jy(t)}

IMl" o

Continuous-Time Modulation13-5

ejwct

X y(t)

Ideal lowpassfilter

H(w)

WO CO

e jct

w(t)X f(t)

TRANSPARENCY13.5The use of amplitudemodulation with acomplex exponentialcarrier to implementbandpass filtering witha lowpass filter.

X(o)

TRANSPARENCY13.6Spectra associatedwith the system inTransparency 13.5.

(-we - WO0 ) (-W d1 + WO)

Y(o) W(o)

F-

- (JO

I

F(w)

A-

Signals and Systems

13-6

TRANSPARENCY13.7Equivalent frequencyresponse associatedwith the system inTransparency 13.5.

TRANSPARENCY13.8Representation ofamplitude modulationwith a complexexponential carrier interms of amplitudemodulation with twosinusoidal carrierswith a 90* phasedifference.

(71(-Wc - Wo0 ) (-wc + wO)

H, (W)

2 wo

H2(F 2 7

-------------

Continuous-Time Modulation

13-7

c(t) = cos (WCt + Oe)

ei(wt + 0) + e et + 602 2

X(C,)

C(o)

we Ie

I weJC

1 YC )C-je l(W elo

TRANSPARENCY13.9Spectra associatedwith amplitudemodulation with asinusoidal carrier.

Cos (Wct + 0,)

I10 012 10

WC (- -M) (- + WpM)W

C(W)

tie tWeW

W(W)

i-2j9,I I 1 2jOC

TRANSPARENCY13.10Demodulation of anamplitude-modulatedsignal with asinusoidal carrier.

- COM

Signals and Systems

13-8

TRANSPARENCY13.11Block diagram ofsinusoidal amplitudemodulation anddemodulation.

MODULATOR

x(t) y(t)

cos (w et + 0 C)

DEMODU LATOR

y (t)

H (w)

2

-W w w

Lowpass filter

cos (oct + Oc) WM < w < (2oc m

COS Wat

TRANSPARENCY13.12Block diagram forfrequency divisionmultiplexing.

Cos Wc t

X c(W

x (t)

Ya (t)Xa (t)

Xb (t)

COS ob t

Yb (t)w(t)

Continuous-Time Modulation

13-9

X, (W)

-wM WM (A

Xb (W)

-wM W (

X ( M

\M M L

(A.)

W(W)

Ar7in K>- Wa

K>r7~~AWa Lb Mc L

TRANSPARENCY13.13Spectra illustratingfrequency divisionmultiplexing.

TRANSPARENCY13.14Demultiplexing anddemodulation of afrequency divisionmultiplexed signal.

- C

Demultiplexing -

Bandpassfilter COS Wat

Demodulation

Lowpassfilter

Xa (t)

-Wb

-Cb

Signals and Systems

13-10

TRANSPARENCY13.15Demodulation of anamplitude-modulatedsignal with asinusoidal carrier.[Transparency 13.10repeated]

TRANSPARENCY13.16Effect of loss ofsynchronizationin phase in asynchronoussinusoidal amplitudemodulation/demodu-lation system.

Cos (WC t + 0c)

Y (W)

j2i1 el6

C(I- 11 (W,- M) W. (W. + WM) W

ciw)

re~lI

Weit

W(W)

-2jO I~O

- w, I W M2 L

cos (ct + 0c)

(oe - +c)]x(t)

Lowpass filter

cos (oct + 4c)

w(t) = x(t) Cos (Wmt + 6,) cos (ot +

= [cos (0c ~ oc)]x(t) +-I- x(t) cos (2wct + Oc + #c)

lowpass component

Continuous-Time Modulation

13-11

TRANSPARENCY13.17A simple system andassociated waveformfor an asynchronousdemodulation system.

y(t) - IA + x(tI cos wCt

Cos ('tAT A

A NA~ ~NA AAAW4-A-

TRANSPARENCY13.18Modulator associatedwith asynchronoussinusoidal amplitudemodulation. For sucha system the carriermust be irjected intothe output. Thistransparency showstime waveforms forthe asynchronousmodulation system.

IY- -V.N ,Lv -yy V. vA

Signals and Systems

13-12

TRANSPARENCY13.19Frequency spectraassociated with anasynchronousmodulation system.

y(t) - A+ x (t)I cos oct

cos (", t

iTA irA12

X(W)

TRANSPARENCY13.20Single-sidebandmodulation in whichonly the uppersidebands areretained.

Y(W)

sideband sideband sideband sideband

Y, (w)

-C W

~W M W M

-we

Continuous-Time Modulation13-13

v (t)

112

c W

H(w)

-Loc cW

1 - -

A 2

TRANSPARENCY13.21Single-sidebandmodulation in whichonly the lowersidebands areretained.

TRANSPARENCY13.22The use of a highpassfilter to obtain asingle-sideband signal.

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