chapter 4: angle - auburn university
TRANSCRIPT
Chapter 4
: Angle M
odulatio
n
7/13/2010
1
Intro
ductio
nAngle m
odulatio
nThe an
gle o
f the carrier w
ave is varied acco
rding to th
e based
ban
d w
aveFreq
uen
cy modulatio
n: f(t)=
f(m(t)),φ
(t)=co
nsta
nt
Phase m
odulatio
n: f(t)=
consta
nt,φ
(t)=φ(m
(t))
The carrier am
plitu
de rem
ains th
e same
Perfo
rman
ceMore ro
bust to
noise/in
terference
…but at th
e cost o
f transm
ission ban
dwidth
and
complexity
Practical trad
e‐off b
etween
the th
ree
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2
θi (t) =
2πf(t)t +
φ(t)
Basic D
efinitio
ns
The an
gle θ
i (t)of th
e carrier is a functio
n of th
e inform
ation w
ave
If θ
i (t)increases m
onotonically w
ith tim
e, the averag
e freq
uen
cyover an
interval is
Let Δ
t0,the in
stantan
eous freq
uen
cy of s(t),
measu
red in ra
dians p
er second, is
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3
tt+Δt
Interp
retation of S(t)
As a ro
tating phaso
rLen
gth: A
c
Angle: θ
i (t)
Angular velo
city: dθi (t)/d
t
Unmodulated
signal
θi (t) =
2πfc t +
φc
Angle m
odulatio
nA
cco
nstan
t
Mak
ing eith
er fcor φ
ca
functio
n of m
(t)
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4
θi (t)
Two Commonly U
sed M
ethods
Phase m
odulatio
n (P
M)
The in
stantan
eous an
gle is varied
linearly w
ith m
(t)
kp : p
hase sen
sitivityof th
e modulato
r, in ra
dians p
er vo
lt
Freq
uen
cy modulatio
n (F
M)
The in
stantan
eous freq
uen
cy is varied lin
early with
m(t)
kf : freq
uen
cy sensitivity, in
Hertz p
er volt
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5
Relatio
nship Betw
een PM an
d FM
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6
Equivalen
ce of P
M an
d FM
An FM sig
nal w
ith m
(t)can
be reg
arded as a P
M sig
nal w
ith
A PM sig
nal w
ith m
(t)can
be reg
arded as an
FM sig
nal w
ith
Due to
the in
tegral/d
ifferentiative relatio
nsh
ip betw
een
phase an
d freq
uen
cy
All P
M properties can
be d
educed
from th
ose o
f FM, an
d
vice versa
Only n
eed to stu
dy o
ne o
f these tw
o
dt tm
d)]
([
Relatio
nship Betw
een PM an
d FM
(co
ntd.)
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7
PM
FM:
FM
PM:
dt tm
d)]
([
m(t)
m(t)
Illustratio
n
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8
)2
sin(
)2
cos(2
)(
0
tf
f k
df
kt
mm f
t
mf
i
90ophase
shift
AM:
s(t)=m(t)×
cos(2π
fc t)
PM:
FM:
)2
cos(
)(
)(
tf
k
tm
kt
mp p
i
Property 1
: Constan
cy of Tran
smitted
Power
Modulated
signal h
as constan
t amplitu
de
Averag
e transm
itted power is
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9
2
)](
2cos[
22
}1)]
(2
{cos[2
)](
[cos
1
2
0
22
0
2
0
22
c
Tc
c
Tc
T
cav
A
dtt
AT
T A
dtt
T A
dtt
AT
P
s(t)
1 ohm
T
Property 2
: Nonlinearity o
f the
Modulatio
n Process
AM is lin
ear: s(m1 (t)+
m2 (t))=
s(m1 (t))+
s(m2 (t))=
s1 (t)+
s2 (t)
PM an
d FM are n
ot
e.g
., for P
M:
The p
rincip
le of su
perp
ositio
nis vio
lated, sin
ce
Complicates th
e spectral an
alysis and noise an
alysis
…but su
perio
r noise p
erform
ance: d
ue to
nonlin
earity
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10
Property 3
: Irregularity o
f Zero‐
Crossin
gsZero
‐crossin
g: w
hen s(t) cro
sses zero
No lo
nger h
ave a perfect reg
ularity in
their sp
acing
Also attrib
uted
to th
e nonlin
ear characteristics o
f the
modulatio
n process
FM an
d PM: m
(t)resid
es in th
e zero‐cro
ssings o
f the
modulated
wave s(t)
Usin
g zero
‐crossin
gs to
sample m
(t)
AM, m
(t)resid
ues in
the en
velope o
f s(t)
Usin
g peak
s of c(t)
to sam
ple m
(t)
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11
Property 4
: Visu
alization Difficu
lty of M
essage Wavefo
rmAMm(t)
envelo
pe o
f s(t)
PM an
d FM
s(t)
has co
nstan
t amplitu
de
m(t) is in
the fo
rm of zero
‐crossin
gs,
which is h
ard to visu
alize
Also attrib
uted
to th
e nonlin
earity of th
e modulatio
n
process
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12
large m
(t) value
small m
(t) value
Property 5
: Trade‐O
ff of Tran
smissio
n
Bandwidth an
d Noise
Perfo
rmance
Advan
tage
Im
proved
noise p
erform
ance
Man
y noise an
d in
terference are ad
ditive
Angle o
f a sinusoidal w
ave is less sensitive to
additive n
oise
AM: am
plitu
de is sen
sitive to ad
ditive n
oise
Atten
d at th
e expen
se of larg
er transm
ission
ban
dwidth req
uirem
ent
Trad
e‐off
Exch
anging an in
crease in tran
smissio
n ban
dwidth fo
r an im
provem
ent in
noise p
erform
ance
Not fo
r AM: sin
ce transm
ission ban
dwidth is fixed
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13
Example 4
.1: Zero
‐Crossin
gsLinear m
odulatin
g w
ave
a=1 vo
lt/s, carrier fc =1/4 H
z, kp =π/2 rad
ians/vo
lt
Phase m
odulatio
n
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14
For zero
‐crossin
gs:
Example 4
.1: Zero
‐Crossin
gs (contd.)
Freq
uen
cy modulatio
n
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15
For zero
‐crossin
gs:
Example 4
.1: Zero
‐Crossin
gs (contd.)
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16
Frequen
cy Modulatio
nNonlin
ear modulatio
n:
s(t)
is a nonlin
ear functio
n of m
(t)
Hard
for sp
ectral analysis
Analyze th
e simplest case: sin
gle‐to
ne m
odulatio
n
Inform
ation w
ave:
Instan
taneo
us freq
uen
cy:
Freq
uen
cy deviatio
n:
The an
gle θ
i (t)is:
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17
Modulatio
n in
dex: β
Frequen
cy Modulatio
n (co
ntd.)
Modulatio
n in
dex: β
=Δf/fm
The F
M sig
nal is:
Dep
ending on th
e value o
f β, tw
o cases o
f FM
Narro
w‐ban
d FM, fo
r which β
is small co
mpared
to one
radian
W
ide‐b
and FM, fo
r which β
is large co
mpared
to one
radian
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18
Narro
w‐Band FM
FM sig
nal w
ith sin
gle to
ne m
(t)is
For sm
all β, w
e have
Sim
ilar to an AM sig
nal w
ith th
e same m
(t)
Req
uires th
e same tran
smissio
n ban
dwidth as A
M
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19
differen
ce
If |x|<<1
•cos(x) ≈
1•sin
(x) ≈x
Wide‐B
and FM
For arb
itrary β=Δf/fm
, use th
e complex rep
resentatio
n
of b
and‐pass sig
nals
Assu
ming fc
is large en
ough:
The co
mplex en
velope o
f s(t)is
A perio
dic sig
nal w
ith perio
d 1/
fm an
d fu
ndam
ental
frequen
cy fm:
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20
Wide‐B
and FM
(contd.)
The co
efficient is
Jn (β
): the n‐th order B
assel functio
n of th
e first kind an
d
argumen
t β
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21
Wide‐B
and FM
(contd.)
W
e have
Spectru
m
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22
Exact an
alysis, no
approxim
ation
Bessel Fu
nctio
n
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23
For sm
all β:
Observatio
ns
The sp
ectrum of an
FM sig
nal
n=0: a carrier co
mponen
t
Amplitu
de o
f the carrier is J0 (β
)
n>0 an
d n
< 0: an
infin
iteset o
f side freq
uen
cies located
symmetrically o
n both sid
es of th
e carrier at freq
uen
cy separatio
ns o
f fm, 2
fm, 3
fm, …
Unlik
e AM th
at is ban
d lim
ited, in
finite
ban
dwidth
W
hen β
«1, Jn (β
) ≈0, fo
r n>2: sin
gle p
air of sid
e freq
uen
cies narro
wban
d FM
Total tran
smit p
ower:
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24
Example 4
.3Modulatin
g sig
nal
Modulatio
n in
dex
fm
determ
ines th
e spacin
g
βfm or A
mdeterm
ines in
terval 2Δf
Case I: Fixed
fmIncreased
β
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25
The sp
ectrum on th
e rig
ht h
and sid
e of th
e plan
e. There is a
symmetric p
art on th
e left h
and sid
e.
Example 4
.3
(contd.)
Case II: Decreased
fmFixed
Am
An in
creasing
number o
f spectral
lines cro
wding in
to
the fixed
interval
W
hen β
∞, th
e FM ban
dwidth
2Δf
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26
Transm
ission Bandwidth of FM
An FM sig
nal co
ntain
s an in
finite n
umber o
f side
frequen
ciesLossless tran
smissio
n: in
finite b
andwidth
For p
ractical systems:
W
here to
cut: trad
e‐off b
etween
disto
rtion an
d ban
dwidth
efficiency
First a
ppro
ach –
Carso
n’s R
ule
Observin
g tw
o extrem
es:
Larg
e β:B
T 2Δ
f
Small β
:BT
2(Δf+fm)
An em
pirical estim
ate:
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27
Transm
ission Bandwidth of FM
(co
ntd.)
Seco
nd appro
ach
: universal cu
rveAn estim
ate based
on retain
ing th
e maxim
um number
of sig
nifican
tsid
e frequen
cies
Significan
t: whose am
plitu
des are all g
reater than so
me
thresh
old
e.g
.: 1% of th
e unmodulated
carrier amplitu
de
The sep
aration betw
een tw
o freq
uen
cies beyo
nd
which none o
f the sid
e frequen
cy is greater th
an 1%
of
the u
nmodulated
carrierBT =2×
nmax ×fm
nmax: th
e largest o
ne w
ith |Jn (β
)|>0.01
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28
Finding n
max
Tab
le 4.1
7/13/2010
29
Finding n
max(co
ntd.)
The u
niversal cu
rve
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30
Discu
ssions
For a g
eneral m
(t)with highest freq
uen
cy WEstim
ate usin
g a w
orst‐case to
ne‐m
odulatio
n an
alysis
Deviatio
n ratio
D:
D
is equivalen
t to β
in th
e case of sin
gle‐to
ne m
odulatio
n
W
if equivalen
t to fm
in th
e case of sin
gle‐to
ne m
odulatio
n
Carso
n’s R
ule fo
r transm
ission ban
dwidth:
Carso
n’s ru
le: underestim
ate Lower b
ound
Universal cu
rve: overestim
ate Upper b
ound
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31
W
tm
k
W
fD
fm
ax|)
(|
Df
f
Wf
Wf
BT
11
21
2)
(2
Sam
e form as
single to
ne
modulatio
n,
D β
Example 4
.4North America, co
mmercial F
M broad
cast radio
Δf =
75 kHz
Maxim
um au
dio freq
uen
cy W=15 k
Hz
D=Δf /W
=75/15=
5
Carso
n’s ru
le:BT1 =2(Δ
f +W
)=2(75+
15)=180 kHz
Universal cu
rve:From Figure 4
.9, B
T2 =3.2Δ
f=24
0 kHz
An FM rad
io ch
annel is B
T =20
0 kHz
BT1 < B
T < B
T2
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32
FM M
odulato
r Implem
entatio
nVoltag
e‐contro
lled oscillato
rA H
artley oscillato
r
Varacto
r: capacitan
ce dep
ends o
n th
e vo
ltage ap
plied
to its electro
des
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33
Assu
ming a sin
usoidal m
odulatin
g w
ave with fm
FM Dem
odulato
rA direct m
ethod
A freq
uen
cy discrim
inato
r: instan
taneo
us o
utput
amplitu
de is d
irectly proportio
nal to
the in
stantan
eous
frequen
cy of th
e input sig
nal
With an en
velop detecto
r, we g
et:
With a D
C block
er, we g
et:
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34
dt
tm
kt
fA
ts
fc
cF
M)
(2
cos)
(
dt
tm
kt
ft
mk
fA
ts
dt df
cf
cc
FM
)(
2sin
)(
2)
(
)(
2)
(1t
mk
fA
tv
fc
c
)(
)(2
tm
kA
tv
fc
Need
to ch
oose k
f to
avoid phase reversal
FM Dem
odulato
r (contd.)
Indirect m
ethod
Phase‐lo
cked lo
op (P
LL)
A neg
ative feedback
system
Used
for syn
chronizatio
n, freq
uen
cy divisio
n/m
ultip
lication,
frequen
cy modulatio
n, an
d in
direct freq
uen
cy dem
odulatio
n
To gen
erate an r(t)
that track
s the p
hase an
gle o
f s(t)
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35
Nonlinear Effects in
FM
Systems
A nonlin
ear chan
nel w
ith
FM sig
nal:
W
e have:
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36
Nonlinear Effects (co
ntd.)
Need
to sep
arate the d
esired sig
nal w
ith carrier fc
Acco
rding to Carso
n’s ru
le, the n
ecessary conditio
n
for sep
arating th
e desired
FM sig
nal w
ith fc
from th
at with 2fcis
W
ith a b
and‐pass filter cen
tered at fc
and w
ith
ban
dwidth 2Δ
f+2W
, we g
et
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37
Nonlinear Effects (co
ntd.)
Unlik
e AM, F
M is n
ot a
ffected by th
e channel in
duced
amplitu
de n
onlin
earities
W
idely u
sed fo
r micro
wave rad
io an
d satellite system
s
Perm
its the u
se of h
ighly n
onlin
ear amplifiers an
d
power tran
smitters, to
produce a m
ax power fo
r long
distan
ce
But still extrem
ely sensitive to
phase n
onlin
earilities
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38
The Su
perh
eterodyne R
eceiverOther fu
nctio
ns o
f a receiver, in ad
ditio
n to dem
odulatio
n
Carrier‐freq
uen
cy tuning: to
select the d
esired sig
nal (o
r, statio
n)
Filterin
g: sep
arate the d
esired sig
nal fro
m other m
odulated
sig
nals
Amplificatio
n: co
mpen
sate for th
e chan
nel atten
uatio
n
The su
perh
eterodyn
e receiver
A sp
ecial type o
f receiver that fu
lfils all three fu
nctio
ns,
particu
larly the first tw
o, in
an eleg
ant an
d practical fash
ion
Practically all rad
io an
d TV receivers are o
f the su
perh
eterodyn
e typ
e
Consists o
f: radio‐freq
uen
cy (RF) sectio
n, m
ixer and lo
cal oscillato
r, interm
ediate freq
uen
cy (IF) sectio
n, d
emodulato
r, an
d power am
plifier
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39
Basic Elem
ents
Basic d
ifference b
etween
AM an
d FM su
perh
eteerodyn
e receivers
The u
se of an
FM dem
odulato
r such as lim
iter‐frequen
cy discrim
inato
r
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40
Them
e Example: A
nalo
g and Digital
FM Cellu
lar Telephones
Advan
ced M
obile P
hone S
ervice (AMPS), 19
83
The in
itial cellular telep
hone system
in N
orth America
FDMA w
ith 30 kHz ch
annels
A pair o
f chan
nels are assig
ned to each
user, 4
5 MHz ap
art
Analo
g FM fo
r voice tran
smissio
n
Freq
uen
cy‐shift K
eying (F
SK) fo
r data tran
smissio
n (w
ill be
discu
ssed in ch
apter 9
)
FM desig
n
Voice is lim
ited to 3K
, the F
M m
odulato
r is desig
ned w
ith peak
deviatio
n Δf=12 k
Hz
Carso
n’s ru
le: BT =2(Δ
f+W
)=2(12+
3)=30
kHz
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41
Them
e Example: A
nalo
g and Digital
FM Cellu
lar Telephones (co
ntd.)
Pros
Can use h
igh efficien
cy power am
plifiers
Can operate in
saturatio
n w
ithout d
istortin
g th
e en
velope, w
hich is co
nstan
t
Constan
t amplitu
de h
elps co
mbatin
g fad
ing on m
obile
links
Cons
No protectio
n fro
m eavesd
ropping
Not b
andwidth efficien
t
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42
Them
e Example: A
nalo
g and Digital
FM Cellu
lar Telephones (co
ntd.)
Global S
ystem fo
r Mobile C
ommunicatio
ns (G
SM)
Successo
r of A
MPS
Built o
n FM related
advan
tages, b
ut im
proves b
andwidth
efficiency b
y
A m
ore co
mplex m
ultip
lexing strateg
y
Digital rep
resentatio
n of d
ata
A freq
uen
cy ban
d is 20
0 kHz, w
hich is sh
ared by 32 vo
ice calls in
one d
irection
Combinatio
n of F
DMA an
d TDMA
Number o
f calls that an
be su
pported
per u
nit b
andwidth
(30
/200)×32=
4.8 tim
es improvem
ent o
ver AMPS
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43
Them
e Example: A
nalo
g and Digital
FM Cellu
lar Telephones (co
ntd.)
7/13/2010
44
Summary
Angle m
odulatio
n
FM (freq
uen
cy) and PM (p
hase)
Equivalen
ce of th
e two
Properties
Hard
for sp
ectral analysis
Study sin
gle to
ne F
M fo
r insig
hts
An em
pirical ru
le: Carso
n’s ru
le for ap
proxim
ate evaluatio
n
of th
e transm
ission ban
dwidth of F
M B
T
Determ
ined by th
e modulatio
n in
dex β
or th
e deviatio
n ratio
D
for n
onsin
usoidal F
M
Constan
t amplitu
de: ro
bust to
noise an
d in
terference
Trad
e‐off b
etween
transm
ission ban
dwidth an
d noise
perfo
rman
ce
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45