1057: Principles of Communication System (I)twins.ee.nctu.edu.tw/courses/commu_07/Lecture 03A-...
Transcript of 1057: Principles of Communication System (I)twins.ee.nctu.edu.tw/courses/commu_07/Lecture 03A-...
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-1
1057: Principles of Communication System (I)
1057: Principles of 1057: Principles of Communication System (I)Communication System (I)
Lecture 3Lecture 3--I Basic Modulation I Basic Modulation TechniquesTechniques
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-2
Modulation• A process to translate the information data, the message
signal, to a new spectral location depending on the intended frequency for transmission.
• Modulation techniques– The choice is influenced by the characteristics of the message
signal, the characteristics of transmitted channel, the desired performance, the use to be made o the transmitted data, and the economic factors,…
• Basic modulation techniques– Analog modulation
• Continuous-wave modulation– Linear Amplitude Modulation– Frequency Modulation– Phase Modulation
• Pulse modulation
m(t) xc(t)
modulated carrier
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-3
Linear and Angle Modulation• A general modulated carrier can be represented by
where ωc is carrier frequency• Once the carrier frequency is specified, only two parameters
are candidates to be varied in the modulation process– The instantaneous amplitude: A(t)– The instantaneous phase deviation: φ(t)
• When A(t) is linearly related to the modulating signal, it is called the linear modulation
• When φ(t) is linearly related to the modulating signal, it is called the frequency or phase modulation (or just the angle modulation)
)](cos[)()( tttAtx cc φω +=
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-4
Linear Double-Sideband (DSB) Modulation
• Definition:• Spectrum of a DSB signal
ttmAtx ccc ωcos)()( =
πω2
),(21)(
21)( c
cccccc fffMAffMAfX =−++=
fC
fC+W fC-W f
XC(f)
W -W f
M(f)
Upper sideband(USB)
Lower sideband(LSB)
ℑ
messagesignal
fC
fC+W fC-W f
XC(f)
Upper sideband(USB)
Lower sideband(LSB)
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-5
DSB Coherent Demodulation• Coherent (synchronous)
demodulator– The receiver side knows
(in advance) exactly the phase and frequency of the received signal
ttmAtmAtttmAttxtd
cCC
ccCcc
ωωωω
2cos)()( cos2]cos)([cos2)()(
+=⋅=⋅=
desired part High freq. noise
- What if the receiver reference is not coherent?
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-6
DSB ModulationPut it together…
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-7
Noise in Modulation Systems• Details in Chap. 6• A phase-locked loop (PLL) is considered in 3.4• A simplified analysis for a time-varying phase error
2cos(ωct+θ(t))
xr(t) d(t)
))(2cos()()(cos)( ))(cos(cos)(2)(
tttmAttmAttttmAtd
cCC
ccC
θωθθωω
++=+⋅=
1)(cos1 )(cos)()( ≤≤−= tttmtyD θθ
varying !!
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-8
A Phase Coherent Demodulation Carrier Generator
( )2
Narrowband BPF at 2fc
2÷f xr(t)
cos2ωct cosωct
ttmAtmAttmAtx cCCcCr ωω cos2)(21)(
21cos)()( 22222222 +==
DC
0 2fc -2fc
FT(m2(t))
Narrow BPF
f
Frequency divider
Carrier recovery circuit
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-9
Remarks• The spectrum of DSB signal does not contain a
discrete spectral component at the carrier frequency unless m(t) has a DC component.
• DSB systems with no carrier frequency component present are often referred to as suppressed carrier (SC) systems.
• If the carrier frequency is transmitted along with DSB signal, the demodulation process can be rather simplified.
• Alternatively, let’s see the following amplitude modulation (AM) scheme.
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-10
Amplitude Modulation (AM)• A DC bias A is added to m(t) prior to the modulation
process– The result is that a carrier component is present in the
transmitted signal• Definition
ttamAtAtmAtx
cnc
ccc
ωω
cos)](1[ cos)]([)(
+=
′+=
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-11
Remarks
• The parameter a is known as the modulation index• Coherent AM DSB demodulation
– precise, but it requires carrier recovery circuit• Incoherent detection
– Envelope detection
-fc fc 0 0
AM
ttamAtx cnCc ωcos)](1[)( +=
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-12
Envelope Detection
mn(t): the normalized message
A
tma t
)(min=
a: the modulation index
• The modulation index is defined such that if a=1, the minimum value of Ac[1+amn(t)] is zero– a <1, it results in Ac[1+amn(t)] >0 for all t
• All the information is just the envelop.• The envelop detection is a simple and straightforward
technique
ttamAtx cnCc ωcos)](1[)( +=
,)(min
)()(tm
tmtmt
n =
m(t): the original message
A: the DC bias
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-13
Remarks• The time constant RC of the envelop detector is
an important design parameter.• The appropriate RC time constant is related to
the carrier frequency fc and to the bandwidth W of the original signal m(t)– 1/fc << RC << 1/W, between then and must be well
separated from both
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-14
Power Efficiency of AM• Suppose that m(t) has zero mean, then the total power
contained in the AM modulator output is
• The power efficiency is defined by the power ratio of the input information to the transmitted signal
])([21
])()(2[21
2cos)()]([21)()]([
21
cos)()]([)(
222
222
2222
2222
tmAA
tmtmAAA
tAtmAAtmA
tAtmAtx
C
C
cCC
cCc
+′=
++′=
′++′+=
′+=
ω
ω
%100)(1
)(%100
)(
)(Efficiency
22
22
22
2
×+
=×+
≡≡tma
tma
tmA
tmE
n
n
⟨⋅⟩ denotes the time average value
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-15
Remarks• If the signal has symmetrical value, i.e. |minm(t)|=|maxm(t)|,
then |mn(t)|≤1 and hence ⟨mn2(t)⟩≤1.
– If a≤1, the maximum efficiency is 50%, e.g. the square wave-type
– For a sine wave, ⟨mn2(t)⟩=1/2, for a=1, the efficiency is 33.3%
– If we allow a>1,• Efficiency can exceed 50%, (a→∞, the efficiency=100%)• But, the envelope detector is precluded.
• The main advantage of AM is– A coherent reference is not necessary for demodulation as long
as a≤1• The disadvantage of AM
– The DC value of the message signal m(t) cannot be accurately recovered. (mixed with carrier)
%)100()(1
)(%)100(
)(
)(22
22
22
2
tma
tma
tmA
tmEfficiencyE
n
n
+=
+≡≡
Figure 3.4
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-16
Single-Sideband Modulation• In DSB, either the USB and the LSB have equal
amplitude and odd phase symmetry about the carrier frequency.– Each sideband contains sufficient information to
reconstruct m(t)– Bandwidth utilization is not efficient– Single-sideband (SSB) modulation is considered
• SSB modulation– A more complex signal processing technique (X)– It reduces the bandwidth of the modulator (O)– Two methods
• (Bandpass) Filtering easy to understand, but hard to implement• Hilbert transform or frequency transformation technique
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-17
Filtering SSB Modulation
Sideband filtering
• An ideal passband filter is necessary• The (very) low frequency component will be encapulated
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-18
Generation of Lower-Sideband Filter
• From the inverse Fourier transform…
)]sgn()[sgn(21)( ccL fffffH −−+=
)(2
)(2
)( cC
cC
DSB ffMAffMAfX −++=
)]sgn()()sgn()([4
)]()([4
)]sgn()()sgn()([41
)]sgn()()sgn()([41
)()()(
ccccC
ccC
ccccC
ccccC
LDSBc
ffffMffffMA
ffMffMA
ffffMffffMA
ffffMffffMA
fHfXfX
−−−+++
−++=
−−+−+−
+−+++=
⋅=
Xc(f)
XDSB(f)
ttmAffMffMAc
ccc
c ωcos)(2
)]}()([4
{1 =−++ℑ−
ttmAeejtmA
etmjetmjA
ffffMffffMA
cCtfjtfjC
tfjtfjC
ccccc
cc
cc
ωππ
ππ
sin)(ˆ2
)](21)[(ˆ
2
])(ˆ)(ˆ[4
)}sgn()()sgn()({4
22
22
1
=−=
−=
−−−++ℑ
−
−
−
Hilbert Transform
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-19
Time-Domain Representation of SSB Modulation
ttmAttmAtx cC
cC
c ωω sin)(ˆ2
cos)(2
)( +=
ttmAttmAtx cC
cC
c ωω sin)(ˆ2
cos)(2
)( −=
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-20
Hilbert Transform• Consider a filter that simply phase-shift all frequency
components of its input by -½π, that is
⎪⎩
⎪⎨
⎧
<−=>
=0,10,00,1
)sgn(fff
f
tth
π1)( =
fjfH sgn)( −=
t
h(t)
j
-j
f
H(f)
π/2
f
∠H(f)
-π/2
Not abs-integrable, since the value is infinite at t→0
⎩⎨⎧
<−>
=−
0,0,
)(fefe
fGf
f
α
α
αconsiderProof:
)sgn()(lim0
ffG =→ αα
∫ ∫∞
∞−
−−
+=−==ℑ
0
0
22221
)2(4);()]([
ttjdfeedfeetgfG ftjfftjf
παπα παπα
αthen
tj
tj
tgfG
ππαπ
α
α
ααα
=+
=
=ℑ
→
→
−
→
220
0
1
0
)2(4lim
);(lim)]([lim
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-21
Hilbert Transform
• Example 2.26
)(txt
thπ1)( = )(ˆ tx
∫∫∞
∞−
∞
∞−
−=
−= τ
πτττ
τπτ dtxd
txtx )(
)()()(ˆ
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-22
Properties of Hilbert Transform• The function is defined as the Hilbert transform of x(t),
then • The energy (or power) in a signal x(t) and its Hilbert
transform are equal.
• A signal and its Hilbert transform are orthogonal
• If c(t) and m(t) are signals with nonoverlapping spectra, where m(t) is lowpass and c(t) is highpass, then
)(ˆ tx)()(ˆ̂ txtx −=
)(ˆ tx2222222
)()()sgn()()()](ˆ[)(ˆ fXfXfjfXfHtxfX =−==ℑ=
∫∫ ∫
∞
∞−
∞
∞−
∞
∞−
==
=
0)()sgn(
)(ˆ)()(ˆ)(
2
*
dffXfj
dffXfXdttxtx
)]([)()]()([ tcHTtmtctmHT =
by Parseval’s Theorem
Check spectra density:
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-23
Proof:
∫ ∫∞
∞−
∞
∞−
′+
−−
′′=
ℑℑ=
fdfdefCfM
fCfMtctmtffj )(2
11
)()(
)]([)]([)()(π
where we assume that M(f)=0 for |f|>W and C(f’)=0 for |f’’|<WThen
)(ˆ)(
)]sgn()[()(
)]sgn()[()(
)]sgn()[()(
)]sgn()[()(
][)()()]()([
22
22
)(2
)(2
)(2
tctm
fdefjfCdfefM
fdfdefjfCefM
fdfdefjfCfM
fdfdeffjfCfM
fdfdeHTfCfMtctmHT
tfjftj
tfjftj
tffj
tffj
tffj
=
′′−′=
′′−′=
′′−′=
′′+−′=
′′=
∫ ∫∫ ∫∫ ∫∫ ∫∫ ∫
∞
∞−
∞
∞−
′
∞
∞−
∞
∞−
′
∞
∞−
∞
∞−
′+
∞
∞−
∞
∞−
′+
∞
∞−
∞
∞−
′+
ππ
ππ
π
π
π
By(2.282)
W
f+f ’=0
f
f ’
A
B
sgn(f+f ’) = 1
sgn(f+f ’) = -1
Example 2.27
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-24
Analytic Signal• Definition :
– for any real signal x(t), the analytic signal of x(t) is
• Then)(ˆ)()( txjtxtxp +=
⎩⎨⎧
<>
=
+=−+=
0,00),(2
]sgn1)[()](sgn[)()(
fffX
ffXffXjjfXfX p
)(ˆ)()( txjtxtxn −=
⎩⎨⎧
<>
=
−=
0),(20,0
]sgn1)[()(
ffXf
ffXfX n
Not even…
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-25
Equivalent Bandpass Signal Representation
• Bandpass signal:)(~ fX
B/2-B/2
2A
ff0
X(f)
-f0
( ))(2cos)()( 0 ttftatx θπ +=
( )
tftxtftxtfttatftta
ttftatx
IR 00
00
0
2sin)(2cos)( 2sin)(sin)(2cos)(cos)(
)(2cos)()(
πππθπθ
θπ
−≡−=
+=
ℑ
⎪⎩
⎪⎨
⎧
=
+=− )
)()((tan)(
)()()(1
22
txtxt
txtxta
R
I
IR
θwhere
inphase quadrature
)( of envelopcomplex thecalled is )(~ where})(~Re{})(Re{)( 00 2)(2
txtxetxetatx tfjttfj πθπ == +
A
carrier
zero-frequency parts of x(t), lowpass signals
complex envelop
)()()()(~ )( tjxtxetatx IRtj +== θ
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-26You may check the other direction !!
Remarks• The complex envelope of an arbitrary bandpass signal is not
a real existing signal. It is just an equivalent expression.
)()()(~)()(
)(ˆ)(
Re
02
02
txtxtxtxtx
txjtxP
e
e
I
R
tfj
tfj
⎯⎯⎯ ⎯←⎯⎯⎯ →⎯
⎯⎯ ⎯←⎯⎯ →⎯
↔+− π
π
tftxtftxtftxtftxtxtftxtftxtx
IRIR
IR
0000
00
2cos)(2sin)()2cos)((2sin)()(ˆ2sin)(2cos)()(
ππππππ
+=−−=−=
tfjtfjIR
tfjI
tfjR
IRP
etxetjxtxetjxetx
tftfjtxtfjtftxtxjtxtx0000 2222
0000
)(~)]()([)()(
]2sin2cos)[(]2sin2)[cos()(ˆ)()(ππππ
ππππ
=+=+=
−++=+=
tfjP etxtx 02)()(~ π−=
Proof:
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-27
Bandpass Systems• Physical
• Equivalent lowpass system
h(t)input
BP output
Bandpass
input Complex envelopoutput
Complex envelope
real(real BP)
(complex envelope)
)(tx
)(~ th
)()()( thtxty ∗=
)(~ tx)(~)(~)(~2 txthty ∗=
tfjtfj
tfjtfjtfj
etheth
ethethethth
00
000
2*2
*222
)(~21)(~
21
])(~[21)(~
21})(~Re{)(
ππ
πππ
−+=
+==
tfjtfjtfj etxetxetxtx 000 2*22 )(~21)(~
21})(~Re{)( πππ −+==
Note:
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-28
∫∫
∫∫
∫
∫
∞
∞−
−∞
∞−
−
∞
∞−
−∞
∞−
−−−∞
∞−
−
∞
∞−
−+−+
−+−=
⎟⎠⎞
⎜⎝⎛ −+−⎟⎠⎞
⎜⎝⎛ +=
−=∗=
λλλλλλ
λλλλλλ
λλλλλ
λλλ
λππλππ
ππ
λπλπλπλπ
detxhedetxhe
dtxhedtxhe
detxetxeheh
dtxhthtxty
fjtfjfjtfj
tfjtfj
tfjtfjfjfj
0000
00
0000
4*24*2
**22
)(2*)(22*2
)(~)(~41)(~)(~
41
)(~)(~41)(~)(~
41
)(~21)(~
21)(~
21)(~
21
)()()()()(
0)}(~)2(~{)(~)(~)(~)(~ *0
1*44* 00 =−−ℑ=∗=− −∞
∞−∫ fXffHtxethdetxh tfjfj πλπ λλλQ
( ) ( )( ){ }
})(~Re{
)(~)(~Re21
)(~)(~41)(~)(~
41
)(~)(~41)(~)(~
41)(
0
0
00
00
2
2
**22
**22
tfj
tfj
tfjtfj
tfjtfj
ety
etxth
txthetxthe
dtxhedtxhety
π
π
ππ
ππ λλλλλλ
=
∗=
∗+∗=
−+−=
−
∞
∞−
−∞
∞− ∫∫
Example 2.29
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-29
Physical Implementation of BP System
hR(t)
hI(t)
hI(t)
hR(t)
+
-
2yR(t)
+
+
2yI(t)
xR(t)
xI(t)
)(~)(~)(~2 txthty ∗=
tftytftytytftxtftxtxtfthtfthth
IR
IRIR
00
0000
2sin)(2cos)()(2sin)(2cos)()( ,2sin)(2cos)()(
ππππππ
−=−=−=
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-30
From Analytic Signal to SSB Signal• Mp(f): the positive-
frequency portion of M(f)
• Mn(f): the negative-frequency poriton of M(f)
• Apply the frequency-translation theorem to both the Mp(f) and Mn(f)– We obtain the upper-
sideband SSB signal and lower-sideband SSB signal, respectively
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-31
Coherent SSB Demodulation• Coherent detection with possibly phase error θ(t)
LPF
Kcos(ωct+θ(t))
d(t) xc(t) yd(t)
fc -fc
f
Xc(f)
2fc -2fc
f
LPF tfK 02cos π×
( )
)](2sin[)(ˆ)](2cos[)()(sin)(ˆ)(cos)(
))(cos(4sin)(ˆcos)(2
))(cos(4)()(
tttmAtttmAttmAttmA
ttttmttmAtttxtd
cCcCCC
cccC
cc
θωθωθθ
θωωωθω
+±++=
+×±=+×=
m
)(sin)(ˆ)(cos)()( ttmttmtyD θθ m=
if K=4,
message crosstalk
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-32
Remarks• If θ(t)≠0, the first term is a time-varying
attenuation of the message signal; while the second term is crosstalk.
• In general, there exist both frequency error Δfand phase error θ(t) in the local carrier, then– For LSB:
– For USB:( ) ( ))(2sin)(ˆ)(2cos)()( tfttmtfttmtyD θπθπ +Δ−+Δ=
( ) ( ))(2sin)(ˆ)(2cos)()( tftttmtfttmtyD θπθπ +Δ++Δ=
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-33
Carrier Insertion with Envelope Detector
( )
ttbtta
ttmAtKtmA
tKttmttmAte
cc
cC
cC
cccC
ωω
ωω
ωωω
sin)(cos)(
sin)(ˆ2
cos)(2
cossin)(ˆcos)(2
)(
±≡
±⎟⎠⎞
⎜⎝⎛ +=
+±=
• A complicated envelope detector is necessary
⎪⎩
⎪⎨
⎧
=
+=
−
)()(tan)(
)()()(
1
22
tatbt
tbtatR
θdefine
⎩⎨⎧
==
)(sin)()()(cos)()(
ttRtbttRta
θθ
that is,
22
)(ˆ2
)(2
)()( ⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ +== tmAktmAtRty CC
D ktmAty CD +≈ )(
2)(
K large enough
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-34
Example 3.2•
⎩⎨⎧
+−=+−=
)3sin(9.0)2sin(4.0)sin()(ˆ)3cos(9.0)2cos(4.0)cos()(
111
111
ttttmttttm
ωωωωωω
)](cos[)(]sin)(ˆcos)([2
)( tttRttmttmAtx cccC
c θωωω +=±=
⎪⎪⎩
⎪⎪⎨
⎧
±=
+=
−
)()(ˆ
tan)(
)(ˆ)(2
)(
1
22
tmtmt
tmtmAtR C
θ
})()(ˆ
{tan))(( 1
tmtm
dtdtt
dtd
cc−±=+ ωθω
The instantaneous frequency of xc(t)
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-35
Remarks• Problems of SSB modulation
– The ideal (sharp cutoff) sideband filter– The poor low-frequency response
• Vestigial-sideband (VSB) filter– A filter that is mean-shifted conjugate anti-symmetric about the
carrier fc– An example:
• Let Hβ(f) be an LP anti-symmetric filter• Define the VSB filter H(f)
⎩⎨⎧
<++−−>−−−
=0),()(0),()(
)(fffHffUfffHffU
fHcc
cc
β
β
Hβ(f)
-1/2
1/2
β -β f
Hβ(f) = - Hβ(-f) ; Hβ(f) = 0 for |f| > β.
H(f)
1/2 U(f-fc)
fc+β f
fc-β fc
• Two advantages of VSB filter– The design of sideband filter is simplified– The low-frequency response is improved
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-36
Vestigial-Sideband Modulation•
f fc -fc
M(f-fc) M(f+fc)
f
M(f)
f fc -fc
M(f-fc)⋅H(f) M(f+fc)⋅H(f)
f
DSB
VSB
LPF
cos(ωct)
d(t)xc(t) yd(t)
( ) )()()()( fHffMffMfX ccc ⋅−++=
( ) ( ) )()2()(21)()()2(
21
)}({)(
cccc ffHffMfMffHfMffM
tdfD
−−+++++=
ℑ=
Demodulation
LPFafter )]()([)(
21)( ccD ffHffHfMfY −++=
Commun. I Lecture3 - Basic Modulation Techniques ([email protected])
3-I-37
Example
tBtA
tAte
cc
c
)cos(21)cos()1(
21
)cos(21)(
22
1VSB
ωωωωε
ωωε
+++−+
+=
tBtAtm 21 coscos)( ωω +=
tBtB
tAtAte
cc
cc
)cos(21)cos(
21
)cos(21)cos(
21)(
22
11DBS
ωωωω
ωωωω
−+++
−++=
The bandwidth required for VSB over the required for SSB/DSB is slightly increased by an offset
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Frequency Translation and Mixing• The process of multiplying a message by a
periodic signal is called mixing.– The mixing effect: frequency translation– Example: A mixer
ttmttmte )2cos()(cos)()( 212 ωωω ±+=
The undesired term is removed by filter
The problem of the mixer:signal at the image frequency
ttk )2cos()( 21 ωω ±Let’s see the input signal:
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Image Frequency
The image frequency must be eliminated !!
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Image Frequency in Superheterodyne Receiver
• The fact is that it is hard to built a narrow bandpass filter at high frequency• The superheterodyne receiver has two amplification and filtering sections
prior to demodulation– A tunable RF filter followed by a fixed IF filter
• If we are attempting to receive a signal having carrier frequency ωc, we will also receive a signal at ωc+2ωIF if the local frequency is ωc+ωIF (or receive a signal at ωc−2ωIF if the local frequency is ωc−ωIF )
• The image frequency can be eliminated by the RF filter, which needs not be narrowband.
Tunable
Fixed
The mixer translates the input frequency ωc to the IF frequency ωIF
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Remarks
• At the antenna: the desired signal with carrier ωc• After the RF filter: only the desired signal at ωc can go through (a wide BPF)• After the mixer: the desired signal at ωIF• After the IF filter: only the desired signal at ωIF can go through (a narrow
BPF) • The IF frequency is almost fixed• Two choices:
– Low-side tuning– High-side tuning
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Smaller tuning range of LO is preferred.
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Angle Modulation• The information is embedded in the angle (either in phase
or in frequency) of the sinusoid– Constant amplitude– General form:
– The phase deviation:
– The frequency deviation:
• Phase modulation (PM):
• Frequency modulation (FM):
dttd )(φ
)](cos[)( ttAtx cCc φω +=
)(tφ
)()( tmKt P ⋅=φ
)()( tmKdt
tdF ⋅=
φ
0000
)(2)()( φααπφααφ +=+= ∫∫t
td
t
tF dmfdmKt
Kp: deviation constant
fd: freq. deviation constant
)](cos[)( tmKtAtx PcCc += ω
])(2cos[)(0∫+=
t
tdcCc dmftAtx ααπω
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Example
Phase changes
frequency changes
)](cos[)( tmKtAtx PcCc += ω
])(2cos[)(0∫+=
t
tdcCc dmftAtx ααπω
PM
FM
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NarrowBand Angle Modulation•
ttAtAetjAeA
tjeA
tjtjeA
eeAttAtx
cCcCtj
Ctj
C
tjC
tjC
tjtjCcCc
cc
c
c
c
ωφωφ
φ
φφ
φω
ωω
ω
ω
φω
sin)(cos])(Re[
)]}(1[Re{
!2))(()(1Re
}Re{)](cos[)(2
)(
−=+=
+≈⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡+++=
=+=
L
If |φ(t)| is much less than unity
Tayler series expansion..
carrier message×sinωc
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If the input is m(t)=Acosωmt
The carrier and the resultant of the sidebands for narrowband angle modulation with sinusoidal modulation are in phase quadrature.
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Frequency-Domain Behavior of Angle Modulation
• Consider the sinusoidal signal– Note that, any message signal can be decomposed into
combinations of sinusoids– WLOG, we may ssume that φ(t)=βsinωmt, and xc (t)=
Accos(ωct+βsinωmt), then
– Then
}Re{)( sin tjtjcc
mc eeAtx ωβω=
Periodic with period 2π/ωm
)(21
21
)sin(
sin
0
sin
0
0
βπ
πω
π
π
β
ωπ
ωπ
ωωβωωβ
nxnxj
tjntjmT tjntjn
Jdxe
dteedteeT
C m
m
mmmm
≡=
==
∫
∫∫
−
−−
−
−−
Its Fourier series coefficients
Bessel function of the 1st kind of order n and argument β
∑∞
−∞=
⋅=n
tjnn
tj mm eJe ωωβ β )(sinby Fourier series
∑∑∞
−∞=
∞
−∞=
+==n
mcnCn
tjnn
tjCc tnJAeJeAtx mc )cos()(})(Re{)( ωωββ ωω
∑∑∞
−∞=
∞
−∞=
+++−−=n
cmncn
cmncc nJAnJAfX )()()()()( ωωωδβωωωδβ
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Frequency-Domain Behavior of Angle Modulation
The spectrum has components at the carrier frequency and has an infinite number of sidebands separated from the carrier frequency by integer multipliers of modulation frequency ωm
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About the Bessel Function Jn(β)• Jn(β) is real-valued function•
•
• When β is small (β<<1) narrowband Angle Modulation
•
⎩⎨⎧
−==
−
−
oddn ),()(evenn ),()(
ββββ
nn
nn
JJJJ
)()(2)(
)(2)()(
11
11
βββ
β
ββ
ββ
−+
+−
−=⇒
=+
nnn
nnn
JJnJ
JnJJ
(by definition)
(proved by induction)
2for 0)( and ,2)( ,1)( 10 ≥≈≈≈ nJJJ n ββββ
1)(2 =∑∞
−∞=nnJ β Note: for angle modulation, we have
carrier nulls: the β values s.t. Jn(β) = 0
Table3.2
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Remarks• The above analysis is based on the assumption that
φ(t)=βsinωmt, and we did not specify the modulator type– PM this means that m(t)=Asinωmt and β=kpA
– FM this means that m(t)=Acosωmt and β=2πfdA/ωm=fdA/fm
• Power in Angle modulated signal∫=
t
d dmft ααπφ )(2)(
( ) ( ) 22222
21sin2cos
21
21)sincos()( CmcCCmcCc AttAAttAtx =++=+= ωβωωβω
Constant transmitter power !!
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Bandwidth of Angle Modulation Signal
• For large n, , hence, for fixed β,
• The bandwidth of an angle modulation signal is infinite, strictly speaking.
• Define the power ratio Pr
• Since the values of Jn(β) become negligible for sufficient large n, the bandwidth of an angle modulation signal can then be determined by an acceptable power ratio, says k, then
!2)(
nJ n
n
nββ ≈ 0)(lim =
∞→βnn
J
∑∑
=
−= +===k
nn
C
k
knnC
r JJA
JAkP
1
220
2
22
)(2)(
21
)(21
power totalpower components ββ
β
98.0for ,)1(22 ≥+≈= rmm PfkfBW β
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General Case• For arbitrary m(t), a general accepted expression for BW is
through the definition ratio D
• For PM– Special case
• For FM– Special case
• The bandwidth is BW≈2(D+1)W
)(
)(max21
)( ofBW deviationfrequency Peak
tm
t
BWdt
td
tmD
φπ=≡
)](cos[)( ttAtx cc φω +=
)()( tmkt P=φtAtm mωsin)( = ,)(max mPt
AKdt
td ωφ=
)()( tmkdt
tdf=
φ))(max(
)(
tmBW
fDt
tm
d=
,sin)( tAtm mω=
AkD p=
m
d
fAfD =
Carson’s rule
1. D<<1, BW≈2W, narrowband angle modulated signal2. D>>1, BW≈2DW=2fd[max|m(t)|] , wideband angle modulated signal
D plays the same role as βfor sinusoidal signal
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Narrowband-to-Wideband Conversion
• Indirect Frequency Modulation--Two stages for generating wideband FM– Narrowband FM– Frequency multiplier
• The mixer output• After BPF
))(cos()( 0 ttAtx C φω +=
))(cos()( 0 tntnAty C φω +=tte LOLO ωcos2)( =
)]()cos[()]()cos[()( 00 tntnAtntnAte LOCLOC φωωφωω +−+++=
LOc
LOc
nn
ωωωωωω
−=+=
0
0 or ,
))(cos()( tntAtx cC φω +=Note: one can use Carson’s rule to determine the BW of the BPF if the transmitted signal is to contain 98% of the power in xc(t)
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• Direct Frequency Modulation: voltage-controlled-oscillator (VCO)• This circuit has an oscillation frequency when x(t) slowly varies
∫+≈t
ccc dxfCCtft ααππθ )(
222)(
0
Narrowband-to-Wideband Conversion
)(cos)( tAtx ccc θ=
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Demodulation of Angle Modulated Signal
• Frequency discriminator– A device that yields an output proportional to the
frequency deviation of the input– Received signal– The output of an ideal discriminator
– For FM
– For PM: Integration of the discriminator output yields a signal proportional to m(t)
• FM discriminator followed by an integrator
))(cos()( ttAtx ccr φω +=
dttdKty DD)(
21)( φπ
= KD: discriminator constant
,)(2)( ∫=t
d dmft ααπφ )()( tmfKty dDD =
linear
)()( tmkKty pDD =
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FM Discriminator
•
• The output of envelope detector
• To reduce the channel noise effect, one applies the following
)](cos[)( ttAtx ccr φω +=
)](sin[)()()( ttdt
tdAdt
tdxte ccCr φωφω +⎟
⎠⎞
⎜⎝⎛ +−==
⎟⎠⎞
⎜⎝⎛ +=
dttdAty cc)()( φω This is always positive, if t
dttd
c ∀−> ,)(φω
CDdCcD AKtmfAdt
tdAty ππφ 2)(2)()( =→==
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Ideal Differentiator•
dtd xr(t) e(t)
H(f)=j2πf
E(f) Xr(f)
H(f)
f
j
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Realization of Discriminator (1)• Time delay block
• RC network
)()()( τ−−= txtxte rr
dttdxte r )()(lim
0=
→ ττ
dttdxte r )()( τ≈
fRCjfRCj
fCjR
RfHπ
π
π21
2
21)(
+=
+=
12 if ,2)( <<≈ fRCfRCjfH ππ
RCAK CD π2=
A highpass filter
For f=10MHz, KD is very small
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Realization of Discriminator (2)• By using BPF
– Linear region ≈ differentiator– Disadvantages
• Small linear region• DC bias (H(f) should be 0 at fc)
• Balanced Discriminator – Advantages
• Wider linear range• No DC bias
f
|H(f)|
fc
H1(f): BPF at fc+Δ
H2(f): BPF at fc-Δ
Envelope detector
Envelope detector
xr(t)
y1(t)
y2(t)
yD(t) +
-
)()()( 21 fHfHfH −=
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