10.1 signal transmission in communication systems

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

The slides contai the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall 2003. Prepared by Professor Zoran Gajic 10–1

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


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


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.


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


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).


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).


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.


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.


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


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


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.


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.


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


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


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


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


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


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.


|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


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.


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


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�


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�


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.


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 .


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


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.


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


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.


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 +

+


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


Since theproduct inthe frequency domaincorresponds to the convolution in the

time domain, wehave

+

Also

Since then by theduality property, we have

so that


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.


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.


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.


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


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


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


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


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


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.


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.


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


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


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


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


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).


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.


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


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’).


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);


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


−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)


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


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


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)


Similarly, it can beshown that thefrequency magnitude spectrum of the signal

cosmod

sinmod

contains onlythe lowerfrequency sidebands, thatis

cosmod

sinmod

0 0

0 0


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


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


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


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.


10.1 signal transmission in communication systems

Documents