chapter 8: transmission baseband
TRANSCRIPT
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Chapter 8
: Base
band D
igita
l Transm
ission
6/8/20
101
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Intro
ductio
n
Tran
smissio
n of d
igital sig
nal
Over a b
aseban
d ch
annel (C
hap
ter 8)
local co
mmunicatio
ns
Over a b
and‐pass ch
annel u
sing m
odulatio
n (C
hap
ter 9)
netw
ork
Chan
nel‐in
duced
transm
ission im
pairm
ents
Chan
nel n
oise, o
r receiver noise
Interferen
ce: sometim
es treated as n
oise
Intersym
bol in
terference (IS
I)
Digital d
ata has a b
road ban
dwidth w
ith a sig
nifican
t low‐freq
uen
cy conten
t
Man
y chan
nels are b
andwidth lim
ited: d
ispersive, u
nlik
e low‐pass filter
Each
received pulse is affected
by n
eighborin
g pulses
ISI
Majo
r source o
f bit erro
rs in m
any cases
Solutio
ns to
be stu
died in th
is chap
ter
Noise: m
atched filter
maxim
ize the sig
nal n
oise level at th
e receiver
ISI:
Pulse sh
aping
minim
ize the IS
I at the sam
plin
g points
Equalizatio
n
compen
sate the resid
ual d
istortio
n fo
r ISI
6/8/20
102
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6/8/20
103
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Transm
ission Im
pairm
ent: N
oise
Therm
al noise: g
enerated
by th
e equilib
rium
fluctu
ations o
f the electric cu
rrent in
side th
e receiver circu
it Due to
the ran
dom th
ermal m
otio
n of th
e electrons
Modeled
as an Additive w
hite G
aussian
noise (A
WGN)
Noise sp
ectral den
sity: No = KT(w
atts per h
ertz), where K
is the B
oltzm
ann’s co
nstan
t K = 1.38
0×10−23, an
d T
is the
receiver system noise tem
peratu
re in kelvin
s ([K] = [°C
] +
273.15)
If b
andwidth is B
Hz, th
en th
e noise p
ower
is N = BKT
Alw
ays exists
Other so
urces o
f noise: in
terference
6/8/20
104
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Additive N
oise
6/8/20
105
http
://files.amourau
x.web
node.co
m/20
0000047‐3fd
9440d34
/resint_eeg
2.jpg
received sig
nal
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Transm
ission Im
pairm
ent: ISI
Line co
des
Map
ping 1’s an
d 0’s to
symbols
Ran
dom process, sin
ce 1’s and 0’s are ran
dom
Power sp
ectrum (S
ection 5.8
)Rep
resentatio
n in th
e frequen
cy domain
The n
ominal b
andwidth of th
e signal is th
e same o
rder o
f mag
nitu
de as 1/T
ban
d is cen
tered aro
und th
e orig
in
Mism
atch betw
een sig
nal b
andwidth B
san
d ch
annel
ban
dwidth B
c
If B
c ≥Bs , n
o problem
If B
c < B
s , the ch
annel is d
ispersive, th
e pulse sh
ape w
ill be
chan
ged an
d th
ere will b
e ISI
6/8/20
106
baseb
and
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Power Sp
ectra of Several Lin
e Codes
6/8/20
107
•Freq
uen
cy axis norm
alized w
ith T
b
•Averag
e power is n
orm
alized to unity
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Transm
ission Im
pairm
ents d
ue to
Lim
ited Channel B
andwidth
Each
received sym
bol m
ay be w
ider
than th
e transm
itted one, d
ue to
loss o
f high freq
uen
cy componen
ts
Overlap
betw
een ad
jacent sym
bols: IS
I
Lim
it on data rate: u
se guard
time b
etween
adjacen
t symbols
Or n
eed to sh
ape th
e pulses to
cancel IS
I at samplin
g points
6/8/20
108
From D
ata Communica
tions a
nd Netw
orking, B
ehrouz A
. Forouzan
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Match
ed Filter –
The P
roblem
Meth
odology:
Cope w
ith th
e two typ
es of im
pairm
ents sep
arately
First assu
me an
ideal ch
annel an
d only co
nsid
er noise
e.g
., low data rate o
ver a short ran
ge cab
le
No problem
of IS
I
Tran
smitted
pulse g
(t)for each
bit is u
naffected
by th
e tran
smissio
n excep
t for th
e additive w
hite n
ose a
t the receiver fro
nt
end
6/8/20
109
Basic p
roblem of d
etecting a p
ulse tran
smitted
over a ch
annel th
at is corru
pted
by ad
ditive w
hite
noise at th
e receiver front en
d
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Match
ed Filter
Received
(or, in
put) sig
nal: x(t) =
g(t) +
w(t), 0
≤t≤T
T: an
arbitrary o
bservatio
n in
tervalg(t): rep
resents a b
inary sym
bol 1 o
r 0w(t): w
hite G
aussian
noise p
rocess o
f zero m
ean an
d power
spectru
m den
sity N0 /2
Output sig
nal: y(t) =
x(t) h(t) =
g0 (t) +
n(t)
6/8/20
1010
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Detectio
n of R
eceived Sign
al
6/8/20
1011
s1 (t)
tT
0
s0 (t)
tT
0
tT
0
tT
0
1T
T
dtt
s0
1)
(
T
dtt
s0
0)
(
Optim
al detectio
n tim
e
Optim
al detectio
n tim
e
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Match
ed Filter (co
ntd.)
Problem
Find h(t)
to m
aximize th
e peak
pulse sig
nal‐to
‐noise
ratio at th
e samplin
g in
stant t=
T:
If G
(f)
g(t), H
(f)
h(t), th
en g
0 (t)
H(f)G
(f). W
e can derive g
0 (t)by in
verse Fourier tran
sform
:
The in
stantan
eous sig
nal p
ower at t=
Tis:
6/8/20
1012
power
noise
output
Average
signaloutput
in thepow
er
ousInstantane
E
)(
2
2
0
(t)
n Tg
df
ftj
fG
fH
tg
)2
exp()
()
()
(0
22
0)
2exp(
)(
)(
)(
df
fTj
fG
fH
Tg
Instan
taneo
us
power
Why sq
uare?
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Match
ed Filter (co
ntd.)
The averag
e noise p
ower
The p
ower sp
ectral den
sity of th
e output n
oise n
(t)is
The averag
e noise p
ower is
The p
eak pulse sig
nal‐to
‐noise ratio
is
6/8/20
1013
df
fH
Ndf
fS
tn
N
20
2)
(2
)(
)(
E
20
)(
2)
(f
HN
fS
N
dff
HN
dffT
jf
Gf
H
20
2
)(
2
)2
exp()
()
(
Find H
(f)
h(t) th
at m
axim
izes η
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Match
ed Filter (co
ntd.)
Sch
warz’s in
equality
Therefo
re, we h
ave
6/8/20
1014
If an
d
The eq
uality h
olds if an
d only if:
when
Complex
conjugatio
n
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Match
ed Filter (co
ntd.)
Excep
t for th
e factor k∙exp
(‐j2πfT), th
e transfer fu
nctio
n of th
e optim
al filter is the sam
e as the co
mplex co
njugate o
f the
spectru
m of th
e input sig
nal
k: scales th
e amplitu
de
exp
(‐j2πfT): tim
e shift
For real sig
nal g
(t), we h
ave G*(f)=
G(‐f): tim
e inversed
The o
ptim
al filter is found by in
verse Fourier tran
sform
Match
ed filter: m
atch
ed to th
e signal
A tim
e‐inversed
and delayed
version of th
e input sig
nal g
(t)
6/8/20
1015
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Properties o
f Match
ed Filters
Match
ed filter:
Received
signal:
Noise p
ower:
6/8/20
1016
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Properties (co
ntd.)
Maxim
um peak
pulse sig
nal‐to
‐noise ratio
Observatio
ns
Indep
enden
t of g
(t): removed
by th
e match
ed filter
Signal en
ergy (o
r, transm
it power) m
atters
For co
mbatin
g ad
ditive w
hite G
aussian
noise, all sig
nals th
at have th
e same en
ergy are eq
ually effective
Not tru
e for IS
I, where th
e signal w
ave form m
atters
E/N
0 : signal en
ergy‐to
‐noise sp
ectral den
sity ratio
6/8/20
1017
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Example 8
.1 M
atched
Filter for
Rectan
gular P
ulse
Rectan
gular p
ulse fo
r g(t)
Match
ed filter
6/8/20
1018
)2 1
(rect
)(
T tA
tg
)2 1(
rect
)2 1
(rect
)(
)(
T tkA
T
tT
kA
tT
kgt
h
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Example 8
.1 M
atched
Filter for
Rectan
gular P
ulse (co
ntd.)
Output g
o (t)
Max o
utput k
A2T
occu
rs at t=
T
Optim
al samplin
g
instan
ce
Im
plem
ented
usin
g th
e integ
rate‐and‐dump
circuit
6/8/20
1019
)(
)(
)(
th
tg
tg
o
Sam
plin
g tim
eInteg
rator is resto
re to
initial co
nditio
n
k=1/A
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Effect of N
oise
6/8/20
1020
William
Stallin
gs, D
ata and Computer C
ommunicatio
ns, 8
/E, P
rentice H
all, 2007.
Bit erro
r rate (BER) =
2/15=13.3%
Line co
ding:
‐“1”: ‐5 vo
lts‐“0”: 5 vo
lts
A properly
chosen
decisio
n
thresh
old
Sam
plin
g
instan
ce
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Probability o
f Error d
ue to
Noise
Assu
me p
olar n
onretu
rn‐to‐zero
(NRZ) sig
nalin
g1: p
ositive rectan
gular p
ulse, +
A
0: n
egative rectan
gular p
ulse, ‐A
Additive w
hite G
aussian
Noise w
(t)of zero
mean
and
power sp
ectral den
sity N0 /2
Received
signal is
The receiver h
as prio
r knowled
ge o
f the p
ulse sh
ape, n
eed
to decid
e 1 or 0 fo
r a received am
plitu
de in
each sig
nalin
g
interval 0
≤t≤tb
6/8/20
1021
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Probability o
f Error d
ue to
Noise
(contd.)
Sam
pled valu
e y, with
thresh
old λ,
6/8/20
1022
received 0
sym
bol ,
if
received 1
symbol
,
if
y y
Two kinds o
f errors
01
01
1 ‐
1 ‐
‐How to quantify resid
ual B
ER?
‐How to ch
oose th
reshold to m
inim
ize BER?
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Consid
er the C
ase When
Symbol 0
Was Tran
smitted
Receiver g
ets x(t) = ‐A + w(t), fo
r 0≤t≤Tb
The m
atched filter o
utput, sam
pled at t=
Tb , is th
e sam
pled valu
e of a ran
dom variab
le Y
Since w
(t)is w
hite an
d Gau
ssian, Y
is also Gau
ssian
with m
ean E[Y]=–A, an
d varian
ce
6/8/20
1023
See P
age 19
Gaussia
n
distrib
utio
n ca
n be
completely
determ
ined by it
mean and va
riance
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When
0 W
as Transm
itted (co
ntd.)
Since w
(t)is w
hite G
aussian
,
The varian
ce is
Yis G
aussian
with m
ean µ
Y =–A
, variance σ
Y2=N
0 /(2Tb )
The co
nditio
nal p
robab
ility den
sity functio
n (P
DF) o
f Y,
conditio
ned on th
at symbol 0
was tran
smitted
, is
6/8/20
1024
T
N
Ay
TN
yy
fb
Y
Y
Y
Y/
)(
exp/
1
2
)(
exp2 1
)0|
(0
2
02
2
Stan
dard
PDF of G
aussian
r.v., with m
ean µ
Yan
d varian
ce σY2
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When
0 W
as Tran
smitted
(co
ntd.)
W
hen no noise, Y
=‐A
W
ith noise, d
rifts away fro
m –
AIf less th
an λ, o
utput 0 (n
o bit
error)
If larg
er thanλ, o
utput 1 (b
it erro
r occu
rs)
6/8/20
1025
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When
0 W
as Transm
itted (co
ntd.)
Assu
me sym
bol 1 an
d 0 are eq
ual lik
ely to be
transm
itted, w
e choose λ
=0, d
ue to
symmetry
Defin
e , we h
ave
Eb : th
e transm
itted sig
nal en
ergy p
er bit
Eb /(N
0 /2) : (signal p
ower p
er bit)/(n
oise p
ower p
er Hz)
6/8/20
1026
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When
0 W
as Transm
itted (co
ntd.)
Defin
e Q‐Functio
n:
The co
nditio
nal b
it error
probab
ility when 0 w
as tran
smitted
is
6/8/20
1027
u
Q‐fu
nctio
n, see P
age 4
01
x=[‐3:0
.1:3];for i=
1:length(x)
Q(i)=
0.5*erfc(x(i)/sq
rt(2));en
dplot(x,Q
)
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When
1 W
as Transm
itted
6/8/20
1028
Receives: x(t)=
A+w(t), 0
≤t≤Tb
Yis G
aussian
with µ
Y =A, σ
Y2=N
0 /(2Tb )
W
e have
The co
nditio
nal b
it error rate is
Choosin
g λ=0, an
d defin
ing , w
e have
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Bit Erro
r Probability (o
r, Bit Erro
r Rate –
BER
)
6/8/20
1029
Bit E
rror R
ate (BER) is
Dep
ends o
nly o
n E
b /N0 , th
e ratio of th
e transm
itted sig
nal
energ
y per b
it to th
e noise sp
ectral den
sity
Noise is u
sually fixed
for a g
iven tem
peratu
re
Energ
y plays th
e crucial ro
le tran
smit p
ower
W
hat are th
e limitin
g facto
rs?Battery life, in
terference to
others, d
ata rate requirem
ent
00
01
10
0
10
22
2 12
2 1
}ed
transmitt
is {1
Pr
}ed
transmitt
is {0
Pr
N EQ
N EQ
N EQ
Pp
Pp
PP
P
bb
be
e
ee
e
Wider p
ulse
Higher am
plitu
de
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BER
6/8/20
1030
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Intersym
bol In
terference
The n
ext source o
f bit erro
rs to be ad
dressed
Hap
pen
s when th
e chan
nel is d
ispersive
The ch
annel h
as a frequen
cy‐dep
enden
t (or, freq
uen
cy‐selective) am
plitu
de sp
ectrum
e.g
., ban
d‐lim
ited ch
annel:
passes all freq
uen
cies |f|<W with
out d
istortio
n
Block
s all frequen
cies |f|>W
Use d
iscrete pulse‐am
plitu
de m
odulatio
n (P
AM) as
example
First exam
ine b
inary d
ata
Then co
nsid
er the m
ore g
eneral case o
f M‐ary d
ata
6/8/20
1031
![Page 38: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/38.jpg)
Example 8
.2: Th
e Disp
ersive
Natu
re of a Telep
hone C
hannel
Ban
d‐lim
ited an
d disp
ersive
6/8/20
1032
Block high frequ
encies:
cut‐o
ff at 3.5 k
Hz
Block dc
![Page 39: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/39.jpg)
Example 8
.2: Th
e Disp
ersive natu
re of a Telep
hone C
hannel
Conflictin
g req
uirem
ents fo
r line co
ding
High freq
uen
cies block
ed
need
a line co
de w
ith a n
arrow
spectru
m
polar N
RZ
But p
olar N
RZ has d
c
Low freq
uen
cies block
ed
need
a line co
de th
at has n
o dc
Man
chester co
de
But M
anch
ester code h
as high freq
uen
cy
6/8/20
1033
![Page 40: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/40.jpg)
Example 8
.2: Th
e Disp
ersive natu
re of a Telep
hone C
hannel (co
ntd.)
Previo
us p
age: d
ata rate at 1600 bps
This p
age: d
ata rate at 3200 bps
6/8/20
1034
![Page 41: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/41.jpg)
Eye Pattern
An operatio
nal to
ol fo
r evaluatin
g th
e effects of IS
I
Syn
chronized
superp
ositio
n of a
llpossib
le realizations o
f the sig
nal view
ed w
ithin a p
articular sig
nalin
g in
terval
6/8/20
1035
http
://mem
bers.ch
ello.nl/~
m.heijlig
ers/DAChtm
l/digco
m/d
igco
m.htm
l
![Page 42: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/42.jpg)
Eye Pattern
(contd.)
Eye o
pen
ing: th
e interio
r region of th
e eye pattern
6/8/20
1036
![Page 43: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/43.jpg)
Interp
reting Eye P
atternThe w
idth
of th
e eye open
ing
Defin
es the tim
e interval o
ver which th
e received sig
nal can
be
sampled w
ithout erro
r from IS
I
The b
est samplin
g tim
e: when th
e eye is open th
e widest
The slo
pe
The sen
sitivity of th
e system to tim
ing erro
rs
The rate o
f closu
re of th
e eye as the sam
plin
g tim
e is varied
The h
eightof th
e eye open
ing
Noise m
argin of th
e system
Under severe IS
I: the eye m
ay be co
mpletely clo
sed
Im
possib
le to avo
id erro
rs due to
ISI
6/8/20
1037
![Page 44: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/44.jpg)
Example 8
.3
6/8/20
1038
The ch
annel h
as no ban
dwidth
limitatio
n: th
e eyes are open
Ban
d‐lim
ited ch
annel: b
lurred
reg
ion at sam
plin
g tim
e
![Page 45: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/45.jpg)
Eye Pattern
on Oscillo
scope
6/8/20
1039
http
://www.m
ypriu
s.co.za/p
cmx2_
processo
r1.ht
m
![Page 46: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/46.jpg)
![Page 47: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/47.jpg)
Baseb
and Binary P
AM System
Input seq
uen
ce: {bk | b
k = 0 or 1}
Tran
smitted
signal:
The receiver filter o
utput:
where:
Assu
me p
(t)is n
orm
alized, p
(o)=1, u
sing μ
as a scaling facto
r to
account fo
r amplitu
de ch
ange d
urin
g tran
smissio
n
6/8/20
1040
k
bk
kTt
ga
ts
)(
)(
)(
)(
)(
tn
kTt
pa
ty
kb
k
)
()
()
()
(t
ct
ht
gt
p
)(
)(
)(
)(
fC
fH
fG
fP
![Page 48: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/48.jpg)
Baseb
and Binary P
AM System
(co
ntd.)
For th
e i‐th received
symbol, sam
ple th
e output y(t)
at ti =iT
b , yielding
6/8/20
1041
)(
])
[(
)(
])
[()
(
i
ik k
bk
i
ik
bk
i
tn
Tk
ip
aa
tn
Tk
ip
at
y
Contrib
utio
n of th
e i‐th
transm
itted bit
Resid
ual effect d
ue to
the
occu
rrence o
f pulses
befo
re and after th
e sam
plin
g in
stant t
i : the IS
I
Effect o
f noise,
taken care o
f by
match
ed filter
In th
e absen
ce of b
oth IS
I and noise:
ii
at
y
)(
![Page 49: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/49.jpg)
Nyquist’s C
riterion fo
r Disto
rtionless Tran
smissio
n
Recall th
at:
The p
roblem
: Usu
ally the tran
sfer functio
n of th
e chan
nel h
(t)an
d th
e tran
smitted
pulse sh
ape are sp
ecified (e.g
., p32, telep
hone ch
annel)
to determ
ine th
e transfer fu
nctio
ns o
f the tran
smit an
d receive
filters so as to
reconstru
ct the in
put b
inary {b
k }
The receiver p
erform
sExtractio
n: sam
plin
g y(t)
at time t=
iTb
Deco
ding:
req
uires th
e ISI to
be zero
at the sam
plin
ginstan
ce:
6/8/20
1042
)(
])
[()
(i
ik k
bk
ii
tn
Tk
ip
aa
ty
)(
)(
)(
)(
tc
th
tg
tp
.,0 ,1
])
[(k
i
ki
Tk
ip
b
![Page 50: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/50.jpg)
Nyquist’s C
riterion fo
r Disto
rtionless
Transm
ission (co
ntd.)
Sam
plin
g p(t)
at nTb , n
=0, ±
1, ±2, …
{p(nTb )}
Sam
plin
g in th
e time d
omain produces p
eriodicity in
the freq
uen
cy domain
, we h
ave
On th
e other h
and, th
e sampled sig
nal is
Its Fourier tran
sform is
If the co
nditio
n is satisfied
6/8/20
1043
nb
bnR
fP
Rf
P)
()
(
)(
)(
)(
bn
bnT
tnT
pt
p
dt
ftnT
tnT
pf
Pb
nb
)2
exp()]
()
([
)(
1)0
()
2exp(
)(
)0(
)(
p
dtft
tp
fP
.,0 ,1
])
[(k
i
ki
Tk
ip
b
![Page 51: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/51.jpg)
Nyquist’s C
riterion fo
r Disto
rtionless
Transm
ission (co
ntd.)
Finally w
e have
The N
yquist C
riterion fo
r disto
rtionless b
aseban
d
transm
ission in th
e absen
ce of n
oise: b
bn
b
nb
b
TR
nRf
P
nRf
PR
fP
/1
)(
1
)(
)(
The freq
uen
cy functio
n P(f)
eliminates IS
I for sam
ples
taken at in
tervals Tbprovid
ed th
at it satisfies
bn
bT
nRf
P
)(
P(f): fo
r the o
verall system,
inclu
ding th
e transm
it filter, th
e chan
nel, an
d th
e receive filter
![Page 52: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/52.jpg)
![Page 53: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/53.jpg)
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![Page 56: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/56.jpg)
Ideal N
yquist C
hannel
The sim
ples w
ay of satisfyin
g The N
yquist
Criterio
n is
Where W
=Rb /2=
1/(2Tb ).
The sig
nal th
at produces zero
ISI is th
e sinc
functio
n
Nyq
uist b
andwidth: W
Nyq
uist rate: R
b =2W
6/8/20
1045
Wf
Wf
WW
W f
Wf
P|
|,0
,2 1
2rect
2 1)
(
)2(
sinc2
)2
sin()
(W
tW
t Wt
tp
![Page 57: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/57.jpg)
Ideal N
yquist C
hannel (co
ntd.)
6/8/20
1046
![Page 58: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/58.jpg)
Raised
Cosin
e Spectru
mP(f)
is physically u
nrealizab
leNo filter can
have th
e abrupt tran
sitions at f=
±W
p(t)
decays at rate 1/|t|: to
o slo
w, n
o m
argin fo
r samplin
g tim
e erro
r (see Fig. 8
.16)
Use raised
cosin
e spectru
m: a flat to
p + a ro
lloff p
ortio
n
6/8/20
1047
![Page 59: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/59.jpg)
Raised
Cosin
e Sp
ectrum (co
ntd.)
6/8/20
1048
Rollo
ff factor: α
=1‐f1 /W
Indicates th
e excess ban
dwidth
over th
e ideal so
lutio
n, W
)1(
)1
1()
2(1
1
WW f
Wf
WB
T
W=Rb /2=
1/(2Tb ).
Sam
e property as sin
c(2Wt), b
ut
now it is p
ractical
![Page 60: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/60.jpg)
![Page 61: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/61.jpg)
![Page 62: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/62.jpg)
How to
Design
the Tran
sceiverNyq
uist C
riterion
P(f)
= raised
cosin
e spectru
m
Study th
e chan
nel
find h(t)
Match
ed filter (to
cope w
ith noise)
c(t)an
d g(t)
are sym
metric
solve fo
r c(t)an
d g(t)
6/8/20
1049
)(
)(
)(
)(
tc
th
tg
tp
)
()
()
()
(f
Cf
Hf
Gf
P
)2
exp()
()
(fT
jf
kGf
C
![Page 63: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/63.jpg)
Example 8
.4 Bandwidth
Req
uirem
ent o
f the T1
SystemT1 system
: multip
lexing 24 vo
ice calls, each 4 kHz,
based
on 8‐bit P
CM w
ord, T
b =0.647μs
Assu
ming an id
eal Nyq
uist ch
annel, th
e minim
um
required
ban
dwidth is
In practice, a fu
ll‐cosin
e rollo
ff spectru
m is u
sed w
ith
α=1. T
he m
inim
um tran
smissio
n ban
dwidth is
In Chap
ter 3, if use S
SB an
d FDM, th
e ban
dwidth is
6/8/20
1050
kHz
773
)2
/(1
2/
b
bT
TR
WB
MH
z
544.1
2)
1(
W
WB
T
kHz
96
424
T
B
Digital tran
smissio
n is n
ot b
andwidth efficien
t
![Page 64: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/64.jpg)
Baseb
and M
‐ary PAM Tran
smissio
nBaseb
and M‐ary P
AM system
M
possib
le amplitu
de levels, w
ith M
>2
Log2 (M
) bits are m
apped to one o
f the levels
6/8/20
1051
![Page 65: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/65.jpg)
Baseb
and M
‐ary PAM Tran
smissio
n
(contd.)
Sym
bol d
uratio
n: T
Signalin
g rate: R
=1/T
, in sym
bols p
er second, o
r bau
ds
Binary sym
bol d
uratio
n: T
b
Binary d
ata rate: Rb =1/T
b
Sim
ilar proced
ure u
sed fo
r the d
esign of th
e filters as in th
e binary d
ata case
6/8/20
1052
MT
Tb
2log
M
RR
b2
log
![Page 66: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/66.jpg)
Eye Pattern
for M
‐ary Data
Contain
s (M‐1) eye o
pen
ings stack
ed up vertically
6/8/20
1053
![Page 67: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/67.jpg)
Tapped
‐Delay‐Lin
e Equalizatio
nISI is th
e majo
r cause o
f bit erro
r in baseb
and
transm
issions
If ch
annel h
(t)or H
(f) is known precisely, o
ne can
desig
n
transm
it and receiver to
mak
e ISI arb
itrarily small
Find P(f)
find G(f)
find C(f)
However in
practice, h
(t)may n
ot b
e known, o
r be k
nown
with erro
rs (i.e., time‐varyin
g ch
annels)
Cau
se residual d
istortio
nA lim
iting facto
r for d
ata rates
Use a p
rocess, eq
ualizatio
n, to
compen
sate for th
e intrin
sic residual d
istortio
nEqualizer: th
e filter used
for su
ch process
6/8/20
1054
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![Page 69: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/69.jpg)
Tapped
‐Delay‐Lin
e FilterTotally (2N
+1) tap
s, with w
eights w
‐N , …w
1 , w0 , w
1 , …w
N
6/8/20
1055
T=Tb : sym
bol d
uratio
n
![Page 70: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/70.jpg)
Tapped
‐Delay‐Lin
e Filter (co
ntd.)
W
e have
and
6/8/20
1056
![Page 71: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/71.jpg)
Tapped
‐Delay‐Lin
e Filter (co
ntd.)
The N
yquist criterio
n m
ust b
e satisfied. W
e have
Den
ote c
n =c(n
T), w
e have
6/8/20
1057
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Tapped
‐Delay‐Lin
e Filter (co
ntd.)
Rem
arks
Referred
to as a zero
‐forcin
g eq
ualizer
Optim
um in th
e sense th
at it minim
izes peak
disto
rtion
(ISI)
Sim
ple to
implem
ent
The lo
nger, th
e better, i.e., th
e closer to
the id
eal co
nditio
n as sp
ecified by th
e Nyq
uist criterio
n
For tim
e‐varying ch
annels
Train
ing
Adap
tive equalizatio
n: ad
justs th
e weig
hts
6/8/20
1058
![Page 76: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/76.jpg)
Them
e Example –
100Base‐TX
–Tran
smissio
n of 1
00 M
bps o
ver Tw
isted Pair
Fast E
thern
et: 100BASE‐TX
Up to 10
0Mbps
Usin
g tw
o pairs o
f twisted
copper w
ires Categ
ory 5
cable
One p
air for each
directio
n
Maxim
um distan
ce: 100 m
eters
First stag
e: NRZ 4B5B
provid
e clock
ing in
form
ation
Seco
nd stag
e: NRZI
Third stag
e: three‐level sig
nalin
g M
LT‐3
W
ith tap
ped‐delay‐lin
e equalizatio
n
6/8/20
1059
![Page 77: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/77.jpg)
Summary
Two tran
smissio
n im
pairm
ents
Noise
ISI
Im
pact o
f the tw
o tran
smissio
n im
pairm
ents
How to m
itigate th
e effects of tran
smissio
n im
pairm
ents
Match
ed filter
Evalu
ating th
e BER
Eye p
attern
Nyq
uist criterio
ns fo
r disto
rtionless criterio
n
Tap
ped‐delay‐lin
e equalizatio
n
Binary tran
smissio
ns an
d M‐ary tran
smissio
ns
6/8/20
1060
![Page 78: Chapter 8: Transmission Baseband](https://reader031.vdocument.in/reader031/viewer/2022012423/61777965a2f868115d347b22/html5/thumbnails/78.jpg)