15
Discrete-Time
Modulation
The modulation property is basically the same for continuous-time and discrete-time signals. The principal difference is that since for discrete-time signals the Fourier transform is a periodic function of frequency, the convolution
of the spectra resulting from multiplication of the sequences is a periodic convolution rather than a linear convolution.
While the mathematics is very similar, the applications are somewhat different. In continuous time, modulation plays a major role in communications
systems for transmission of signals over various types of channels. That application usually is inherently a continuous-time application. However, in many
modern communication systems, signals may go through various stages and
types of modulation as they move from one channel to another, and often this
conversion from one modulation system to another is best implemented digitally. In their digital form the signals are discrete-time signals, and such transmodulation systems are based on modulation properties associated with discrete-time signals.
In addition to digital modulation systems, the concepts of discrete-time
modulation (and, for that matter, continuous-time modulation also) are useful
in the context of filtering, particularly when it is of interest to implement filters with a variable center frequency. It is often simpler in such situations to
implement a fixed filter (either continuous time or discrete time) and through
modulation shift the signal spectrum in relation to the fixed filter center frequency rather than shifting the filter center frequency in relation to the signal.
For discrete-time signals, for example, from the modulation property it follows that multiplying a signal by (- 1)' has the effect of interchanging the
high and low frequencies. Consequently, by alternating the algebraic sign of
the input signal, processing with a lowpass filter, and then alternating the algebraic sign of the output signal, a highpass filter can be implemented.
In discussing continuous-time modulation in Lecture 13 and discretetime modulation in the first part of this lecture, the emphasis is on a carrier
signal that is a complex exponential or sinusoidal signal. Another important
and useful class of carrier signals is periodic pulse trains that are constant for
some fraction of the period and zero for the remainder. In effect, then, either
15-1
Signals and Systems
15-2
in continuous time or discrete time, modulation with such a pulse train consists of extracting time slices of the modulating signal. In data representation
or 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 amazingly, the original modulating signal can be recovered exactly after pulse-amplitude modulation provided only that the fundamental frequency of the carrier pulse train is greater than twice the highest frequency in the modulating
signal. The modulating signal can then theoretically be recovered exactly by
filtering the pulse-amplitude-modulated signal with an ideal lowpass filter.
Furthermore, this ability to exactly reconstruct the original signal is independent 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 continuoustime case, the "on" time of the carrier pulse train is made arbitrarily small,
with the amplitude increasing proportionately, the carrier then corresponds
to 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 samples. In both cases, then, modulation with the impulse train carrier would correspond to sampling the modulating (input) signal. This leads to an extremely
important concept, referred to as the sampling theorem. The sampling theorem states that a bandlimited signal can be exactly reconstructed from equally spaced time samples provided that the fundamental frequency of the sampler (i.e., the impulse train carrier) is greater than twice the highest frequency
in the signal to be reconstructed. This fundamental and important result, to be
explored further in Lecture 16, provides a major bridge between continuoustime and discrete-time signals and systems.
Suggested Reading
Section 7.5, Discrete-Time Amplitude Modulation, pages 473-479
Section 7.4, Pulse Amplitude Modulation and Time-Division Multiplexing,
pages 469-473
Discrete-Time Modulation
15-3
MARKERBOARD
15.1
Ctn]
e
Cormplex exponentiQL
Carrier
. Sin\usoidal
XE"3
E I (e)c(n-e)d 0
x
Cav-rier
CoMP ex exponentla L
. ScroNous rrodulat.ion-.
C, 3(n -r'en+c)
Asinchronous
,modulakion_
Carrier
S'nusod'
C[r= Cos(en.+ec)
. S I le Side baNd
Pulse Carrier
x[n] c[n]
) dO
X()C(2
2i7
X(2)
TRANSPARENCY
15.1
III\I
I
-27r
-M
0
2r
M
E2
C(92)
2w
27r
-2w+
21r
2w
fl
0
27r
_
c
-27r + 2(c2c
92
c
Y()
27r
2w +
- EZ )
Se
(Oc + EM)
27r
Spectra associated
with discrete-time
amplitude modulation
with a complex
exponential carrier.
Signals and Systems
15-4
c[n]
cos 2cn
=
iei cn + 1 e
2
TRANSPARENCY
2
15.2
X(2)
Spectra associated
with discrete-time
amplitude modulation
with a sinusoidal
carrier.
iI
cjn
C(2)
I
I
I
2c
-2c
-T
2
Y(Q)
A1A
_r-2
iT
2r
cos 2n
y[nI
w[n]
TRANSPARENCY
15.3
Spectra associated
with demodulation
of an amplitudemodulated signal with
a sinusoidal carrier.
Y(M)
l2
W(W)
F1
4
2
22C
2&2r
S
nT
Discrete-Time Modulation
15-5
TRANSPARENCY
15.4
System and spectra for
sinusoidal amplitude
demodulation.
(-1
)
TRANSPARENCY
15.5
The use of amplitude
modulation to
implement highpass
filtering with a
lowpass filter.
(-1)
Lowpass filter
H(2)
x~n]
x
y[n]
y1p [n]
oE2
oE
x
XhpE]
Signals and Systems
15-6
X(92)
TRANSPARENCY
-2
15.6
Spectra associated
with the use of
modulation and
demodulation to
implement highpass
filtering using a
lowpass filter.
2n7
-f f
C(£2)
21r
31T
-I
-31r
2
Y(Q)
-Ir
-
Q .
2
7r
xI (E2)
it
NA
NA
p(t)
TRANSPARENCY
15.7
Continuous-time
amplitude modulation
with a pulse carrier.
y(t)
x (t)
x (t)
y(t)
iI
I
j
I
I
I
Discrete-Time Modulation
15-7
TRANSPARENCY
15.8
x[n]
Discrete-time
amplitude modulation
with a pulse carrier.
y[n]
p[n]
p[n]
fiT
see
Sam
N
0 1* * - M
(N + M)
X(W)
-
WM
WM
27rA/T
w
-W
TRANSPARENCY
15.9
Transparencies 15.9
and 15.10 illustrate
spectra associated
with the system of
Transparency 15.7.
Shown here are the
input spectrum and
spectrum of a pulse
carrier.
Signals and Systems
15-8
X(W)
TRANSPARENCY
15.10
Input spectrum and
output spectrum.
WM
-wM
Y(w)
-
~-LWM
WM
L p
t
TRANSPARENCY
15.11
Time division
multiplexing using
amplitude modulation
with a pulse carrier.
X1 (t)
X
1 (t)
t
x 2 (t)
Y2 (t)
y (t)
t
x3 (t)
Y3 (t)
Discrete-Time Modulation
15-9
p(t)
x(t)
TRANSPARENCY
15.12
Time waveforms
associated with
amplitude modulation
with a pulse carrier.
y(t)
x (t)
y(t)
ri
0
t
X(W)
TRANSPARENCY
15.13
Spectra associated
with amplitude
modulation with a
pulse carrier.
~WM
WM
c
Y(W)
A/T
-LO
~WM
M
p
Signals and Systems
15-10
TRANSPARENCY
15.14
Amplitude modulation
with a pulse carrier
with the pulses chosen
to have unit area.
r
p (t)
x(t)
TRANSPARENCY
15.15
Amplitude modulation
with a pulse carrier in
the limit as the pulse
width approaches zero
and the pulse area
remains unity. This
corresponds to
amplitude modulation
with an impulse train
carrier.
x(nT)
I
I
I
I
I
IT
t
Discrete-Time Modulation
15-11
X(w)
C
M
M
(A
P(w)
2r
T
2-,
- oS
W,
2o,
o
TRANSPARENCY
15.16
Transparencies 15.16
and 15.17 illustrate
spectra associated
with impulse train
modulation. Here, the
frequency of the
modulating impulse
train is chosen large
enough so that the
individual replications
of the input spectrum
do not overlap.
T
wM(WM
,
jM
TRANSPARENCY
15.17
The carrier
fundamental
frequency is chosen
such that the
individual replications
of the input spectrum
overlap.
Signals and Systems
15-12
p(t)=
TRANSPARENCY
15.18
Modulation and
demodulation with an
impulse train carrier.
x(t)
MARKERBOARD
15.2
Samkin Theoreix%
etually space4 Samples
oS x(t)
x(Kr) n%=o,ti,tz,...
X(=0&DlI >LO~A
Then. X(t) Aniquelk recoerable
with a loupass filter
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Signals and Systems
Professor Alan V. Oppenheim
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.