amplitude modulation dsb-scihayee/teaching/ee5765/ece5765_chapter_4_5.pdfamplitude modulation...
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Amplitude Modulation – DSB-SC
Modulation: )()( Mtm
)]()([2
1)cos()()(mod ccc MMttmtS
)(tm
)cos( tc
)cos()( ttm c
Modulating signalModulated Signal
Modulating carrier
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)(M
BB
c
Modulation: )()( Mtm
)]()([2
1)cos()()(mod ccc MMttmtS
)(mod S
c
2
1
1
Bc Bc Bc Bc
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Demodulation: )(cos)()( 2 ttmtS cdem
)]2cos()()([2
1ttmtm c
)]2()2([4
1)(
2
1)( ccdem MMMS
Amplitude Demodulation – DSB-SC
)cos( tc
)cos()( ttm c
Modulated signalDemodulated Signal
Modulating carrier
)(tSdem
Lowpass Filter
1. Upper and Lower
Sidebands
2. Relationship of carrier
freq and signal bandwidth
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c
)(mod S
c
2
1
Bc Bc Bc Bc
Demodulation: )(cos)()( 2 ttmtS cdem
)]2cos()()([2
1ttmtm c
)]2()2([4
1)(
2
1)( ccdem MMMS
)(mod S
c2
2
1
Bc 2Bc 2Bc 2c2BB Bc 2
Lowpass Filter
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Example of DSB-SC Modulation
Let )cos()( ttm m
)]()([)( mmM
)cos()()(mod ttmtS c
)cos()cos( tt cm
])cos()[cos(2
1tt mcmc
then
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)(cos)()( 2 ttmtS cdem
)(cos)cos( 2 tt cm
))2cos(1)(cos(2
1tt cm
)2cos()cos(2
1)cos(
2
1ttt cmm
Example of DSB-SC Demodulation
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Frequency Conversion or Mixing
How to change carrier frequency from one frequency to another?
)()( Mtm
)]()([2
1)cos()()(mod ccc MMttmtS
c
)(mod S
c
2
1
Bc Bc Bc Bc
The Goal is to change toc I
)cos()( ttm c )cos()( ttm I
conversionupif cI
conversiondownif cI
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tIc )cos(2
)cos()( ttm c
Signal with a carrier frequencyFrequency Converted Signal
Bandpass FilterOption 1
tttmtx Icc )cos(2)cos()()(
])cos()cos(2)[( tttm Icc
])2cos())[cos(( tttm IcI
)(tx
I
)(X
I
1
Ic 2Ic 2
Bandpass Filter
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tIc )cos(2
)cos()( ttm c
Signal with a carrier frequencyFrequency Converted Signal
Bandpass FilterOption 2
tttmtx Icc )cos(2)cos()()(
])cos()cos(2)[( tttm Icc
])2cos())[cos(( tttm IcI
)(tx
I
)(X
I
1
Ic 2Ic 2
Bandpass Filter
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Switching Modulators
AM modulation can be obtained by not only multiplying with pure sinosoidal signal
but by any periodic signal with fundamental frequency c. Any periodic signal can
be represented as:
)cos()(0
nc
n
n tnCt
)cos()()()(0
nc
n
n tntmCttm
)cos()()()()()()(2
10 nc
n
nnc tntmCtCtmtmCttm
Need a Bandpass Filter
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t
w(t)
....]5cos5
13cos
3
1[cos
2
2
1)( ttttw ccc
.....]5cos)(5
13cos)(
3
1cos)([
2)(
2
1)()( ttmttmttmtmtwtm ccc
)]()([1
)(2
1)(mod cc MMMS
)]3()3([3
1cc MM
)].....5()5([5
1cc MM
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Modulating Signal Modulated Signal
BPF
Switch
Can be realized with a diode (a basic
switching element) or combination of diodes
Demodulation still needs to be synchronous
~
tccos
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)cos()()cos()()mod( ttmtAtS ccAM
Regular Amplitude Modulation without carrier suppression
)cos()]([ ttmA c
)]()([)]()([2
1)()( ccccAMdem AMMS
Important condition for envelop detection
A+m(t) ≥ 0 for all t
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Let mp is the maximum peak value of m(t), then
pmA
A
mp
10
Let’s define modulation index
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Power of Carrier vs. Power in Sidebands
)cos()()cos()()mod( ttmtAtS ccAM
2
2APc )(
2
1 2 tmPs
)(
)(22
2
tmA
tm
PP
P
P
P
sc
s
total
useful
and
Power Efficiency
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)cos()()cos()()mod( ttmtAtS ccAM
)cos()( tBtm m
)cos()cos()cos()()mod( ttBtAtS cmcAM
)cos()]cos([ ttBA cm
Let
ABA
B
A
mp
)cos()]cos([)()mod( ttAAtS cmAM
)cos()]cos(1[ ttA cm
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Example of Tone modulation => max efficiency = 33%
2
2APc )(
2
1 2 tmPs and
2
2APc
22
1 22 APs
2
2
222
22
22
AA
A
PP
P
P
P
sc
s
total
useful
1%3333.012
1
for
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Generation of AM signals
Generation of AM signals is the same as generation f AM-SC
signals, the only difference is that m(t) is replaced by [A+m(t)]
Demodulation of AM signals
Two commonly used methods are:
1. Rectifier Detection – synchronous demodulation
2. Envelop Detection – very simple detection
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Rectifier Detector
)cos()]([1 ttmAv c+
-)()}cos()]({[2 twttmAv c
2v
1v
....]5cos5
23cos
3
2cos
2
2
1)[cos()]([ tttttmA cccc
termsfrequencyhighertmA )]([1
LPF
3v
)]([1
3 tmAv
4v
)(1
4 tmv
....]5cos5
23cos
3
2
2
1)[cos()]([cos
2*)cos()]([ ttttmAtttmA ccccc
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Envelop Detector
+
-
1vCharges Discharge
CR
Time Constant = RC
c
RC
1
BRC
2
1
3v 4v
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Frequency Division Multiplexing
1c
SingnalFDM
1c
1
2c 3c 4c2c3c4c
Ch. 1 Ch. 2 Ch. 3 Ch. 4
Bandwidth/Channel needed = 2B
(where B is the bandwidth of the signal)
Superheterodyne AM Receiver?
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Amplitude Modulation – SSB-SC
Modulation: )()( Mtm
)]()([2
1)cos()()(mod cccDSB MMttmtS
)]()()()([2
1mod ccccSSB MMMMS
c c
2
1
Bc Bc Bc Bc
)(M
BB
)(mod DSBS
1
)( cM
)( cM )( cM
)( cM
same for real signals
=> waste of bandwidth
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c c
2
1
Bc Bc Bc Bc
)(mod DSBS
)( cM
)( cM )( cM
)( cM
c c
2
1
Bc Bc Bc Bc
)(mod SSBS
)( cM )( cM
c c
2
1
Bc Bc Bc Bc
)(mod SSBS )( cM )( cM
AM-SSB can potentially double the useable channel bandwidth
What we need to accomplish this?
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)(tm
)cos( tc
Modulating signalSSB Signal
Modulating carrier
Sharp cut-off BPF
c c
2
1
Bc Bc Bc Bc
)(mod DSBS
A sharp cut-off bandpass filter!
Not practical to make!
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More Practical Bandwidth Saving
Amplitude Modulation Formats
• Vestigial Side Band (VSB)
increases almost 50% effective bandwidth
used in TV
Same as SSB but uses a practical bandpass filter
• Quadrature Amplitude Modulation (QAM)
Doubles the effective bandwidth
Uses in limited bandwidth channel applications
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c c
2
1
Bc Bc Bc Bc
)(mod SSBS
c c
2
1
Bc Bc Bc Bc
)(mod VSBS
c c
2
1
Bc Bc Bc Bc
c c
2
1
Bc Bc Bc Bc
AM-Single Side Band
Bandwidth needed = B
AM-Vestigial Side Band
Bandwidth needed ~ 1.5B
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Quadrature Amplitude Modulation
)sin()()cos()()( 21 ttmttmtS ccQAM
2
~
)(1 tm
)(2 tm
tccos
)(tSQAM
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Quadrature Amplitude Demodulation
)sin()()cos()()( 21 ttmttmtS ccQAM
2
~ tccos
LPF )(1 tx
)(2 tx
)(tSQAM
LPF
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)cos()]sin()()cos()([)( 211 tttmttmtx ccc
)sin()()cos()()( 21 ttmttmtS ccQAM
)cos()sin()()(cos)( 2
2
1 tttmttm ccc
)cos()sin()()2cos()(2
1)(
2
1211 tttmttmtm ccc
will be filtered by lowpass filter
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)sin()]sin()()cos()([)( 212 tttmttmtx ccc
)sin()()cos()()( 21 ttmttmtS ccQAM
)(sin)()sin()cos()( 2
21 ttmtttm ccc
)2cos()(2
1)(
2
1)sin()cos()( 221 ttmtmtttm ccc
will be filtered by lowpass filter
)(2
1)2cos()(
2
1)sin()cos()( 221 tmttmtttm ccc
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Concept of Generalized Angle
)(cos)( tAt
)cos()( 0 tAt cSpecial Case
dt
dti
)(
t
i dxxt )()(
Instantaneous Frequency
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Phase Modulation
)cos()( tAt c
)()( tmktt pc
)](cos[)( tmktAt pcPM
)()( tmkdt
dt pci
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Frequency Modulation
)cos()( tAt c
)()( tmkt fci
t
fc dxxmkt )]([)(
t
fc dxxmkt )(
])(cos[)(
t
fcFM dxxmktAt
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PM and FM - Inseparable
PM)(tm dxxm )(
)(tFM
FM)(tm
)(tm
dt
d )(tPM
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Example 1
)(tm
)(tm
1
-1
2x10-4
t
t
20,000
-20,000
5102 fk
10pk
MHzfc 100
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Example 2
)(tm
t
1
-1
5102 fk
2
pk
)(tm
t
)(2 t
)(2 t
MHzfc 100
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Bandwidth of FM Signal
])(cos[)(
t
fcFM dxxmktAt
])()(
t
dxxmta
)]([)(ˆ taktj
FMfcAet
)(tajktj fc eAe
Let’s define
and
then )](ˆRe[)( tt FMFM
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)()(ˆ tajktj
FMfc eAet
)](!
...)(!2
)(1[ 2
2
tan
kjta
ktajkAe n
n
fnf
f
tj c
]sin)][cos(!
...)(!2
)(1[ 2
2
tjttan
kjta
ktajkA cc
n
n
fnf
f
)](ˆRe[)( tt FMFM
...]sin)(!3
cos)(!2
sin)([cos 3
3
2
2
ttak
ttak
ttaktA c
f
c
f
cfc
If m(t) is band limited to B, then Bandwidth of an(t) is nB
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...]sin)(!3
cos)(!2
sin)([cos)( 3
3
2
2
ttak
ttak
ttaktAt c
f
c
f
cfcFM
0.1)( tak fif
]sin)([cos)( ttaktAt cfcFM then
]sin)([cos)( ttmktAt cpcPM Similarly
Narrow Band FM and PM (NBFM or NWPM)
Bandwidth of narrowband FM or PM is 2B
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Generation of NBFM or NBPM
]sin)([cos)( ttaktAt cfcFM
]sin)([cos)( ttmktAt cpcPM
tA ccos
NBPM)(tm
~
2/
DSB-SC
Modulator
tA csin
tA ccos
NBFM)(tm
~
2/
DSB-SC
Modulator
tA csin
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]sin)([cos)( ttaktAt cfcFM
Distortion Estimate in NBFM Signals
)](cos[)( tttAE c
)(1)( 22 taktE f
)]([tan)( 1 takt f
where
Amplitude distortion can be
minimized by bandbass limiter
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Phase Distortion
)]([tan)( 1 takt f
)(1
)()()(
22 tak
taktt
f
f
i
)(1
)(22 tak
tmk
f
f
...])()(1)[( 4422 taktaktmk fff
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Example of Tone Modulation
...])()(1)[()( 4422 taktaktmkt fffi
ttm m cos)(
m
mtta
sin)(
m
fk
...]sinsin1(cos 4422 ttt mmmm
)sin1(cos)( 22 ttt mmmi
)2
]2cos1[1(cos 2 t
t mmm
B
FLet’s define
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tt mm
mm
3cos4
cos)4
1(32
)2
]2cos1[1(cos 2 t
t mmm
)2
2cos
21(cos
22 tt m
mm
2
2coscoscos
2cos
33 tttt mmm
mm
mm
4
)3cos(coscos
2cos
33 tttt mmm
mm
mm
13cos4
cos3
fortt mm
mm
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B4 B4
pfc mk pfc mk)( kfc tmk
)( ktmpm
pm
t
)(tm
pf mk2
B2
1
t
Wide Band FM (WBFM)
c
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Wide Band FM (WBFM)
)2(2 BfBFM
2
pf mkf
Bf BBFM 4if then
This bandwidth is slightly more than the actual because this
corresponds to the staircase approximation of the signal
Let’s make the adjustment, we know that
Which is not right from our NBFW analysis,
we know that
BBFM 2
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Wide Band FM (WBFM)
Bf fBFM 2
B
f )1(2 BBFMif we define then
)(2 BfBFM
Now, if then
)2(2 BfBFM Therefore,
Should change to
And if Bf BBFM 2then
WBFM
NBFM
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Wide Band PM (WBPM)
)(2 BfBPM
2
ppmkf
)1(2 BBPM
)()( tmkt pci
B
f
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Example 1
)(tm
)(tm
1
-1
2x10-4
t
t
20,000
-20,000
5102 fk
10pk
1. What is the bandwidth of
corresponding FM or PM
signals ?
2. What if m(t) is doubled in
magnitude?
3. What if the period is doubled?
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Understanding the Spectrum of FM
)]([)(ˆ taktj
FMfcAet
We previously defined
Also, we know
ttm m cos)(
m
mtta
sin)(
)]sin([
)(ˆt
ktj
FM
mm
fc
Aet
fpf kmk
m
fk
B
fNow
Let’s use the example of tone modulation
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)]sin([)(ˆ ttj
FMmcAet
][)sin( tjtj mc eAe
n
tj
n
tj mm eCe sin
Now
Where )(2
/
/
sin
n
tjntmn JdteeC
m
m
mm
tjn
n
n
tj
FMmc eJAet
)()(ˆ
Bessel Function
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tjn
n
n
tj
FMmc eJAet
)()(ˆ
)()(
tntj
n
nmceJA
)](ˆRe[)( tt FMFM
Remember we were interested in
)cos()()( tntJAt mc
n
nFM
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tA ccos
)(tm
2/
DSB-SC
Modulator
tA csin
OSC
200kHz
Frequency
Multiplier
x64
Frequency
Multiplier
x48
Frequency
Converter
Power
Amplifier
OSC
10.9MHz
Hzf
kHzfc
25
2001
kHzf
MHzfc
6.1
8.121
kHzf
MHzfc
6.1
9.11
kHzf
MHzfc
8.76
2.911
Wideband FM Generation - Indirect
NBFMFrequency
MultiplierWBFM
)(tm
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Wideband FM Generation - Direct
Voltage Controlled Oscillator
LC
10
can be varied with bias voltage
can be varied with current through coil
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Demodulation of FM
])(cos[)(
t
fcFM dxxmktAt
]})(cos[{)(
t
fcFM dxxmktAdt
dt
])(sin[)]([
t
fcfc dxxmkttmkA
Envelope
Detection
)(tFM
dt
d )]([ tmkA fc
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Demodulation of PM
)](cos[)( tmktAt pcPM
)]}(cos[{)( tmktAdt
dt pcPM
)](sin[)]([ tmkttmkA pcpc
Envelope
Detection
)(tFM
dt
d
)]([ tmkA pc
)(tmAk p
)(tmAk pC
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Demodulation of FM – Phase Lock Loop
VCO
)(H)(tx
)()(0 tmte )](sin[ ttA ic
)](cos[ 0 ttB c
)(0 tcecVCO
)](cos[)](sin[)( 0 ttttABtx cic
)]}()(2sin[)]()({sin[2
00 tttttAB
ici
t
fi dmkt )()(
Suppressed by loop filter, )(H
)()(0 tcet o
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So the effective input to the loop filter will be
))]()(sin[2
)( 0 ttAB
tx i
)()()( 0 ttt ie where
)](sin[2
tAB
e
)()()(0 ttt ei
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)]()([ 0 tcet o
)()()(0 ttt ei
])()([
t
fi dmkt )()()( tdmkt e
t
fo
)()( tmkt fo
For small error
Now
)()( tmc
kte
f
o
For FM Signal
t
fo dmkt )()(
Taking derivative on both sides
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)]()([ 0 tcet o
)]()([ tmkt pi )()()( ttmkt epo
)()( tmkt po
For small error
Now
)()( tmc
kte
p
o
For PM Signal
You need an integrator after PLL for PM demodulation
)()( tmkt po
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Interference in Angle Modulated Signals
tItAtr cc )cos(cos)(
Desired signal Interfering signal
ttIttItA ccc sinsincoscoscos
)](cos[)( tttE dcr
ttIttIA cc sinsincos)cos(
22 )sin()cos()( tItIAtEr
tIA
tItd
cos
sintan)( 1
where
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tA
It
A
Itd sinsintan)( 1
tIA
tItd
cos
sintan)( 1
When I<<A
)](cos[)()( tttEtr dcr
Remember original signal
tA
Ityd sin)(
tA
Ityd
cos)(
For PM
For FM
After passing through PM or FM demodulator