Post on 10-Mar-2020
ERG2310A-II p. II-33
Intermediate Frequency (IF)Image frequency
ERG2310A-II p. II-34
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
ERG2310A-II p. II-35
Amplitude Modulation: SSB
SSB:
SSB:
ERG2310A-II p. II-36
Amplitude Modulation: SSB
Generation of SSB Signals
-first generate a DSB signal, then suppress one of the sidebands by filtering.
ERG2310A-II p. II-37
Amplitude Modulation: SSB
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
ERG2310A-II p. II-38
Amplitude Modulation: SSB
Generation of SSB Signals : Phase-Shift Method
ERG2310A-II p. II-39
Amplitude Modulation: SSB
Demodulation of SSB Signals
The synchronous detector will properly demodulate SSB-SC signals
ERG2310A-II p. II-40
Amplitude Modulation: SSB
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 =
ERG2310A-II p. II-41
Amplitude Modulation: SSB
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
ERG2310A-II p. II-42
Amplitude Modulation: SSB
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.
ERG2310A-II p. II-43
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)
ERG2310A-II p. II-44
Amplitude Modulation: VSB
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 ++−=
ERG2310A-II p. II-45
Amplitude Modulation: VSB
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
ERG2310A-II p. II-46
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.
ERG2310A-II p. II-47
Amplitude Modulation: VSB
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
ERG2310A-II p. II-48
Amplitude Modulation: VSB
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.
ERG2310A-II p. II-49
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
ERG2310A-II p. II-50
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
ERG2310A-II p. II-51
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)
ERG2310A-II p. II-52
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)
ERG2310A-II p. II-53
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
ERG2310A-II p. II-54
Angle Modulation
FM and PM Waveforms
ERG2310A-II p. II-55
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.
ERG2310A-II p. II-56
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
−≈
−≈
∫
ERG2310A-II p. II-57
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)
ERG2310A-II p. II-58
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,
ERG2310A-II p. II-59
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,
ERG2310A-II p. II-60
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