chapter 3. amplitude modulation. start with the sinusoidal modulating wave. single-sideband...
TRANSCRIPT
Chapter 3. Amplitude Modu-lation
• Start with the sinusoidal modulating wave.
Single-Sideband Modulation
2
)2cos()( tfAtm mm
)13.3(])(2cos[2
1])(2cos[
2
1
)2cos()2cos(
)()()(
tffAAtffAA
tftfAA
tmtctS
mcmcmcmc
mcmc
DSB
)14.3(])(2cos[2
1)( tffAAtS mcmcUSSB
)15.3()2sin()2sin(2
1)2cos()2cos(
2
1)( tftfAAtftfAAtS mcmcmcmcUSSB
)16.3()2sin()2sin(2
1)2cos()2cos(
2
1)( tftfAAtftfAAtS mcmcmcmcLSSB
)17.3()2sin()2sin(2
1)2cos()2cos(
2
1)( tftfAAtftfAAtS mcmcmcmcSSB
• Generalize (3.17) to periodic message signals
Single-Sideband Modulation
3
)18.3()2cos()( n
nn tfatm
)19.3()2sin()2sin(2
1)2cos()2cos(
2
1)( tfatfAtfatfAtS n
nnccn
nnccSSB
)20.3()2sin()(
nnn tfatm
)21.3()2sin()(2
)2cos()(2
)( tftmA
tftmA
tS cc
cc
SSB
• Hilbert transform
– and are Hilbert transform pair.
• Given a Fourier transformable and its Hilbert transform , the SSB modulated wave is defined by
Single-Sideband Modulation
4
m t m̂ t h t
ˆˆ , m t m t h t M f M f H f
1Hilbert transform: sgn , H f j f h t
t
1 1sgn
2 2c c c cf f f f j f f f f fj
)23.3()2sin()(2
)2cos()(2
)( tftmA
tftmA
tS cc
cc
Single-Sideband Modulation
5
SSB Modulation
6
• Frequency discrimination method
SSB Modulation
7
• Phase discrimination method
SSB Modulation
8
• Frequency translation
Vestigial Sideband Modulation
9
• Motivation– SSB works well for a message signal with an energy gap
centered around zero frequency.– However, the spectra of wideband signals contain signif-
icant low frequencies.– DSB-SC requires a transmission bandwidth equal to
twice the message bandwidth.
• Vestigial sideband (VSB) modulation– Instead of completely removing a sideband, a trace of
vestige of that sideband is transmitted, the name vesti-gial sideband.
– Instead of transmitting the other sideband in full, almost the whole of this second band is also transmitted.
WfB vT
VSB Filter
10
DSB-SC
SSB
VSB
cfcf W cf W
)26.3(for,1)()( WfWffHffH cc
VSB Modulator
11
)26.3(for,1)()( WfWffHffH cc
Example: Sinusoidal VSB
12
• Let the upper side-frequency at as well as its im-age at be attenuated by the factor .
• From (3.26), the lower side-frequency must be at-tenuated by the factor .
Coherent Detection of VSB
13
)2cos()()( ' tftsAtv cc
)32.3()]()([2
1)( '
ccc ffSffSAfV
)33.3()()]()([2
1)( fHffMffMAfS ccc
)34.3()()]()2([2
1)( cccc ffHfMffMAffS
)35.3()()]2()([2
1)( cccc ffHffMfMAffS
)36.3()]()2()()2([4
1
)(4
1)(
'
'
cccccc
cc
ffHffMffHffMAA
fMAAfV
Baseband Representation of Modulated Waves
14
• Baseband: The band of frequencies representing the original signal as delivered by a source of in-formation
• Consider a modulated wave
– where and – : in-phase component, : quadrature component– and are orthogonal to each other.
• The complex envelope of is defined by
– Fictitious signal to simplify signal processing operations on bandpass signals
Baseband Representation of Modulated Waves
15
• Complex carrier wave
• Modulated wave can be represented by
Baseband Representation of Modulated Waves
16
Baseband Representation of Bandpass Filters
17
• : Transfer function of a bandpass filter– Mid-band frequency: – Bandwidth:
• : Transfer function of the complex low-pass fil-ter
– Keep the part of that corresponds to positive frequen-cies; let denote this part.
– Shift to the left along the frequency axis by an amount equal to .
– Scale it by the factor 2.
• Actual output is determined from the formula
(3.44)0for ),(2)(~
ffHffH c
)45.3()2exp()(Re)(~
tfjtyty c
Baseband Representation of Bandpass Filters
18