Week2(AM-II)

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KNE334:

Communication Systems 1

Amplitude Modulation-II

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Summary of week one lectures

-Introduction to communication systems-Major components: Source, Transducer, Modulator/Demodulator,

Transmitter/Receiver, Channel

-Types of modulation

-Channel features

-Amplitude modulation

-Carrier, message wave, and mixer, modulation depth-Spectrum of modulated signal

-Lower sideband and upper sideband, carrier power and sideband power of

modulated signal. Modulation index.

Channel behavior

Power of sideband versus power of carrier band

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Outline of this week content

AM demodulation and envelope detector design

Phasor representation for signal distortion and delay

analysis

Further discussions on delay and distortion, analysisof delay by equivalent filtering technique, square law

detection, and synchronised detection.

AM receivers

(Demodulation steps)

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Demodulation of AM Signals

Envelope detection of an AM signal

This is conceptually the simplest way of recovering the original signal from an AM

wave.

Since we utilise the envelope of the AM wave, we need the modulation to be less

than 100%.

LPF

Tuned circuit (c)

LPF

Figure 4: Rectification of an AM signal

Low pass filter

R

Diode to prevent back flow and cut off negative half of signal

Adjustable capacitor

Antenna

Envelope detector

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In fact, the action of the diode is equivalent to multiplication by a square wave d(t)

of frequency C, with two levels of amplitude: 0 or 1.

AM Wave

Square wave

(d(t))

Tc/2

This is an even signal.

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Analytically, we can expand the square wave d(t) as a Fourier series

1 2 2 2 2( ) cos( ) cos(3 ) cos(5 ) cos(7 )

2 3 5 7c c c c

d t t t t t

1 2 2( ) 1 cos( ) cos( ) cos( ) cos(3 )2 3cos( )1 1 1 1

cos( ) cos(3 )2 3

1 1cos(3 ) cos( )

3 2 6

m c c c

mc c

c c m

y t m t t t t

m tt t

mt t

= DC + wanted signal + harmonic distortion.

Notice that d(t)is an even function symmetric to zero, with balanced values of zero and

one. Taking the first three terms of this Fourier series the output is given by:

Thus the output of the rectifier contains a DC signal, the wanted signal and a number

of unwanted harmonics (at multiples of the carrier frequency) which can be removed

by low pass filtering.

Diode eliminated part of signal

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Practical envelope detection

In practice the filtering is achieved by the addition of a capacitor to form the circuit

below

RC

Figure 6: Simple Radio circuit (Crystal detector)

The RC time constant (the inverse of cut-off frequency of RC LPF circuit) is

required to be many times longer than the period of the carrier.

i.e. we need to filter out the carrier and unwanted harmonics around C, 2C,

3C

etc.

CC

C

C f

fTRC 2,1=>>

RC circuit

(Fc = cutoff frequency)

Additional component : capacitorRefer previous image to understand.Compare.Combined with the resistor and diodeit serves as a filter.

This is an important design consideration.

Envelope detector.

Antenna

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On the other hand RC cannot be made too long or it will begin to affect the signal

as well. This means that

wherefmaxis the highest frequency present in the input or modulating signal.

The envelope of the AM wave is given by

which has a slope given byThe discharge of the capacitor is given by

with an initial slope equal to at t = 0.

max

1RC

f

Figure 7: The effect of diode conduction

CR

Figure 8: Practical envelope detector for AM

)cos(1' tmEE m

)sin( tmE mm /' t RCE e

RCE/'

Capacitors charge quickly and discharge slowly.Discharging curve

Ac

Discharge slope must be greater than message slope to trap the envelope.

E' = The part where the capacitor voltage meets the incoming signal voltage.

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If the envelope of the wave is decaying more rapidly than the RC network can

discharge, we obtain distortion.

Figure 9: Examples of distortion caused by the RC filter

Thus for no envelope distortion we need the signal from the RCfilter to decay

faster than the envelope for all tgiving

)sin('

tEmRC

Emm for all t

This gives )cos(1 )sin(1 tm tmRC mm

m

for all t. Calculate the maximum value of right hand side, which occurs when

cos(mt) = m, giving

We cannot recover the envelope from an AM wave when m = 1.

2

max

11

mm

RC

(The slopes are from the previous slide.)

We need RC. This is to design the circuit.

Time varying.

Maximum of signal f(t) is obtained when f'(t) = 0 for a period. Stationary point if not at ends, is found and used.

Distortions

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Other methods of amplitude detection

An AM signal can also be demodulated by multiplying (or mixing) it with an

un-modulated carrier signal.

This carrier must be synchronousi.e. have the same frequency and phase

as AM carrier.DESIGN CONSIDERATIONS:

1)It should not track the carrier signal. (it has much lower frequency)

RC>>Fc

2)It must be able to track the envelope.

I))slope at discharging >> slope of signal II) RC< ([1-m^2]. / mw

Modulated signalY(t) = E(1+m.cos(wmt))cos(wct)

Cos (wct). (Unmodulated carrier wave)

y(t)Demodulated signal.

y(t) = Y(t).cos(wct) =E(1+m.cos(wmt) ) {cos (wct)}^2

= E/2 + E/2 m.cos(wmt) + mE/2 1/2 [ cos (2wct - wmt) + cos (2wct + wmt) ]

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AMPLITUDE MODULATIONII

Phasor Representation of an AM wave

A simple way of looking at the behaviour of an AM wave is interms of rotating vectors or phasors.

If we consider the carrier to be a phasor of lengthACrotating at

frequency c, and we modulate it with a sinusoidal tone of depth

of modulation mand frequency m, this can be expressed

mathematically as:

x(t) =ACcos (ct) + 0.5ACmcos((cm)t) + 0.5AC mcos((c+m)t)

We can draw this as a phasor diagram:

AC

0.5m AC

0.5m ACm

m LSBUSB

Carrier. (Zero phase)

Clockwise positive USB

Counterclockwise negative LSB

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The envelope can thus be determined graphically as:

time

envelopeamplitude

Based on the idea thata vector can be divided

into cosine and sine

components in x and y

directions, respectively,

we can consider the

combination of cosineand sine components

using vector operations.

Phase change become a

rotation in vector space,

and two phasor

components can beadded together using

vector addition.

Notice that the rotating

parts are still time-

varying ( ).mt

Varying with respect tocarrier amplitude.

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Effects of Distortion on an AM wave

Amplitude distortion caused by sideband removing

Since the envelope of the AM wave depends on the vector sum of the two

sidebands it is apparent that if one sideband is attenuated relative to the other

there will be a distortion in the envelope.

Consider an AM wave with m=2/3 and modulating signal cosSt

AC

1/3 AC

1/3 ACm

m

If the transmission channel completely removes the LSB, the phasor amplitude isgiven by

A(t)

Ac

3 )(t

2 2( ) [ cos( )] [ sin( )]

3 3

10 2cos( )

9 3

c cc m m

c m

A AA t A w t w t

A w t

No longer does the envelope match the

original modulating signal!

We usually assume transmitted signal is message signal. But it is distorted in practice.

The LSB and USB are always such that the

sum is horizontal. If one vector is smaller,the sum is not horizontal. It deviates fromthe axis.Cos(wct) - in phase component.Represents real axis.Sin(wct) - quadrature component/out ofphase component. Represents imaginaryaxisof Fourier transform.

Carrier Sum of sidebands

Magnitude

Phase change

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Amplitude distortion

Another effect of losing one sideband is the introduction of a periodic

fluctuation in the phase of the output

tcos3

1+sin3

11tan)t(

mCC

mC

AAtA

tcos+3

tsin1-tan=m

m

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Carrier and Envelope Delay

Consider a single frequency signal being transmitted through the channel. Let

the frequency be r/s, the total phase shift between input and output radians and the time for transmission tseconds. We have = t. The quantity t

is known as the phase or carrier delay of the channel.

If the carrier delay is measured over a propagation path such as a transmission

line, the distance travelled divided by tgives the phase velocity of the wave.

For systems where tincludes the propagation delay through filters or other

circuits, we can not relate tto the velocity of propagation.

The phase delay however is not necessarily the signal delay. In order to transmit

information, either the amplitude or angle of the sinusoid must be varied in

sympathy with the information.

Consider an AM wave with sinusoidal modulation. The transmitted signal is:

t

mt

mtAttmAtx mcmccccmc )cos(

2)cos(

2)cos()cos())cos(1()(

The delay varies with time as well.

Basically, delay causes proportional change in phase as well. The disturbance, if large could impact thereceiver end heavily. This should be prevented.

Radians

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This signal contains 3 frequencies, let the phase delay of the signals be , ,

and +radians respectively.

Phase

Shift

Frequency cm c cm

Figure: Total phase shift through transmission channel.

The received signal is:

( ) cos( ) cos{( ) } cos{( ) }2

cos( ) cos{ } cos{ }

1 cos( ) cos( )

c c c m c m

c c c m

c m c

my t A t t t

A t m t t

A m t t

Thus the envelope of the carrier has a phase delay of corresponding to a time

delay of .mdt

Carrier signal phase shiftUSB

LSB

2CosAcosB = cos(A-B) + cos(A+B)

The phase shift causes a delay, but the resultantsignal isn't affected much.

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For no distortion, we require tdto be constant giving . The quantity

is called the group or envelope delay.

So for distortionless transmission, we require (we also require theamplitude of the received signal to be independent of ).

When we are considering transmission through a channel we can think in terms

of the velocity of propagation. If the signal travels a distance of Lin a time T, the

velocity of propagation is , but the time is related to the phase shift

and frequency, . For a carrier phase delay of at a frequency of the

carrier velocity is and is known as the phase velocity.

The velocity of the envelope is and is known as the group velocity.

m td

m

T

Lv

T

c

cp

Lv

m

g

Lv

Delay due to transmission.

Tx Rx

Time

T=0. T=0

(T stands for one time period/cycle.)

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AM Modulation as Complex Convolution

Amplitude modulation can also be seen as a convolution operation in the

frequency domain.

Given a time domain signalf(t), the Fourier spectrum is defined as

If we consider two time domain signalsfm(t) andfc(t) with Fourier spectra Fm(w)

and Fc(w) respectively, the frequency spectra of the product of the 2 time

functions (aka mixed signal) involves the convolution of their individual spectra:

Let . Then

( ) ( ) j tF f t e dt

)()()( tftftf cm

( )

1( ) ( ) ( ) ( ) ( )

2

1 1( ) ( ) ( ) ( )

2 21

( ) ( )2

j t j t j t

m c m c

j t

m c m c

m c

F f t f t e dt f t F d e dt

f t e dt F d F F d

F F

This is similar to the analysis of a linear system.

Considering as a channel function.

m(t) AM Modulating filter Modulated signal.

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Impulse

Response

h(t)

x(t)

X(f)

y(t)

Y(f)

dthxty )()()(In time domain: In frequency domain: )()()( HXY

On the other hand, if in frequency domain we can obtain:

Baseband

Carrier

AM spectrum

( ) ( ) ( ),r t p t q t

That is: Multiplication

of two signals in the

time domain is

equivalent to convolve

their spectra and

scaling by

1( ) ( ) ( ) ( ) ( ) .

2R P Q p v q v dv

12

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Now consider the effects of passing the AM wave through a channel having a constant

magnitude response but a phase shift (f) (a function of frequency) over the pass band

i.e.

wherefmis the highestfrequency present in the modulating signal. The effect of

demodulation is to move the AM signal down to the base-band as shown below

( )( ) j fCH c m c mH f Ke f f f f f

(f)

f

(f-fc)

fHCH(f)

HLP(f)

cj f fLPH (f) = Ke

The equivalent low-pass spectrum (transfer function of

combined demodulator and channel) is given by:

Modulation so far shifted the real frequency bands to other bands, I.e; wc. Moved to passband.

And constant gain

PassbandBaseband

We receive this at receiver. By applying low pass filtering, we get this (ideally)

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Now we can shift the channel input spectrum of the signal to the low-pass region in

the same manner. Let be the bandpass carrier modulated by an AM

wave and . is the demodulated signal envelope.

)cos()()( ttAtx cmBP

)()( tAtx mLP ( )( ) ( )cj f fLP LPY f Ke X f

If we assume that the channel phase shift is , aand bconstants, then)(2)( bfaff c

bffbabfbfafff cccc 2)(2)(2)(

and 2 (( ) ) ( ) 2 21( ) ( ) ( ) .

c cj a b f bf j a b j bf j bf

LP LP LP LPY Ke X f Ke X f e K X f e

Since delaying a signal in time is equivalent to a phase shift proportional to

frequency of its spectrum). It leads to

)()( 1 btxKty LPLP

Specifically, The envelope is delayed by bseconds and is termed the group (orenvelope) delay.

The channel output is

the band-pass output is given by

The carrier delay is aseconds and is termed the phase (or carrier) delay.

2 ( ) ( )cj af bfBP BPY Ke X f

)](cos[)()( atbtKxty cLPBP

Shifter.

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Foldover Distortion

It is apparent that the bandwidth of an AM wave is equal to 2W Hz, where WHz is the

highest frequency present in the signal. If the carrier frequency is not greater than W

then foldover (overlapping) distortion occurs, as the modulated signal spectrum is the

convolution of the carrier spectrum and signal spectrum.

W 0

+W f

|S(f |

Base-band modulating signal

fc< W

fc> W

Foldover

distortion

Example of foldover distortion

Example of no foldover distortion

0 W fc

0 fc W

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Foldover Distortion

It is also important to realise that the upper and lower sidebands are derived from the

positive and negative frequencies of the original signal s(t). Since this is usually true for

any real signal:

and as a consequence the upper and lower sidebands contain the same information.

There are a number of coding schemes which transmit only one of these sidebands in

order to reduce the bandwidth required to transmit the signal.

In such case (that only one side band is transmitted), the foldover frequency range is

also reduced to half of the double bands.

)()( * fSfS

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The Square Law Detector

Essentially all that is required by square law detector for this system is to pass the AM

wave through a non-linear device, e.g.

Because of the non-linearity of the transfer characteristics the positive and negative

parts of the carrier are amplified to a different extent. Thus when the output

waveform is averaged there is a signal which bears a resemblance to the original signal.

)()( 2

tkxty

In fact, let the modulated signal be

ttmAAtx cc cos)](1[)( 0

The square law output becomes:2

0

2

0 0

2 22 2

( ) { [1 ( )]cos }

{ 2 [1 ( )]cos

[1 ( )] [1 ( )] cos(2 )}2 2

c c

c c

c cc

y t k A A m t t

k A A A m t t

A Am t m t t

Thus after filtering out the DC and

high frequencies c and 2c,we

obtain

)](2

1)([)( 22 tmtmkAts co

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The Square Law Detector

Points to note with this type of demodulation:

The non-linearity does not have to be a square law. Any non-linearity having even-

function symmetry will do.

It is possible for demodulation to occur when it is unintended, eg., by passing the signal

through a non-linear amplifier.

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Synchronous Detection

We have already seen that AM modulation can be achieved by multiplying the base

band signal by the carrier, and that has the effect of shifting the baseband signal up in

frequency.

Put simply if multiplication by a carrier can move a signal up in frequency it can also

move it down.

AM wave

= DC + signal + unwanted high frequency terms

The DC term appears when the carrier frequency is synchronized with the signal

frequency.There are a number of problems with implementing this practically which will be

discussed in relation to DSB-SC later in this unit.

tttmty cc coscos)](1[)(

)]2cos(1[2

1)](1[ ttm c

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AM Receivers

Before finishing with conventional AM modulation it is useful to discuss the two

receivers which have practical or commercial significance: the tuned radio frequency

(TRF)receiver and the superheterodyne receiver.

The TRF receiver

The TRF is an example of a simple "logical" receiver but is not now considered except

for a fixed frequency receiver.

demodulator

Audio

amplifier

1stRF

amplifier

2ndRF

amplifier

3rdRF

amplifier

Station selection

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The TRF receiver

The main problem is with feedback from the output of the amplifiers to the input.

Assuming a gain of about 40,000 through the amplifiers it does not require much stray

feedback to cause instability in the receiver.

Another problem is that the filters have to be variable in frequency. It is extremely

difficult to design bandpass filters which maintain a constant bandwidth over a wide

frequency range. Typically, AM broadcasting ranges from 540 - 1640 kHz a significant

variation.

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Superheterodyne Receiver

The problems in building an effective TRF receiver lead to the design of the

superheterodyne receiver. The basic principle of this receiver is to shift the carrier of

the incoming signal onto a fixed frequency known as the IF (intermediate frequency).

Once this is done the design of the following stages is simplified considerably.

Station

select

MixerRF

amplifier

IF

amplifier

Demodulator

Audio

amplifier

Local

oscillator

Carrier frequencyfcLocal oscillator frequencyflo

IF frequencyfif (455kHz for standard AM) ifclo fff

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Superheterodyne Receiver

The RF component is to limit the bandwidth of the signal reducing noise and providing

rejection of the image frequencies which can also produce unwanted signals at the

intermediate frequency. Since mixers are inherently noisier devices than RF amplifiers,

adding some gain in the RF stage also improves the noise figure of the receiver.

The heart of the "superhet" receiver is in the mixing component at which a signal

centred around an arbitrary carrier frequency , is shifted to the fixed intermediate

frequencyfif. In practice this is achieved by making . If we now apply the

incoming carrier frequency to the mixer the output is given by

fc

ifclo fff

( )cos cos cos cos( )

cos(2 ) cos2

c c lo lo c lo c c if

c loc if if

A t t A t A A t t

A At t

from which is easy to extract the wanted signal at the intermediate frequency.

However, If a signal with a frequency is present at the input to the

mixer then this will also produce output at the intermediate frequency. Morespecifically,

ifcc 2'

' ' '

'

( )cos cos cos( 2 ) cos( )

cos(2 3 ) cos2

c c lo lo c lo c if c if

c loc if if

A t t A t A A t t

A At t

Thus there exists an unwanted intermediate frequency known as theimage

frequency: ifc 2

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Effects of mixing on the wanted and image frequencies

fc fLO

Inputs tothe mixer

-fi fi 2fcfi

Input to IF amplifier, fi is the intermediate

frequency

Unwanted case:

A signal centred around

the image frequency alsoproduces two replicas of

the input spectrum and

again one is centred

aroundfif.

Inputs tothe mixer

fLO fc 2fi(image frequency)

-fi fi 2fc 3fi

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Effects of mixing on the wanted and image frequencies

In practice it is common to choose the intermediate frequency in such away that the image frequency lies outside the receiver bandwidth and

can be rejected by the RF stage filter.

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AM Receiver Example

An AM receiver is required to work with a carrier over the range 600 - 1500kHz. What

is a reasonable value of the intermediate frequency to ensure that no problems with

image frequencies occur in the receiver? The bandwidth of the signal is 5kHz at

baseband.

We consider the case where the image frequency is above the carrier frequency we

need to ensure that:> 1505.Notice that = = ,which leads

to , wherefsis the input signal frequency. Since the lowest value offsis 595kHz, we need> max(

2)=

9

2 = 455.

On the other hand, it is not good practice to have the IF frequency within the range of

the receiver. Thus, we choose the intermediate frequency to be

455

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Summary of the spectra in an AM "superhet" receiver

f

Audio output

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Problems with the "superhet" receiver

The main problem with the superhet receiver is its potential for spurious responses to

signals centred on frequencies other thanfc.

We have already seen that the image frequency can usually be rejected by the RF

stage.

Nonlinearities can also cause unwanted signals to appear at the output.

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Week2(AM-II)