p. II-33webpages.eng.wayne.edu/ece4700/Lecture Notes/lecture-ssb.pdf · 2003-11-14 · ERG2310A-II...

ERG2310A-II p. II-33

Intermediate Frequency (IF)Image frequency


Amplitude Modulation: SSB

DSB modulation:

By mixing with a sinusoidal carrier at ωc rad/sec, half of this spectral density is translated up in frequency and centered about ωc and half is translated down to (- ωc ).

Since each half contains all the information about the signal, the original signal can be recovered again from either the upper or lower pair of sidebands by an appropriate frequency translation. So only the upper or the lower pair of sidebands is required to transmit.

Such kind of modulation is called single-sideband (SSB) modulation. It is efficient because it requires no more bandwidth than that of the original signal and only half of the corresponding DSB signal.

Doubling of the bandwidth of a given signal



SSB:

SSB:



Generation of SSB Signals

-first generate a DSB signal, then suppress one of the sidebands by filtering.



Consider the modulating signal x(t) is tj metx ω=)(and let the carrier signal be .tj ce ω

Multiplying, we get tjtjtj cmc eeetx ωωω =)(

Taking the real part, we have

tttteeeeee

cmcm

tjtjtjtjtjtj cmcmcm

ωωωω

ωωωωωω

sinsincoscos}Im{}Im{}Re{}Re{}Re{

−=−=

Because this represents the upper sideband, we write

ttttts cmcmSSB ωωωω sinsincoscos)( −=+

Similarly, by using , the lower sideband is tj metx ω−=)(ttttts cmcmSSB ωωωω sinsincoscos)( +=−

In general, we can write ttxttxts ccSSB ωω sin)(ˆcos)()( ±=m

where is that signal obtained by shifting the phase of x(t) by 90° at each frequency.

)(ˆ tx

Generation of SSB: Phase-Shift Method

}{ tj ce ωℑ0 ω

}{ tj me ωℑ

ωm

0 ωωc

}{ tjtj mc ee ωωℑ

0 ωωc ωc+ωm

]}{Re[ tjtj mc ee ωωℑ

0 ωωc ωc+ωm−ωc-ωm



Generation of SSB Signals : Phase-Shift Method



Demodulation of SSB Signals

The synchronous detector will properly demodulate SSB-SC signals



Given that the incoming SSB-SC signal is ttxttxts ccSSB ωω sin)(ˆcos)()( ±=m

Let the locally generated carrier signal be ],)cos[()( θωω +∆+= ttc c

where (∆ω) is the frequency error and θ is the phase error.

]})2sin[(])){sin[((ˆ]})2cos[(])){cos[((])cos[(]sin)(ˆcos)([)()(

21

21

θωωθωθωωθωθωωωω

+∆+−+∆+∆+++∆=+∆+±=

tttxtttxtttxttxtcts

c

c

cccSSB

m

m

After passing through a low-pass filter, the output xo(t) becomes

])sin[()(ˆ])cos[()()( 21

21 θωθω +∆+∆= ttxttxtxo m Distorted !

If ∆ω=0 and θ =0, then )()( 21 txtxo =



Example:

When an SSB (upper/lower-sideband) is received and fed into the following demodulator:

IF Amp LPFsSSB±(t) xo(t)

cos(2π fct) cos(2π fdt)(10.000-10.003 MHz)(SSB-SC at 20 MHz)

20 MHz

3kHz

-20 MHz

Upper sideband SSB-SC

10.003 MHz10 MHz

+fc-fc fc=30.003MHz

fd=10.003MHz

20 MHz-20 MHz

Lower sideband SSB-SC

10.003 MHz10 MHz

+fc-fc fc=30.000MHz

fd=10.000MHz

0

0

Freq.

Freq.

3kHz

+fd

-fd

-fd

+fd



Single Sideband-Large Carrier (SSB-LC) Signals

An expression for an SSB-LC signal isttxttxtAts cccc ωωω sin)(ˆcos)(cos)( m+=

The original signal x(t) can always be recovered from s(t) using synchronousdetection.

If the carrier is large, however, envelope detection can also be used.

The envelope can be written as

2

2

2

222 )(ˆ)()(21)](ˆ[)]([)(

ccccc A

txAtx

AtxAtxtxAtr +++=++=

If the carrier is much larger than the SSB-SC envelope, we have

).()(1)(21)( txAAtxA

AtxAtr c

cc

cc +=

+≈+≈

Thus after discarding the dc term introduced by carrier, the SSB-LC signal canthen be demodulated correctly using an envelope detector.


Amplitude Modulation: VSB

In VSB modulation, one passband is passed almost completely whereas only a residual portion of the other sideband is retained in such a way that the demodulation process can still reproduce the original signal.

The partial suppression of one sideband reduces the required bandwidth fromthat required for DSB but does not match the spectrum efficiency of SSB.

If a large carrier is also transmitted, the desired signal can be recovered usingan envelope detector.If no carrier is sent, the signal can be recovered using a synchronous detectoror the injected carrier method.

Vestigial sideband (VSB) modulation is a compromise between DSB and SSB.

ωc−ωc 0 ω

SDSB(ω)

ωc−ωc 0 ω

SVSB(ω)



Generation of VSB Signals

The filtering operation can be represented by a filter H (f) that passes some of the lower (or upper) sideband and most of the upper (or lower) sideband.

)()]()([)( 21

21 ωωωωωω HXXS ccVSB ++−=



The spectral density of the received vestigial-sideband signal is

)()]()([)( 21

21 ωωωωωω HXXS ccVSB ++−=

The output of the synchronous detector is

LPcVSBo ttstx ]cos)([)( ω=

[ ] [ ][ ]LPcc

LPcLPco

HHXHXHXX

)()()()()()()()(

41

41

41

ωωωωωωωωωωωω

−++=−++=

For faithful reproduction of x(t), we require that

.)]()([ mLPcc HH ωωωωωω ≤=++− constant,

Demodulation of VSB Signals

LPF xo(t)

cos ωct

sVSB(t)

Synchronous Demodulation


Amplitude Modulation: VSB.)]()([ mLPcc HH ωωωωωω ≤=++− constant,

By letting the constant be 2H(ωc) :

mcLPcc HHH ωωωωωωω ≤=++− ),2 ()]()([

Thus, H(ω) exhibits odd symmetry around the carrier frequency ωc. The sum of the values of H(ω) at any two frequencies equally displaced above and below ωc is unity.



Synchronous Demodulation of VSB Signals

ωc−ωc 0 ω

SVSB(ω)

ωc−ωc 0 ω

Xd(ω)

−2ωc 2ωc

ωc−ωc 0 ω

Xo(ω)After LPF

After mixer

Received VSB signal



Example of VSB Signals: Television signal

Television picture signal has nominal bandwidth of 4.5MHz

If DSB modulation is used, it requires at least 9MHz for each TV channel.So, VSB modulation is used so that the whole TV signal is confined to about 6MHz.


Angle Modulation

A continuous-wave (CW) sinusoidal signal can be varied by changingits amplitude and its phase angle.

)](cos[)()( tttAts c φω +=

Amplitude modulation:

Keep θ(t) constant and varies A(t) proportionally to x(t).

Angle modulation:

Keep A(t) constant and varies [ωct+φ(t)] proportionally to x(t).

To carry a message signal x(t):

[ ] )(cos)()( txttAts oc ∝+= A(t) where φω

[ ] [ ] )()(cos)( txtttAts cc ∝++= (t) where φωφω


Angle Modulation

Phasor RepresentationThe phasor representation of a constant-amplitude sinusoid is shown asfollows

A: magnitude of the phasor

θ(t): phase angle

oc ttt θφωθ ++= )()(ωi(t) : instantaneous angular rate

ωi(t)

θ(t)

dttd

dttdt ci

)()()( φωθω +==

00)()( θττωθ += ∫

t

i dt


Angle Modulation

If the phase φ(t) is varied linearly with the input signal x(t) , we have

0)()( θωθ ++= txktt pc

where ωc , kp , θ0 are constants.

As the phase is linearly related to x(t), this type of angle modulationis called phase modulation (PM) with

(t)(t) .e. xi ∝φ

and , φ(t) is called instantaneous phase deviation.

])(cos[)( opccPM txktAts θω ++=

dttdxk

dttd

dttdt pcci

)()()()( +=⇒+== ωωφωθω i(PM)


Angle Modulation

If the instantaneous frequency ωi proportional to the input signal, we have

x(t)dt(t)d ∝⇒+= φωω )()()( txkt fcFMi

where ωc , kf are constants.

As the frequency is linearly related to x(t), this type of angular modulation is called frequency modulation (FM) , with

dttd

dttdt ci

)()()( φωθω +==Q

and; is called instantaneous frequency deviation.dt(t)d φ

])(cos[)(0

o

t

fccFM dxktAts θττω ++= ∫

∫ ++=t

of dxkt0

)( θττωθ c(t)


Angle Modulation

Phase Modulation (PM) Frequency Modulation (FM)

])(cos[)(0

o

t

fccFM dxktAts θττω ++= ∫

)()()( txkt fcFMi += ωω

])(cos[)( opccPM txktAts θω ++=

dttdxk pc)(+=ωω i(PM)

x(t)dt(t)d ∝φ

(t)(t) x∝φ

Instantaneous angular rate

Modulated signal

Proportionality


Angle Modulation

FM and PM Waveforms


Angle Modulation: Fourier spectra

[ ][ ]{ }

{ })(

)(

)(cos)(

tjtj

ttjc

cc

ee

eA

ttAts

c

c

φω

φω

φω

cARe Re

=

=

+=+

Consider an angle-modulated signal

Expand ejφ(t) in a power series, gives

++−−=

++−−+=

L

LL

tttttttA

ntjttjeAts

ccccc

nntj

cc

ωφωφωφω

φφφω

sin!3

)(cos!2

)(sin)(cos

!)(

!2)()(1Re)(

32

2

The signal consists of an unmodulated carrier plus various amplitude-modulated terms, such as φ(t)sinωct, φ2(t)cosωct, φ3(t)sinωct, …, etc.

Hence the Fourier spectrum consists of an unmodulated carrier plus spectra (sidebands) of φ(t), φ2(t), φ3(t), …, etc., centered at ωc.


Angle Modulation: Narrowband

If |φ(t)|max << 1 , then, by neglecting the higher-power terms of φ(t) in the s(t) , gives

ttAtAts cccc ωφω sin)(cos)( −≈

which is called the narrowband (NB) angle-modulated signal.

tdxkAtAts

ttxkAtAts

c

t

fcccNBFM

cpcccNBPM

ωττω

ωω

sin)(cos)(

sin)(cos)(

0

−≈

−≈

∫


Angle Modulation: NBFM (Sinusoid)

Consider tatx mm ωcos)( =For FM,

tkatxk

mfmc

fci

ωωωω

cos

)(

+=

+=

where kf is the frequency modulation constant; typical units are in radiansper second per volt.

Define a new constant called the peak (maximum) frequency deviation,

fmka=∆ωthus, we have

tmci ωωωω cos∆+=

Narrowband FM (NBFM)


Angle Modulation: NBFM (Sinusoid)The phase of this FM signal is

tt

tt

d

dt

mc

mm

c

t

mc

t

i

ωβω

ωω

ωω

θττωωω

θττωθ

sin

0sin

]cos[

)()(

00

00

+=

+∆+=

+∆+=

+=

∫∫

θ0 is set to zero for convenience.

wheremωωβ ∆=

Thus, the resulting FM signal is

{ }

)sinsin(sin)sincos(cos)sincos(

Re)( )(

ttAttAttA

eAts

mccmcc

mcc

tjcFM

ωβωωβωωβω

θ

−=+=

=

For narrowband FM (NBFM) , β is very small so thatttt mmm ωβωβωβ sin)sinsin(,1)sincos( ≈≈

ttAtAts cmcccNBFM ωωβω sinsincos)( −=

The parameter β is called the modulation index of the FM signal.

Thus,


Angle Modulation: NBPM (Sinusoid)

tatx mm ωsin)( =The phase of this PM signal is

Consider

tttakt

txktt

mc

mmpc

pc

ωβωωωθωθ

sin

0sin

)()( 0

+=

++=

++=θ0 is set to zero for convenience.

The resulting PM signal is

{ }

)sinsin(sin)sincos(cos

)sincos(Re)( )(

ttAttAtaktA

Aets

mccmcc

mpcc

tjPM

ωβωωβωωω

θ

−=

+==

Narrowband PM (NBPM)

ttt mmm ωβωβωβ sin)sinsin(,1)sincos( ≈≈

ttAtAts cmcccNBPM ωωβω sinsincos)( −=

For narrowband PM (NBPM), β is very small,

mpak=β

Thus,


Angle Modulation: NB (Sinusoid)

In summary, if the message signal x(t) is a pure sinusoid, that is,

for FM

PMfor

tata

txmm

mm

=ωω

cossin

)(

Then, tt mωβφ sin)( =

where

∆==

mm

mf

mp

akak

ωω

ωβ

for PM

for FM

It is only defined for sinusoidal modulation.

If φ(t) has a bandwidth of WB, the NB angle-modulated signal will have a bandwidth of 2WB.

Note that β is known as modulation index for angle modulation and is the maximum value of phase deviation for both PM and FM.

∆ω : peak frequency deviation

p. II-33webpages.eng.wayne.edu/ece4700/Lecture Notes/lecture-ssb.pdf · 2003-11-14 · ERG2310A-II...

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