Eee3086f 506 AM DSB LC Ajw

AM DSB-LCA.J.Wilkinson, UCT EEE3086F Signals and Systems II506 Page 1 May 18, 2009

EEE3086FSignals and Systems II

2009

Andrew [email protected]

http://www.ee.uct.ac.za/people/ajw.phpDepartment of Electrical Engineering

University of Cape Town


5.3 Double Sideband Large Carrier (DSB-LC) Amplitude Modulation

5.3.1 DSB-LC modulation and demodulation5.3.2 Topologies for generating DSB-LC5.3.3 Power and efficiency of DSB-LC


5.3.1 Double Sideband Large Carrier (DSB-LC) Amplitude Modulation and Demodulation

DSB-LC is sometimes called AMmodulation, as in AM radio.

To hear an AM radio broadcast, tune to Cape Talk 567 kHz on the medium wave

radio band.

A.J.Wilkinson, UCT AM DSB-LC EEE3086F Signals and Systems II506 Page 4 May 18, 2009

DSB-SC modulation requires a fairly complicated demodulator, involving a mixer and oscillator, which must be synchronised to the transmitter.

SOLUTION: An alternative approach is to design the modulation such that the modulating signal can be recovered purely from theenvelope of the modulated carrier.This can be achieved by ensuring that the signal fed into the mixer is always positive.

Very simple circuits can be used to perform envelope detection. Early radio receivers used this approach.

Standard Medium Wave broadcast AM radio uses this technique. In South Africa, broadcast AM radio uses a 9kHz bandwidth, with radio stations located in the range 540 kHz > 1600 kHz band).


Amplitude Modulation: Large Carrier (DSB-LC)

The signal fed into the mixer can be made always positive by adding a DC offset to f(t):

such that

The modulated carrier can be re-expressed as:

which is equivalent to adding a carrier component to a DSB-SC signal, hence the name DSB-LC (large carrier).

DSB-LC is sometimes referred to as AM as in AM radio.

tAttft ccAM coscos)()( +=

[ ] tAtft cAM cos)()( +=0)( + Atf { })(min tfA


DSB-LC in Frequency Domain

Fourier transforming the DSB-LC signal,

we get:

)()()(21)(

21)( ccccAM AAFF +++++=

[ ] tAtft cAM cos)()( +=

Note that the carrier is present


Block diagram implementations of DSB-LC Signal

tAtftAttft cccAM cos])([coscos)()( +=+=

1) EITHERAdd DC offset to f(t)prior to mixing

2) ORCreate DSB-SC and add carrier

)(tf

A tccos

)(tAM

tccos

)(tf

A

)(tAM


1) Addition of DC offset and multiply to produce DSB-LC

t

Atf +)( )(2)( AF +

W 0 W

tccos

t

c c

{ }tAttf cc coscos)( +FCarrier Envelope

tAttf cc coscos)( +

LCDSB t

A

0

c c

{ }tccosF

(weight)2 ADC offset Aadded to f(t)

)(21

cF A

21


2) Addition of Carrier to DSB-SC to produce DSB-LC

t

)(tf

)(tf

SCDSBc c

W2

)0(21 F

{ }ttf ccos)(F

)(F)0(F

W 0 W

W2

Consider DSB-SC waveforms:

ttf ccos)(

t


2) cont

tA ccos

CarrierADD

c c

{ }tA ccosFA A

c c

{ }tAttf cc coscos)( +FCarrier Envelope

tAttf cc coscos)( +

LCDSB t

t


Demodulation of DSB-LC AM

DSB-LC signals are very easy to demodulate using a simple envelope detector.

DSB-LC is suited to audio applications e.g. speech or music,which do not contain a DC component.

An envelope detector recovers f(t) + A. If f(t) contains no DC component, then passing f(t) + A through a

BPF eliminates A, recovering f(t).

The simplest AM receivers receiver the signal from an antenna, pass it through a BPF, amplify and then into an envelope detector, followed by a bandpass filter.


Simple DSB-LC AM Receiver

tunablebandpassfilter

amplifier

Antenna

The tunable BPF selects a radio station. AM signal demodulation using an envelope detector and band pass filter.

BPF BPF Audiooutput

Envelopedetector


Demodulation using an Envelope Detector

)(tAM

)()(0 tftv

Envelope detector Bandpass filter

)(0 tv)(tve

Atftve + )()(Envelope

)(tAM

CR


Choosing the RC time constant

)(tAM

Envelope detector

)( tv eCR

The RC time constant T is the time it takes for voltage todrop to 1/e ~ 0.37 of its charged value. Clearly,RC = T (carrier period) would be too short. A value of RC somewhere in the range T < RC < 1/B, where B is thebandwidth of the modulating signal would give a smooth result.

T = 2/c

RC too small

RC about rightRC too large

)(tve


Modulation Index m

In DSB-LC AM, the signal f(t) causes the amplitude of the carrier to fluctuate. The modulation index is a number which quantifies the degree of amplitude modulation of the carriers amplitude.

Considering

The modulation index is defined as the fractional fluctuation in the amplitude of the carrier, i.e.

ttfAttftAt cccAM cos)]([cos)(cos)( +=+=maxmax )( ftff

AmplitudeCarrierAmplitudeCarrier PeakEnvelope =m => Envelope peak = A(1+m)


Modulation index m

Envelope

t

AMax value = A(1+m)

Min value = A(1-m)

Note: The modulation index should be less than 1. i.e. m0.

ttfAtftAt ccAM cos)]([)(cos)( +=+=

Case: m=0.9 shown


Modulation index m

Modulation index:

Thus

Also,A

mAAmplitudeCarrier

AmplitudePeakm == SC DSB

AAm

AmplitudeCarrierAmplitudeCarrierPeakEnvelopem

AM =

=}max{

)1(}max{ mAAmAPeakEnvelope AM +=+==


Modulation index: Over-modulated case

m1(cant recover f(t))

t

t

t

t

t

cant recover f(t)from envelope

(most powerefficient)

Over-modulated

Okay

t

Envelope


Modulation Index m for sinusoidal modulation

For a sinusoidal modulating signal: DSB-LC signal is

Modulation Factor or Index

Define

tAttktAttft

ccm

ccAM

coscoscoscoscos)()(+=

+=

%100% = mModulation

tktf mcos)( =

mAkORAkm == /A

AkAm +== )(AmplitudeCarrier

AmplitudeCarrier - envelope LCDSBPeak


5.3.2 Topologies for generating Double Sideband Large Carrier (DSB-LC) Amplitude

Modulation


Generation of DSB-LC Signal

Block Diagrams:ttfAtftAt ccAM cos)]([)(cos)( +=+=

)(tf

A tccos

)(tAM

tccos

)(tf

A

)(tAM


Generation of DSB-LC Signal

Practical Implementations:Chopper (switch) type modulatorsExploiting non-linear characteristics of devices

i.e.

L++= )()()( 221 teateati


Chopper Modulator (DSB-LC)

Simply chop and pass through cBPF @

Chop

))(( Atf +

)(tf

A c

)(tAMc

BPF@

{ }Atf +)(F

00 0c

c3c

c3BPF

Chop rate

Spectrum after chopper


Another method Chop and filter.

Rectifier Method: also works if one replace switch with a diode.

Chopper Modulator (DSB-LC)

~

)cos)(( tKtf c+

~

c

BPF@)(tftK ccos

c

R )(0 tV

c


Understanding Chopper / Rectifier Circuit

Voltage across is

Where is square wave, 010101 In frequency domain

)(tPT

R

)()()cos)(( tVtPtKtf RTc =+

{ } 21)(cos)()( TcR PtktfV += F


Understanding Chopper / Rectifier Circuit

)(F

)(TPc

K0 c

KSum of Signal +Carrier

c3c 0 c

c3

c 0 c)(RV

c 0 c)(0 V BPFAfter

)(Output

c3c3


DSB-LC Using Non-Linear Devices

Passing the sum through a non-linear device (e.g. diode) generates DSB-LC spectral replicas in frequency spectrum.

Consider Circuit:

RtitVR )()( = v

)cos)(( tKtf c+

i

Non-linear Diode characteristic

~~

BPF)(tf

tK ccosR )(0 tV

)( tv

)(tVR)(ti

The voltage drop across R is proportional to current through diode:



The non-linear voltage to current relationship is modeled by a power series:

For (small voltage drop across R) tKtftv ccos)()( +

)(tvVR



ttKRfatKRatV cc cos)(2cos)( 210 +=At the output of the bandpass filter we obtain the desired DSB-LC signal:

tKaKatKa cc 2cos221cos

222

222

2 +=

)()(21)(2 FFtf

The term:

is located at baseband, and has bandwidth of 2B Hz.

c 0 c c2

The term: gives a DC termand a 2c term

c2BPF


5.3.3 Power calculations and Efficiency of DSB-LC AM


Carrier and Sideband Power in AM

In DSB-Large Carrier AM, the carrier is included to allow simple demodulation via envelope detection. This allows cheap receivers to be constructed.

The carrier component does not however convey any information about f(t).

The additional power required to transmit the carrier in large-carrier AM, makes it less power efficient than DSB-SC.

In the following section we shall examine power calculations in amplitude modulation, and determine the efficiency of DSB-LC AM, which is a function of the modulation index m.

In the analysis, we shall assume that f(t) has no DC component.


Carrier and Sideband Power in AM

+ 021)(2 tf

212 = A

ttAfttftAt ccc 222222 cos)(2cos)(cos)( ++=+

PowerCarrier

BandsSideInPower 0)( =tfSince

tP cP= + sP

tAttft ccAM coscos)()( +=where and f(t) varies slowly compared to cosct.The power in the signal (mean square value) is given by

Let0)( =tf


Note:

Similarly

0

2cos)(21)(

21

2cos)(21)(

21cos)( 2

=+=

+=

ttftf

ttftfttf

c

cc

This is a DSB-SC termat 2c andhas no DC component.f(t) has no

DC component

)(21

2cos)(21)(

21

2cos)(21)(

21cos)(

2

22

2222

tf

ttftf

ttftfttf

c

cc

=

+=

+=


Power Considerations: Frequency Domain

For finite energy signals, we analyze the energy spectral density

In this case, it we are analysing a power signal, and require treatment in terms of the power spectral density.

2)(=ESD

{ })(

21)()(

21)(

coscos)()(

cccc

ccAM

FAFA

tAttf

+++++=+= F

0c c

A A)()(21

cF + )(21

cF


Power Considerations: Frequency Domain

For power calculations we consider the power spectral density shown below:

where of for which

)(S

...)( DSPS f = )(tf

+

= dStf f )(21)(2

0c c

2/2A 2/2A)(

41

cfS + )(41

cfS )(S


Note

The in the Fourier Transform becomesin the PSD.

To see how it arises, let

Thus, using a result from linear system theory,

21)(where)()(

)(21)(

=+=

+=

HFH

FG

c

c

)(41)()()( 2 cfcfg SSHS +=+=

)(21

cF +

)(41

cfS +


Obtaining the Power from the PSD

Carrier Power

Sideband Power

Total Power

)(21

21

)]()([21

22 tfAPPP

dSSP

sct

sct

+=+=

+=

)(21)(

41)(

41)(

21 222 tftftfdSP ss =+==

222

21]

22[

21)(

21 AAAdSP cc =+==


Transmission Efficiency

sc

s

t

s

PPP

PP

+=== PowerTotalsidebands theinPower

The transmission efficiency of large-carrier AM is definedas the ratio (sometimes expressed as %)

)()(

)(21

21

)(21

22

2

22

2

tfAtf

tfA

tf

+=+=

Substituting,


Transmission Efficiency for a Sinusoidal Modulating Signal

Let tmAtf mcos)( =tAttmAt ccmAM coscoscos)( +=

2

21 APc =

2222

41)(

21

21)(

21 AmmAtfPs ===

2

2

222

22

241

21

41

mm

AmA

Am

PPP

PP

sc

s

t

s

+=+=+==

Transmission Efficiency

Sideband power:

Carrier power:


Example: High Power AM Radio Broadcast Transmitter

A DSB-LC AM transmitter transmits with carrier power of 40 kW and a modulation index of 0.707For a sine wave modulating signal of maximum amplitude, calculate:a) Total power radiated

kWmPPmP

AmAtfAP

ccc

t

50)2

707.01(40)2

1(21

21

21

21)(

21

21

222

22222

=+=+=+=

+=+=


b) Transmission efficiency (comment: DSB-LC is inefficient!)

c) Peak voltage if driving into a 50 load (being the input impedance of a radiating antenna)

If driving a 50 load, the carrier power is

{ } voltsAmmAAtMax AM 3414)1()( =+=+=

voltsRPARAP cc 20004000050222

1 2 ====

20.0707.02

707.02 2

2

2

2

=+=+=+== mm

PPP

PP

sc

s

t

s

The peak voltage is then


EEE3086FSignals and Systems II

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Eee3086f 506 AM DSB LC Ajw