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10.1 Signal Transmission in Communication Systems
The main role of a communication system is to transmit signals (information) from
the source of information(system input) to the user, destination (system output).
The transmission is doneover acommunication channelusing atransmitterand a
receiver. A simplified basic communication system is presented in Figure 10.1.
Source, sender
Input
Information
Transmitter
Transmitted signal
Channel
Noise
Original
baseband
signal
Receiver
Received signal
Output
Information
User, destination
Reconstructed
(estimated)
signal
Figure 10.1: Basic communication system
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The original signal, usuallycalled, thebasebandsignal (this namewill be
justifiedafter weexplain the modulationconcept) is first transformed into the signal
convenientfor transmission(called thetransmittedsignal) using thetransmitter. The
transmittersends sucha signal asan electrical oroptical (electromagnetic)signal
over a communicationchannel, which representsa physical medium convenient
for propagationof electromagnetic waves (low signalattenuation and distortion).
Communication channels canbe guided media (such as copperwire or optical
fiber cable channels)or free-space channels (such as satellite or wireless (radio)
channels). The role ofthe receiver is to convertreceived signals, theoretically, into
baseband signals and passthem to the user. Dueto channel attenuation, distortion,
and noise, the receiver produces a signal that is only similar but not identical to
the baseband signal. Such a signal is called estimated or reconstructed signal. The
estimated signalcan be slightly different thanthe originally sent signal (baseband
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signal) especiallyfor voice and video transmissions,since the human eye and ear
areunable todetect small errors.However, in the case when we transmit data, the
signal transmission mustbe error free.
Modulation
In a standard communication system, the transmitter is a modulator, and the
receiver is a demodulator.The modulator and demodulator togetherare called
modem. We havealready introduced the modulation concept withinthe properties
of the Fourier transform.The modulation property of the Fouriertransform says
the following: Let the signal have the Fouriertransform equal to ,
where . Then, theFourier transform of themodulated signal, definedas
c c c, is given by
c c c
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Since denotes thesignal spectrum,it can beseen that thespectrum of
the modulated signalis shifted leftand right by c, as represented in Figure10.2.
The frequency c is called thecarrier frequency. The originalsignal spectrum
is the baseband signalspectrum, and theother twospectra in Figure 10.2 are the
modulatedsignal spectra. Thisjustifies the namethe baseband signal. Due to
the magnitude spectrumsymmetry, the positive frequencies carry allinformation
contained in the given signal.We can make twoobservations from Figure 10.2.
(1) The spectrum of the modulatedsignal is doubled comparing tothe spectrum
of the baseband signal. It contains theupper frequency sidebandand thelower
frequency sideband, each having the bandwidth equal to the bandwidth of the
baseband signal.
(2) Due to frequency translation, the negative frequencies come into the picture,
and they form the lower frequency sideband.
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Hence, theamplitude modulationprocedure presented requiresdoubling in the
spectrumrequirements (wasteof the frequencyband). In Section 10.5, we will
study a techniquethat remedies thisproblem.
ω
ω−ω0
ω+ω0
1
2
|X(j( ))|+ |X(j( ))|
1
2
|X(j )|ω
0
ω−ω
max
ωmax
ωmax
ω +0
ωmax
ω −0 ω0
ωmax
−ω +0
ωmax
−ω −0
−ω0
0
Figure 10.2: The spectrum of the original and modulated signals
The modulation concept indicates one extraordinary possibility that the same
channel can be used to simultaneously transmit several signals by appropriately
shifting their spectra such thatthey do not overlap in the frequency domain. Note
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that thesignals mayoverlap in thetime domain. In Figure 3.8, we have considered
a telephone networkthat transmits manytelephone signals (calls) simultaneously.
Spectrum of the original signal
f
[kHz]
0 4
0 4
f
[kHz]
8 12 NN-
1
Spectrum of a sequence of modulated signals
Figure 3.8: Transmission of telephone modulated signals over the same channel
It can beobserved from Figure 3.8 that the users share the frequency band. If we
assume that the channel frequency bandwidth is equal toBW and that the channel
must serve users (baseband signals), then we see that each user has reserved all
the times a part of the channel frequency band equal toBW . Such a channel
sharing is calledfrequency division multiplexing(FDM).
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Another channelsharing techniqueused in communicationsystem practice is the
time divisionmultiplexing (TDM), a technique inwhich each user gets the whole
frequencyband ofthe channel, butonly during alimited period of time. In such
a case theusers are switchedon and off according tothe given time schedule.
For example, eachuser uses thewhole channel frequency band during the time
period of , andthey rotate so that each getsa turn after time units(fair
sharing of thechannel). Note that there isno single criteria by whichto judge that
one of thechannel sharing techniques is better, due to the very simple fact that a
channel with alarger frequency bandwidth has ahigher capacity(it can transmit
more units ofinformation per unit of time,it is a faster channel).Hence, there is an
interplay between transmitting at high speeds during short periods of time (TDM)
and transmitting at low speeds all times (FDM).
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Demodulation
The demodulation process is reciprocal to the modulation process. Demodulation
is an operation that reconstructs the original baseband signal from its modulated
signal. Technically speaking, the demodulator has to cut out (filter out) the
frequency bandthat corresponds tothe given baseband signal.
Demodulation can be performedby modulating again the modulatedsignal
c c c
c c
By passing thissignal through a low-pass filterwe can recover the original signal
multiplied by , that is . In Section10.5 we will say moreabout both
the modulation and demodulation procedures. In the remaining part of this section
we will introducedsome notions frequently used in signal transmission.
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Signal to Noise Ratio
As mentioned earlier, channel noise is most often random in nature. Despite the
fact that we will not study channels from the stochastic point of view, we can define
a simple quantity that tells us how much, in average, a given channel is noisy. Let
s denote theaverage signalpower and let n denote theaverage noisepower.
The signal to noise ratio indecibels[dB] is defined by
10s
n
Apparently, the higherSNR the better channel.
Channel Capacity
It can beexperimentally observed that the channelcapacity is directly propor-
tional to its frequency bandwidth. It has also been observed that the higher SNR
implies the higher channel capacity.
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An exact formula thatrelates the channelcapacity in bits per second, channel
frequencybandwidth inHz, and thechannel signal power to noise power ratio was
derivedby Shannon(also known asShannon-Harteley’s formula). Itis given by
BW 2s
n
The formulais valid for channels with Guassiannoise (noise statistics is completely
describedby the first and second ordermoments). In the case ofnon Gaussian noise,
the aboveformula gives only an approximatelower bound.
Optical Fiber Cable
As the waveguide medium of the future, the optical fiber cable has a huge
frequency bandwidth that theoretically can reach several hundreds of THz
(1 terahertzis equal to 12 Hz). It hasalso very low signal attenuation of only
, which means that
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10
inp
out
where inp and out represent, respectively, the input and output signal powers.
In addition, the optical fiber cable has verylow signal distortion. Notethat such
a cable is made of silica glass (dielectric)and that it transports lightsignals,
also calledoptical signals. Similarly to the frequencydivision multiplexing, in
optical communication systems,wavelength division multiplexing(WDM) is used to
transmit simultaneously many signals (eighty or even more) over the same optical
fiber channel. The optical wavelength is defined by , where is the
light speed(equal to 8 in vacuum and r r in a
guided media, r and r are respectively themedium permittivity and permeability
constants) and the frequency of the correspondinglight signal. Note that
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DWDM standsfor dense wavelengthfrequency divisionmultiplexingthat hasoptical
wavelengthchannels denselyspaced every �9 (1 nanometar).
It is interesting to point out that a channel represents a dynamic system that can be
either linear or nonlinear, time invariant or time varying, deterministic or stochastic
(see systemclassification in Section1.4). For example, telephone channels are linear
systems in most cases, wirelesschannels can be considered astime varying linear
systems, fiber optics channels arenonlinear time invariant systems thatare often
linearized (see section on linearizationof nonlinear systems, Section 8.6),satellite
channels are nonlinear. Another classification of channels distinguishes between
bandlimited channelssuch as telephone networks andpower limited channelssuch
as optical fiber and satellitechannels.
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10.2 Signal Correlation, Energy and Power Spectra
In addition to the system frequency bandwidth, the signal power represents another
importantquantity that engineersare particularly concerned with while transmitting
signals. We havealready defined signal energy and power in the time domain in
Section2.3. Here, we present their representations in the frequency domain and
relate them tothe quantity known as the signalcorrelation function.
Devices called signalcorrelators are used to measurepower of incoming signals
in many communication(and signal processing) systems. For example, in wireless
communication systems, correlatorsat the base station measure at all times the signal
power of allmobiles in the base stationarea (cell). Those powers are periodically
adjusted such that eachmobile has sufficient signal powerfor a good quality
transmission, but not so much signal power as to cause unnecessary interference to
the other mobiles that use thesame frequency band.
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Continuous-Time Signal Correlation
The analytical expression for signal correlation is very similar to the convolution
integral, even though signal correlation and signal convolution have completely
different physical meanings.
Correlation of two continuous-time signals1 and 2 is defined by
12
1
�11 2
where is a parameter, . More precisely, 12 is called the
cross-correlation function. Assumingthat the signals 1 and 2 have Fourier
transformsrespectively given by 1 and 2 , that is
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1
1
�11
j!t2
1
�12
j!t
then, wehave
12
1
�11
1
�12
j!(t+�)
1
�1
1
�11
j!t2
j!�
1
�1
�1 2
j!�
Note that �1 1 . The lastformula indicates that 12 and
�1 2 form the Fourier transform pair, thatis
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12�1 2
In the case when 1 2 , we have the definition of the
autocorrelationfunction as1
�1
In this case,we have1
�1
� j!�
1
�1
2 j!�
that is, theautocorrelation function and 2 form the Fouriertransform pair
2
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It can be shownthat the autocorrelationfunction has the following properties:
1) The autocorelation function is even, that is .
2) 1, where 1 stands forthe total signal energy.
3) .
4) is continuous intime (like convolution).
The quantity 2 defines the signalpower at the given frequency so
that 2 is calledthe power spectrum. 2 is alsocalled theenergy
density spectrumfor the reason to be clearsoon. Introducing notation
2
we have
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1
�1
�j!t1
�1
j!�
Note that is a real positiveand even function,that is .
It follows from 1 that
1
!=1
!=�1
It is clear that represents theenergy density inthe frequency domain, which
justifies the density energy spectrum name used for 2 .
If one intends to find the signalenergy in the frequency domainin any frequency
range, say 1 2 , then the knowledge of the signal density energy gives
the following formula
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1 2
!2
!1
�!1
�!2
!2
!1
The last expressionfollows from the fact that is a positiveand symmetric
function of frequency. This formula determines thedistribution of the signal energy
in the frequency domain. Here, we seehow the “negative” frequencies come into
the picture and how the signal energy can be completely expressed in terms of
positive frequencies, which reflects physical reality.
Example 10.1: In this example we will find the frequency range that contains
the given percentage (50%) of the signal energy. Consider the signal frequency
spectrumpresented in Figure 10.3.
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|X(j )|ω
ω-1 0 1
Figure 10.3: The frequency spectrum of a signal
We are looking for the frequency1 such that
1
1
�1
1
�1
2
1
0
2
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Hence, wehave theequality
1
!1
0
2
Using thefollowing change ofvariables , the above integral canbe
easily calculated,which leads to 13 . Note thatin this problem we
have tacitlytaken into account the contributionof “negative” frequencies to the
signal energy.
As a measure of similarityof two signals, the so-calledcorrelation coefficient
can be defined as
1212
11 22
When the correlation coefficient isclose to one then the signals are similar.
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Correlation of Periodic Signals
In the case when signals are periodic, the correlation functions can be obtained
using the Fourier series. Let1 1 and 2 2 ,
, then thecross-correlation function for periodic signalsis defined by
12
T2
�T2
1 2
Using the fact that periodic functions can be expressed using the Fourier series
1
n=1
n=�11 0
jn!0t
2
n=1
n=�12 0
jn!0t0
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we have
12
T2
�T2
1
n=1
n=�12 0
jn!0(t+�)
n=1
n=�12 0
T2
�T2
1jn!0t jn!0�
n=1
n=�1�1 0 2 0
jn!0�
Note that 12 and �1 0 2 0 are the correspondingFourier series
pair, which implies that
�1 0 2 0
T2
�T2
12�jn!0�
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In the case when 1 2 , we can definethe autocorrelation
function for periodic signals as
T2
�T2
which leads to
n=1
n=�10
2 jn!0�
Introducing the notionof the power spectrum 0 02 of a
periodic signal, wehave the corresponding Fourier seriespair
n=1
n=�10
jn!0�0
T2
�T2
�jn!0�
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Note that for periodic signals the autocorrelationfunction is also an even
function. The correspondingspectrum is aneven and positive function. In addition,
definesthe signal energyduring one timeperiod, thatis
n=1
n=�10
n=1
n=�10
2
T2
�T2
2T
This relation alsorepresentsParseval’s theorem for periodicsignals.
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10.3 Hilbert Transform
The Hilbert transform plays an important role in communication systems. It can be
easilyderived using knowledgefrom Chapter 3 about the Fourier transform. There
aretwo forms ofthe Hilbert transform. The first form is valid for causal signals and
the second form holds for real signals. The first form of the Hilbert transform has
applications in linearelectrical circuits and electric power systems,and the second
form of theHilbert transform is used incommunications systems.
Hilbert Transform for Causal Signals
In the following we will show that in the case of causal signals, the Hilbert
transform, in fact, relates the real and imaginary parts of the corresponding Fourier
transform. Such a relationship holds for anycausalreal or complex signal (function)
. Recall thatcausal signals are equal to zero for .
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Due to causality, wehave
Re Im
1
0
�j!t
Causality impliesalso , where is theunit stepfunction. The
applicationof the Fourier transform produces
Using the expressionfor the Fourier transform of the unit step function, we obtain
Re Im
Re Im
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It is known that the convolution ofany signal with the delta impulse signal
producesthat signal.Using this fact,the above equation is simplified into
Re Im Re Im
Re Im
Equating the realand imaginary parts inthe last equation, we have
Re Im
and
Im Re
These formulas relate the real and imaginary parts of the Fourier transform of the
causal signal and define the Hilbert transform.
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Using thedefinition of the frequency domainconvolution, the last two formulas
can be written in the following form
Re
1
�1Im
and
Im
1
�1Re
Example 10.2: The unit step signal h is a causal signal whose Fourier
transform hasboth the real and imaginary parts, that is
h
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The imaginarypart of the given Fouriertransform is related to its real part
through the Hilbert transform, that is
1
�1
This form ofthe Hilbert transform has applicationsin linear electrical circuits and
electric powersystems in order to findthe imaginary part of the Fouriertransform
when its real part is known (obtained experimentally) and vice versa to find the
real part from the imaginary part of the Fourier transform. For completeness, it is
presented inthis section, together with the second form of the Hilbert transform,
which has a particular importance for themodulation process in communication
systems.
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Hilbert Transform for Real Signals
The second form of the Hilbert transform is derived for real Fourier transformable
signals. Let , that is
1
�1
j!t
Consider the signal + whose spectrum iszero for negative frequencies and
equal to for positive frequencies,that is
+
1
0
j!t
The following relationship exists between the signals and +
+
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where is the Hilbert transformof defined by
1
�1
This result can beshown asfollows. From theexpression for and , we
have
+ h
where h represents the unitstep function in the frequency domain. We know
that h . Using theduality property, we have
h
h
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Since theproduct inthe frequency domaincorresponds to the convolution in the
time domain, wehave
+
Also
Since then by theduality property, we have
so that
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Finding analyticallythis form of the Hilbert transform requires that the signal
Fouriertransform ismultiplied by and thatthe inverse Fouriertransform
is applied to theresult obtained.This procedure isdemonstrated in the next example.
Example 10.3: The Hilbert transform of the sine function is obtained as0 0 0
0 0 0
Hence, theHilbert transform of the signal 0 is equalto 0 .
Using the definition of the sign function, we have�j �
2
j �2
It can be seen thatfor real signals the Hilbert transform introduces the phase shift
of � for positive frequenciesand the phase shift of � for
negative frequencies.
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The signal + is calledthe positive frequencypre-envelope signalof .
Its main featureis given by its spectrum formula
+
The middlerelation follows from the propertyof the frequency domain unit step
function, thatis + h . Similarly, wecan define the
negative frequencypre-envelope signalof by � . Its
spectrum is
�
Applications of this form of the Hilbert transform in communication systems
will be discussedin Section 10.5 within the single sideband modulation technique.
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10.4 Ideal Filter
Signal filtering plays a very important role in communication systems. Filters
can extract from agiven frequency spectrum either low frequency components
(low-pass filtering)or high frequencycomponents (high-pass filtering) or signal
componentsthat belong to a certain frequency range (band-pass filtering). A filter
can also eliminatecertain components from the signal frequencyspectrum (band-
stop filtering).
It is importantto know that an ideal filter that exactly passes the given range of
frequency components andexactly suppresses the frequency components outside of
that range isnot physically realizable. However, theideal filter has theoretical im-
portance in understanding theinterplay between the time andfrequencies domains.
Moreover, with sight modifications we can construct realizable filters starting with
the frequency characteristics of ideal filters.
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The frequencycharacteristics ofsuch an ideallow-pass filter is presented in
Figure10.4. Thefrequency 0 is calledthe filter cut-off frequency. According to
Figure 10.4 the ideallow-pass filtertransfer function isgiven by
�j!td0
Note that wehave assumed that the phaseof the ideal filter changeslinearly in
frequency, whichcorresponds to the time shiftof the filter input signalsby d (time
shifting property of the Fourier transform).
ω
1
0
|H(j )|ω
ω0
−ω0
ω0
(b)
(a)
ω0
−ω t
d
0
arg{ }
H(j )ω
Figure 10.4: The frequency spectra of an ideal
low−pass filter: (a) magnitude and (b) phase
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In the following, we derive the impulseresponse of the ideal low-pass filter and
showthat suchan impulse responsedoes not correspond to the impulse response of
a causal (realphysical) system. We know thata rectangular frequency domain pulse
hasthe following time domain Fourierequivalent (note thatin this case 0),
seeExample 3.16
2!0
0 0
Using the timeshift property of the Fourier transform,we have
2!0�j!0td 0 0
d
We haveobtained the shifted sinc signalwhose maximum, at d, is equalto
0 d. The waveform is presentboth left and right from the point d, having
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infinite durationin bothdirections, see Figure10.5, where we use MATLAB to plot
the corresponding impulseresponse for 0 and d .
−2 0 2 4 6 8 10−0.5
0
0.5
1
1.5
2
time in seconds
Idea
l filt
er im
puls
e re
spon
se
Figure 10.5: MATLAB plot of the impulse response of an ideal low−passfilter
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It can be concludedthat the ideallow-pass filter impulse response produces a
waveformdifferent fromzero even forthe times ( ) beforethe delta impulse
input is applied tothe filter. That violates thecausality of the filter, that is, its
physical realizability.
Problem 10.15: Direct derivations of theideal filter impulse response
�11
�12!0
�j!td j!t
!0
�!0
j!(t�td)!0
�!0
d
d
d
!=!0!=�!0
0 d
d
00 d
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10.5 Modulation and Demodulation
Signal modulation has been the cornerstone for the development of modern com-
munication theory and its applications. More precisely, the formula
c c c c c c
defines amplitude modulation. Thesignal c c is the carriersignal
with carrier frequency c and carrier amplitude c. The modulatedsignal is
c c . The basebandsignal is also calledthe message signal,
modulating signal,or original signal. The spectraof the original (baseband) and
modulated signalsare presented in Figure 10.2.
There areseveral types of modulation techniques.In addition to amplitude
modulation, wehave frequency modulationand phase modulationtechniques, in
which, respectively, the carriersignal frequency and the carrier signal phase are
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affected bythe basebandsignal. Hence, inthose cases the carrier frequency and
the carrier phasecarry information aboutthe original signal . Frequencyand
phaseamplitude modulation areoutside thescope of thisintroductory chapter on
communicationsystems.
In additionto the sinusoidalcarrier, thetrain of pulsesis used asthe carrier signal.
In the case of thetrain of pulses, we haveagainamplitude modulation(the pulse
magnitude is proportional to theoriginal signal magnitude at thegiven time instant),
pulse duration modulation(the pulsewidth is proportional to themagnitude of the
original signal), andpulse position modulation(the pulse position with respect to the
reference position is determined by the magnitude of the original signal). The above
pulse modulation techniques are specificfor continuous-time (analog) signals. Due
to space limitation and introductorynature of this chapter, continuous-timepulse
modulation techniques will not be discussed.
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For digital signals, wehave thepulse codemodulation technique,in which
digital signals arebinary encoded andthe bits carrying information about signal
magnitudeare transmitted.We will say more aboutthis modulation technique in
the next section,where we presentthe essence ofdigital communicationsystems.
Amplitude Modulation
It can be seenfrom the previous analysis thatthe carrier signal amplitude is
equal to c. Being multipliedby , the carriersignal changes its magnitude
according to c, which is the way the carrier signalcarries information about
the signal . There areseveral variants of the amplitudemodulation technique.
Let us demonstrate on a simple example that the envelope of the modulated
signal cancarry information about the originalsignal. At the same time we will
establish conditionsrequired for such a signaltransmission.
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Example 10.4: Consider asimple signal �th . Its Fourier
transform is given by 2 . The modulated
signal c c c and the original signal arepresented in
Figures 10.6 and10.7, respectively for c and c .
0 1 2 3 4 5 6 7−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
time in seconds
mod
ulat
ed a
nd o
rigin
al s
igna
ls
Figure 10.6: Modulated (solid line) and original
(dashed line) signals for c
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It canbe seenfrom Figure 10.6that the modulated signal in its envelope basically
carries information aboutthe original signal. It is natural to expect that such
information is sufficient for recovery ofthe original signal. However, it follows
from Figure 10.7that in thiscase, the recoveryprocess ofthe original signal from
the modulated signalis very difficult if possible at all.
0 1 2 3 4 5 6 7−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
time in seconds
mod
ulat
ed a
nd o
rigin
al s
igna
ls
Figure 10.7: Modulated (solid line) and original (dashed line) signals for c
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In Figure10.8, wehave presented themagnitude spectrum of the original signal.
It can beseen from thisfigure that the signal has a significant frequency component
at c andalmost negligible frequencycomponent at c .
We candraw a conclusionthat for an easy andaccurate signal recovery from the
modulated signal thecarrier frequency must bemuch higher thanthe frequency of
any significant spectral component of the signal.
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
angular frequency in rad/s
mag
nitu
de s
pect
rum
of x
(t)
Figure 10.8: The magnitude spectrum of the original signal
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Amplitude Modulation with a Transmitted Carrier
Note that in Example 10.4 the signal �th is positive for all .
If changes its sign for some, then the modulated signal c c
will change the phase at that time, such that its envelope will be distorted and it
will no longer preserve the shape of the original signal. To prevent this problem,
we can define the modulated signalusing a slightly different modulationformula
a c c c c c c
a c c a c c
where a is an arbitrary constant called either theamplitude sensitivityor index of
modulation. By choosingthis constant such that
a a
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the envelopeof the modulated signal willhave the shape of the original signal
and hence carryinformation about theoriginal signal at all times. This property
will facilitate the use of simplemodulators for signalmodulation and simple
demodulators(envelope detectors)for signal reconstruction.
The frequencydomain price forsuch a time domain convenience is the presence
of two additional deltaimpulses in the frequency spectrumof the modulated signal.
Since c may have alarge value, such as inthe case of the modulator knownas
the switching modulator,a considerable amount of poweris wasted in this kind of
modulation known asdouble sideband with transmitted carrier modulation(DSB-
TC). Theoriginally considered modulation technique ( c c ) does not
require anindependent carrier transmission. It is known asdouble sideband with
suppressed carriermodulation(DSB-SC).
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In both DSB-SC andDSB-TC, the lowerand upper signal frequency sidebands
are transmitted. Sincethe signal informationis completely contained in either
the upper or lower frequency sideband,we conclude thatthese two modulation
techniqueswaste asignificant amount ofthe channel’s frequencyband. Exactlyhalf
of the frequencyband can besaved by transmitting only the lower or upper signal
frequency sideband.This can be facilitated bythe modulation technique known as
single sideband (SSB) modulation.Theoretical foundations for SSB modulation lie
in the Hilbert transform considered in Section 10.3.
Switching Modulator and Envelope Detector (Demodulator)
Amplitude modulation with the transmitted carrier and corresponding demodu-
lation are easily performed by usingpretty simple electrical devices known as the
switchingmodulatorand envelope detector (demodulator).They are presented in
Figures 10.9 and 10.10.
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R
l
v (t)
out
x(t)
~
A
c
cos( )
ω t
c
v (t)
i
Figure 10.9: Switching modulator
R
s
C
R
l
v (t) x(t)
out
x (t)
mod
Figure 10.10: Envelope detector
Amplitude modulation for DSB-SC signals requires the use of more complex
modulators. The most common of which is called the ring modulator. As the
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corresponding demodulator,the Costasreceiver is mostlyrecommended. It is
beyondthe scopeof this textbookto go into detail about these devices.
Note that MATLAB has the modulation functionmodulate, which can be
used for any of the above three modulation techniques. Its general form is
xmod=modulate(x,fc,fs,’method’,parameter), where x represents
samples of the originalcontinuous-time signal sampled with thefrequencyfs. fc is
the carrier frequency (c c ). method is eitheramdsb-sc or amdsb-tc
or amssb, denoting respectively the modulation methodused DSB-SC or DSB-TC
or SSB. The choice of theparameter should be such that the modulating signal
is positive with the minimum equal to zero. Theparameter is set to zero for
DSB-SC andSSB. It can also be omitted since its default value is zero. Similarly,
the MATLAB function demod performs demodulation,which can be achieved by
using the following MATLAB statementx=demod(xmod,fc,fs,’method’).
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Problem 10.22
In this problem weuse MATLAB to find the Fourier transform (spectrum) of the
signalpresented inFigure 2.8, findits DSB-SC andDSB-TC amplitude modulated
signalsand plottheir spectra. Notethat the chosencarrier frequencyis
c s , which implies that c c .
Ts=0.01; tf=3; t=0:Ts:tf; tt=0:Ts:1;
xs=-tt+1; x=[zeros(1,1/Ts) xs zeros(1,1/Ts)];
figure (1); subplot(221); plot(t-1,x);
fs=1/Ts; fc=0.1*fs;
xmodSC=modulate(x,fc,fs,’amdsb-sc’);
xmodTC=modulate(x,fc,fs,’amdsb-tc’,0.1);
subplot(222); plot(t-1,xmodSC);
subplot(224); plot(t-1,xmodTC);
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N=length(x)-1; X=Ts*fft(x,N);
XmodSC=Ts*fft(xmodSC,N);
XmodTC=Ts*fft(xmodTC,N);
k=0:1:N/2-1; w=(2*pi*k/N)/Ts
subplot(223); plot(w,abs(X(1:N/2)));
figure (2)
subplot(211); plot(w,abs(XmodSC(1:N/2)));
subplot(212); plot(w,abs(XmodTC(1:N/2)));
The results obtained are presented in FIGURES 10.7 and 10.8. Note that the
signal spectra (Fourier transforms)are evaluated using FFT and the formula
s
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−1 0 1 2−0.5
0
0.5
1
1.5
Time
Sig
nal
−1 0 1 2−1.5
−1
−0.5
0
0.5
1
1.5
Time
DS
B−
SC
sig
nal
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
Frequency [rad/s]
Sig
nal s
pect
rum
−1 0 1 2−1.5
−1
−0.5
0
0.5
1
1.5
Time
DS
B−
TC
sig
nal
FIGURE 10.7 (Solutions Manual)
0 50 100 150 200 250 300 3500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Frequency [rad/s]
Xm
odS
C s
igna
l spe
ctru
m
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
Frequency[rad/s]
Xm
odT
C s
igna
l spe
ctru
m
FIGURE 10.8 (Solutions Manual)
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Single Sideband Amplitude Modulation
Theoretical foundations for the development of single sideband amplitude modu-
lation lie in the Hilbert transform. The single sideband amplitude modulated signal
can be obtained by using the Hilbert transform as follows. Consider the cosine
modulated originalsignal, that is
cosmod 0 0 0
The Hilbert transformof , denoted by , modulated bythe sine signal is
sinmod 0 0 0
The signals and are related through the Hilbert transform so that
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which implies
sinmod 0
0 0 0 0
If we form now the new modulated signal as
modcosmod
sinmod
its frequency spectrumwill be given by
0 0 0 0
Having in mind the expression for the signum function, we see that the spectrum
of the above modulated signal has the following form
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moodcosmod
sinmod
0 0
0 0
This spectrum ispresented in Figure 10.11.
ωmax
−ωmax
ωω
max
ω +0
ω0
ωmax
−ω −0
−ω0
0
Frequency spectra
Figure 10.11: The frequency spectra of the original (dashed
line) and single sideband modulated signal (solid line)
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Similarly, it can beshown that thefrequency magnitude spectrum of the signal
cosmod
sinmod
contains onlythe lowerfrequency sidebands, thatis
cosmod
sinmod
0 0
0 0
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Demodulation of SSB Signals
The original signal can be extracted from a single sideband amplitude modulated
signal by modulating the modulated signal again using the signal of the same
frequency and phase, that is
c c c
c c
c c
c c
The original signal can be easily extracted by using a lower pass filter since
This demodulation technique is calledcoherent demodulation
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10.6 Digital Communication Systems
Nowadays signal transmission in communication systems is mostly done digitally.
The advantage of digital signal transmission techniques is their improved tolerance
to noise. Noise is unavoidably present in all communication channels. The rapid
developmentof digital computer networks, digitalsignal processing, fast electronic
and photonic switching devices during thelast ten years has facilitated powerful
signal transmission techniques that can makedigital communication systems more
efficient than corresponding analog communication systems.
In the introductory Section 1.1.1, we have introduced the concept of discretization
of continuous-time signals withthe given sampling period, which leads to the
formation of discrete-time signals. Thedevice that performs the signal discretization
(sampling) is called thesampler. In addition of being discretized,in digital
communication systems, signals are also quantized (discretized with respect tothe
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magnitude). Thedevice thatperforms such amagnitude quantization is called the
quantizer. Suchan obtaineddiscretized and quantizedsignal is called the digital
signal. Finally, the digital signal obtainedis encoded intoa stream of bits. This
processcomposed ofsampling, quantization, andencoding, is knownas thepulse
codemodulation(PCM) technique. It issymbolically presented in Figure 10.12.
Sampler
Quantizer
Encoder
x(t)
t
x (
t
)
t
d
T
s
dt
x (t)
t
d
q
x
t
transmited
Figure 10.12: Pulse code modulation technique
The transmitter in a digital communication system performs pulse code modu-
lation on an incoming signaland forms the encoded binary signal. The encoded
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binary signalis thensent over acommunication channel as a stream of bits.
In this section, we have presented only the essential idea of digital communica-
tions. Further study of digital communications is beyond the scope of this chapter.
Example 10.5 PCM for Speech Signals
Speech (telephone)signals are sampledevery , which generates
samples per second.Quantization of speech signals isperformed at 7
levels, with eachquantized sample being encoded usingbits (one bit for the
sign). This generates bits per second, commonly denoted
as (kilo bits per second). Hence, while talking on the telephone, each
user(speaker) generates every second. Owingto recent advances in digital
communication networksthat use optical fiber channelssuch a heavy bit stream
can beeasily handled.
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