15Discrete-TimeModulation

The modulation property is basically the same for continuous-time and dis-crete-time signals. The principal difference is that since for discrete-time sig-nals the Fourier transform is a periodic function of frequency, the convolutionof the spectra resulting from multiplication of the sequences is a periodic con-volution rather than a linear convolution.

While the mathematics is very similar, the applications are somewhat dif-ferent. In continuous time, modulation plays a major role in communicationssystems for transmission of signals over various types of channels. That appli-cation usually is inherently a continuous-time application. However, in manymodern communication systems, signals may go through various stages andtypes of modulation as they move from one channel to another, and often thisconversion from one modulation system to another is best implemented digi-tally. In their digital form the signals are discrete-time signals, and such trans-modulation systems are based on modulation properties associated with dis-crete-time signals.

In addition to digital modulation systems, the concepts of discrete-timemodulation (and, for that matter, continuous-time modulation also) are usefulin the context of filtering, particularly when it is of interest to implement fil-ters with a variable center frequency. It is often simpler in such situations toimplement a fixed filter (either continuous time or discrete time) and throughmodulation shift the signal spectrum in relation to the fixed filter center fre-quency rather than shifting the filter center frequency in relation to the signal.For discrete-time signals, for example, from the modulation property it fol-lows that multiplying a signal by (- 1)' has the effect of interchanging thehigh and low frequencies. Consequently, by alternating the algebraic sign ofthe input signal, processing with a lowpass filter, and then alternating the al-gebraic sign of the output signal, a highpass filter can be implemented.

In discussing continuous-time modulation in Lecture 13 and discrete-time modulation in the first part of this lecture, the emphasis is on a carriersignal that is a complex exponential or sinusoidal signal. Another importantand useful class of carrier signals is periodic pulse trains that are constant forsome fraction of the period and zero for the remainder. In effect, then, either

15-1

Signals and Systems15-2

in continuous time or discrete time, modulation with such a pulse train con-

sists of extracting time slices of the modulating signal. In data representationor transmission, this permits, for example, a type of multiplexing referred to

as time division multiplexing since during the "off" part of the pulse train,time slices from signals in other channels can be inserted. Somewhat amaz-

ingly, the original modulating signal can be recovered exactly after pulse-am-

plitude modulation provided only that the fundamental frequency of the carri-

er pulse train is greater than twice the highest frequency in the modulating

signal. The modulating signal can then theoretically be recovered exactly byfiltering the pulse-amplitude-modulated signal with an ideal lowpass filter.

Furthermore, this ability to exactly reconstruct the original signal is indepen-dent of the "duty cycle" of the carrier, i.e., it is theoretically possible no matter

how narrow the "on" time of the pulse train is made. If for the continuous-

time case, the "on" time of the carrier pulse train is made arbitrarily small,with the amplitude increasing proportionately, the carrier then correspondsto an impulse train. For discrete time, the pulse train with the smallest "on"

time would likewise correspond to a periodic train of impulses or unit sam-

ples. In both cases, then, modulation with the impulse train carrier would cor-

respond to sampling the modulating (input) signal. This leads to an extremelyimportant concept, referred to as the sampling theorem. The sampling theo-

rem states that a bandlimited signal can be exactly reconstructed from equal-

ly spaced time samples provided that the fundamental frequency of the sam-

pler (i.e., the impulse train carrier) is greater than twice the highest frequencyin the signal to be reconstructed. This fundamental and important result, to be

explored further in Lecture 16, provides a major bridge between continuous-

time and discrete-time signals and systems.

Suggested ReadingSection 7.5, Discrete-Time Amplitude Modulation, pages 473-479

Section 7.4, Pulse Amplitude Modulation and Time-Division Multiplexing,pages 469-473

Discrete-Time Modulation15-3

e Cormplex exponentiQLCarrier XE"3

Ctn]

. Sin\usoidal C av-rier

. ScroNous rrodulat.ion-.

Asinchronous,modulakion _

x E I (e)c(n-e)d 0

CoMP ex exponentla L

C , 3(n -r' en+c)

S' nusod' Carrier

C [r= Cos(en.+ec)

. S I le Side baNd Pulse Carrier

x[n] c[n] X()C(2 ) dO2i7

X(2)

I III\I-27r -M 0 M 2r E2

C(92)

2w 2w 21r

27r -2w+ fl 0 _ 27r 2w + 92

Y()

TRANSPARENCY15.1Spectra associatedwith discrete-timeamplitude modulationwith a complexexponential carrier.

27r -27r + 2(c2c - EZ ) Se (Oc + EM) 27r

MARKERBOARD15.1

c c

Signals and Systems15-4

TRANSPARENCY15.2Spectra associatedwith discrete-timeamplitude modulationwith a sinusoidalcarrier.

c[n] = cos 2cn

iei cn + 1 e cjn2 2

X(2)

i I

C (2)

I I I-T -2c 2c 2

Y(Q)

A1A_r-2 iT 2r

TRANSPARENCY15.3Spectra associatedwith demodulationof an amplitude-modulated signal witha sinusoidal carrier.

cos 2n

y[nI w[n]

Y(M)

l 2

W(W)

F1

24

2&2r nT S22C

Discrete-Time Modulation15-5

(-1 )

x~n] y[n]x

Lowpass filter

H (2)

o E2 oE

(-1)

y1p [n] XhpE]x

TRANSPARENCY15.4System and spectra forsinusoidal amplitudedemodulation.

TRANSPARENCY15.5The use of amplitudemodulation toimplement highpassfiltering with alowpass filter.

Signals and Systems

15-6

TRANSPARENCY15.6Spectra associatedwith the use ofmodulation anddemodulation toimplement highpassfiltering using alowpass filter.

X(92)

-2 -f f 2n 7

C(£2)

21r

-31r -I 31T 2

Y(Q)

-Ir - 2

xI (E2)

NA it NA

p(t)

TRANSPARENCY15.7Continuous-timeamplitude modulationwith a pulse carrier.

x (t) y(t)

x (t)

y(t)

iI I j I I I

Q . 7r

Discrete-Time Modulation

15-7

y[n]

p[n]

p[n]

0 1* * - M N (N + M)

TRANSPARENCY15.8Discrete-timeamplitude modulationwith a pulse carrier.

TRANSPARENCY15.9Transparencies 15.9and 15.10 illustratespectra associatedwith the system ofTransparency 15.7.Shown here are theinput spectrum andspectrum of a pulsecarrier.

x[n]

fiT

X(W)

- WM WM

27rA/T

-W w

Samsee

Signals and Systems

15-8

TRANSPARENCY15.10Input spectrum andoutput spectrum.

TRANSPARENCY15.11Time divisionmultiplexing usingamplitude modulationwith a pulse carrier.

X(W)

-wM WM

Y(w)

- ~-LWM WM L p

t

X1 (t) 1 (t)X

t

x 2 (t) Y2 (t)

t

y (t)

Y3 (t)x3 (t)

Discrete-Time Modulation15-9

p(t)

x(t) y(t)

x (t)

y(t)

ri0 t

TRANSPARENCY15.12Time waveformsassociated withamplitude modulationwith a pulse carrier.

X(W)

~WM WM c

Y(W)

A/T

-LO ~WM M p

TRANSPARENCY15.13Spectra associatedwith amplitudemodulation with apulse carrier.

Signals and Systems

15-10

TRANSPARENCY15.14Amplitude modulationwith a pulse carrierwith the pulses chosento have unit area.

TRANSPARENCY15.15Amplitude modulationwith a pulse carrier inthe limit as the pulsewidth approaches zeroand the pulse arearemains unity. Thiscorresponds toamplitude modulationwith an impulse traincarrier.

p (t)

x(t)

x(nT)

I I I I I ITt

r

Discrete-Time Modulation15-11

X(w)

P(w)

2rT

2-, - oS W, 2o, o

T

wM(WM , jM

TRANSPARENCY15.16Transparencies 15.16and 15.17 illustratespectra associatedwith impulse trainmodulation. Here, thefrequency of themodulating impulsetrain is chosen largeenough so that theindividual replicationsof the input spectrumdo not overlap.

TRANSPARENCY15.17The carrierfundamentalfrequency is chosensuch that theindividual replicationsof the input spectrumoverlap.

M C M (A

Signals and Systems

15-12

p(t)=

TRANSPARENCY x(t)

15.18Modulation anddemodulation with animpulse train carrier.

MARKERBOARD15.2

Samkin Theoreix%

etually space4 Samples

oS x(t) x(Kr) n%=o,ti,tz,...

X( =0&D lI > LO~A

Then. X(t) Aniquelk recoerable

with a loupass filter

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