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Chapter 7: T
he Transitio
n fro
m
Analog to D
igita
l(of H
aykin & M
ohler)
5/18/20
101
Intro
ductio
n
Analo
g
Contin
uous‐w
ave (CW
) modulatio
n: o
ne o
f the th
ree param
eters of a
sinusosid
al carrier is varied co
ntin
uously
according to th
e inform
ation w
ave
Analo
g to digital
Digital
Discrete tim
e & discrete levels
Theo
retical understan
dings: 19
30s to
1960s
Hard
ware: so
lid state electro
nics, m
icro‐electro
nics, an
d larg
e‐scale integ
ration in th
e 1970
s
Three step
s for th
e transitio
n
Sam
plin
g:
discrete tim
e
Pulse‐am
plitu
de m
odulatio
n
Pulse‐p
ositio
n m
odulatio
n
Quan
tization:
discrete levels
Unifo
rm quan
tization
Non‐unifo
rm quan
tization
Line C
oding: d
ecimal in
tegers (b
ase 10)
binary (b
ase 2)
5/18/20
102
Why D
igitize Analo
g Sources?
Digital system
s are less sensitive to
noise
than an
alog
It is easier to
integ
ratedifferen
t services into th
e same
transm
ission sch
eme
e.g
., analo
g or d
igital T
V in Chap
ter 3
The tran
smissio
nsch
eme can
be relatively in
dep
enden
t of
the so
urce
homogen
eous stream
of 0
’s and 1’s
Digital circu
itryis easier to
repeat an
d less sen
sitive to
physical effects su
ch as vib
ration an
d tem
peratu
re
Sim
pler to
characterize
and typ
ically do not h
ave the sam
e am
plitu
de ran
ge an
d variab
ility as analo
g system
s
5/18/20
103
Why D
igital (contd.)
Alm
ost all tran
smissio
n m
edia m
ay be u
sed fo
r either
digital o
r analo
g tran
smissio
nDigital u
ses med
ia more efficien
tlyMultip
lexing
Med
ia sharin
g is easier to
implem
ent fo
r digital tran
smissio
ns
Source co
ding:
Rem
ovin
g red
undan
cy in digital d
ata
Chan
nel co
ding:
Adding co
ntro
lled red
undan
cy to a d
igital tran
smissio
n
For erro
r detectio
n an
d co
rrection
Other ch
annel co
mpen
sation sch
emes, e.g
., equalizatio
n, are
easier to im
plem
ent fo
r digital tran
smissio
ns
Easier fo
r standard
izationfor in
teroperatib
ility
5/18/20
104
The Sam
pling
Process
Arbitrary sig
nal
g(t)W
ith fin
ite en
ergy: w
hy
Specified
for all
time t: w
hy
Sam
ple in
st. at a unifo
rm rate
Sam
plin
g rate:
Ts= 1/fs
Infin
ite sequen
ce of sam
ples,
One every T
sseco
nds
5/18/20
105
0t
c(t)
The Sam
pling P
rocess (co
ntd.)
Sam
plin
g perio
d: T
s , seconds
Sam
plin
g rate: fs , sam
ples/seco
nd
Instan
taneo
us sam
plin
g: im
pulses at t
weig
hted
by
the co
rresponding g(t)
values –
ideal sam
pled sig
nal
5/18/20
106
ns
s
ns
ns
nTt
nTg
nTt
tg
nTt
tg
tc
tg
tg
)(
)(
)(
)(
)(
)(
)(
)(
)(
Spectral A
nalysis
Fundam
ental th
eory o
f Fourier T
ransfo
rm:
5/18/20
107
Mak
ing a sig
nal p
eriodic in
the tim
e domain has th
e effect o
f samplin
g th
e spectru
m of th
e signal in
the
frequen
cy domain
Duality p
roperty
Sam
plin
g a sig
nal in
the tim
e domain has th
e effect of
mak
ing th
e spectru
m of th
e signal p
eriodic in
the
frequen
cy domain
HW 3.5
Prob
2.12
Duality
: if g(t)
G(f), th
en G(t)
g(‐f)
Duality P
roperty o
f Fourier Tran
sform
Proof:
Exam
ple:
5/18/20
108
Duality
: if g(t)
G(f), th
en G(t)
g(‐f) )
()
(
i.e., ,
)2
exp()
(
:have
we
,
and
variablesing
Interchang
)2
exp()
(
)
2exp(
)(
:T
ransfromF
ourier
inverseto
according),
()
(
Since
fg
tG
dtft
jt
Gf)
g(
ft
dfft
jf
Gt)
g(df
ftj
fG
g(t)
fG
tg
W f
W AW
tc
At
g2
rect2
)2(
sin)
(
Spectral A
nalysis (co
ntd.)
Sectio
n 2.6
: Fourier T
ransfo
rm of P
eriodic S
ignals
Determ
ine th
e Fourier tran
sform of g
δ (t) by ap
plyin
g
the d
uality p
roperty an
d Eqn. (2.8
8)
5/18/20
109
n
ss
sn
snf
fnf
Gf
nTt
g)
()
()
(
ns
snT
tnT
gt
g)
()
()
(
(2.88)
n
ss
nff
Gf
fG
tg
)(
)(
)(
G(f): F
ourier tran
sform of g
(t)
The p
rocess o
f unifo
rmly sa
mplin
g a co
ntin
uous‐tim
e signal o
f finite en
ergy
results in
a perio
dic sp
ectrum with a perio
d eq
ual to
the sa
mplin
g ra
te
Tells u
s the sh
ape
of th
e spectru
m
Spectral A
nalysis (co
ntd.)
Altern
atively, directly tak
e Fourier tran
sform
Delta fu
nctio
n an
d tim
e shiftin
g property
W
e have fo
r
5/18/20
1010
n
ss
nTt
nTg
tg
)(
)(
)(
Discrete F
ourier T
ransfo
rm of g
δ (t) . Can be view
ed as a co
mplex F
ourier
series represen
tatio
n of th
e perio
dic freq
uen
cy functio
n G
δ (f).
1)
(
t
)2
exp()
()
(0
ofT
jf
Ht
th
)2
exp()
(s
sfnT
jnt
t
ns
snfT
jnT
gf
G)
2exp(
)(
)(
See P
age 39
, (2.8
0).
Tells u
s the
expressio
n
Spectral A
nalysis (co
ntd.)
Consid
er a strictly ban
d‐lim
ited g(t), i.e., G
(f)=0, fo
r |f|≥
W
W
ith fs =
2Wan
d T
s =1/(2W
), the sp
ectrum G
δ (f) is
The sp
ectrum G
δ (f)is p
eriodic fs
Also easily seen
from duality:
5/18/20
1011
n
W nfj
W ng
fG
exp2
)(
n
ss
nff
Gf
fG
tg
)(
)(
)(
Perio
dic
Spectral A
nalysis (co
ntd.)
W
e have
if
Or
5/18/20
1012
Wf
Wf
GW
fG
),(
2 1)
(
W
f
Wf
fG
s2
for
0)
(
Wf
WW nf
j
W ng
Wf
Gn
,exp
22 1
)(
The Sam
pling Th
eorem
Exam
ine th
e G(f)
expressio
n
Im
portan
t observatio
ns
The sam
ple valu
es g(n/(2W
)) are specified
for all n
G(f)
is uniquely d
etermined in [‐W
, W]
g(t)
is uniquely d
etermined by in
verse Fourier
transfo
rm of G
(f)
g(t)
is uniquely d
etermined by sam
ple valu
es g(n/(2W
))
5/18/20
1013
otherwise
,0
,exp
22 1
)(
Wf
WW nf
j
W ng
Wf
Gn
The sam
pled seq
uen
ce{g(n/(2W
))} contain
s all inform
ation of
g(t)
Reco
nstru
ct g(t)
Reco
nstru
ct g(t)
from th
e samples g
(n/(2W
))
Interch
anging th
e order o
f sum an
d in
tegratio
n
W
e have
5/18/20
1014
Shifted
sin
cfunctio
n,
Pag
e 8
Tim
e‐shiftin
g
property
Reco
nstru
ct g(t)
(contd.)
Interp
olatio
n fo
rmula
Reco
nstru
ct g(t)
from th
e samples {g
(n/2W
)}
Interp
olatio
n fu
nctio
n: sin
c(2Wt)
5/18/20
1015
h(t)=
sinc(2W
t)g(t)
gδ (t)
Sampling Th
eorem
For strictly b
and‐lim
ited sig
nal o
f finite en
ergy
Nyq
uist rate: fs
= 2W
Nyq
uist in
terval: Ts= 1/ fs
=1/(2W
)
5/18/20
1016
Sampling Th
eorem
(contd.)
Signals w
hich are n
ot b
and‐lim
ited, o
r fs< 2W
Undersam
pled
Aliasin
geffect
High freq
uen
cy componen
t in th
e spectru
m seem
ingly ta
king
on th
e iden
tity of a lo
wer freq
uen
cy in th
e spectru
m of its
sampled versio
n
Solutio
nLow‐pass p
re‐alias filter: to lim
it the b
andwidth of th
e sig
nal to
be sam
pled
Reco
nstru
ction filter: lo
w‐pass in
[‐W, W
], with
transitio
n ban
d fro
m in [W
, fs ‐W]
5/18/20
1017
Example
Undersam
pled sig
nal
5/18/20
1018
Example (co
ntd.)
W
ith pre‐
alias filter
5/18/20
1019
transitio
n ban
d
Low pass
Pulse‐A
mplitu
de M
odulatio
nThe am
plitu
des o
f regularly sp
aced pulses are varied
in proportio
n to th
e corresp
onding sam
ple valu
es of
g(t)
Pulse: rectan
gular
form or so
me o
ther fo
rms
Two sim
ilar techniques:
Natu
ral samplin
g
PAM (flat‐to
p sam
plin
g)
5/18/20
1020
Pulse‐A
mplitu
de M
odulatio
n
(contd.)
Two step
s:
Step
1: get th
e instan
taneo
usly sam
pled sig
nal
Pass it th
rough a lin
ear system w
ith im
pulse resp
onse
h(t) =
rect(t/T‐1/2) (see H
omew
ork 1, P
roblem
2.3)
5/18/20
1021
ns
snT
tnT
gt
g)
()
()
(
s(t)gδ (t)
ns
snT
th
nTg
th
tg
ts
)(
)(
)(
)(
)(
Pulse‐P
ositio
n
Modulatio
n
Pulse d
uratio
n m
odulatio
n
(PDM)
The sam
ples o
f the m
essage
signal are u
sed to vary th
e duratio
nof in
divid
ual p
ulses
A.k.a.: p
ulse‐w
idth
modulatio
n or p
ulse‐len
gth
modulatio
n
Pulse‐p
ositio
n m
odulatio
n
(PPM)
The p
ositio
nof a p
ulse is varied
acco
rding to th
e signal valu
e
PAM
Amplitu
de
modulatio
n
PPM
Angle m
odulatio
n
5/18/20
1022
The fallin
g ed
ges
represen
ts inform
ation
PDM
PPM
Unmodulated
signal
Inform
ation w
ave
PPM (co
ntd.)
g(t‐n
Ts ): a sin
gle p
ulse at n
Ts
m(nTs ): sam
ple to
modulate th
e positio
n of th
e n‐th
pulse
kp : sen
sitivityof P
PM
Sufficien
t conditio
n: d
ifferent p
ulses in
s(t)must b
e strictly n
on‐overlap
ping
5/18/20
1023
n
sp
snT
mk
nTt
gt
s))
((
)(
2|
)(
|m
axs
p
Tt
mk
Them
e Example –
UWB
Im
pulse rad
ioInform
ation is sen
t by m
eans o
f very narro
w pulses th
at are w
idely sep
arated in tim
e
Very n
arrow pulses
very wide b
andwidth
Ultra‐w
ideb
and(U
WB)
The p
ulse
Gau
ssian m
onocycle: th
e first derivative o
f the G
aussian
pulse
τ: p
ulse w
idth ~ betw
een 0.2 an
d 1.5 n
anoseco
nds
5/18/20
1024
UWB (co
ntd.)
Very n
arrow pulses
1 n
anoseco
nd
Very w
ide b
andwidth
W≈5G
Hz
Im
plicatio
ns
For U
WB sig
nal w
ith
limited
power, th
e signal
power in
any p
articular
narro
w ban
d is sm
all
Could be lo
wer th
an th
ermal
noise
security
Falls in
allnarro
w ban
ds
Interferes all
Power sh
ould be lim
ited
5/18/20
1025
UWB (co
ntd.)
Pulse p
ositio
n m
odulatio
n (P
PM)
Nominal sep
aration T
p
“0”–slig
htly early (‐T
c ); “1”–slig
htly late (+
Tc )
Data rate: R
=1/T
p
5/18/20
1026
]1 ,
40[
]
1000 ,
25[
Mbps
Mbps
Rns
nsT
p
The Q
uantizatio
n Process
Human sen
se can detect o
nly fin
ite inten
sity differen
cesOrig
inal co
ntin
uous sig
nal can
be ap
proxim
ated by a sig
nal
constru
cted of d
iscrete amplitu
de selected
on a m
inim
um
error b
asis from an availab
le set
Amplitu
de q
uan
tization
Tran
sform
ing th
e sample am
plitu
de m
(nTs )into discrete
amplitu
de v(n
Ts )tak
en fro
m a fin
ite set of p
ossib
le am
plitu
des
Contin
uous tim
e, contin
uous am
plitu
de
discrete tim
e, discrete am
plitu
de
Mem
oryless q
uan
tization: tran
sform
ation at tim
e t=nTsis
not affected
by earlier o
r later samples
5/18/20
1027
The Q
uantizatio
n Process (co
ntd.)
Mem
oryless q
uan
tization
Signal am
plitu
de is sp
ecified by in
dex k
if it lies insid
e the
k‐th in
terval
L: to
tal number o
f amplitu
de levels
m
k : decisio
n levels, o
r decisio
n th
resholds
Quan
tum
or step
‐size: spacin
g betw
een tw
o decisio
n levels
Output v
k : represen
tation levels o
r reconstru
cted levels
Quan
tizer characteristic: v =
g(m
)
5/18/20
1028
The Q
uantizatio
n Process (co
ntd.)
Midtread
: the o
rigin lies in
the m
iddle o
f a tread of th
e staircase‐like
grap
h
Midrise: th
e orig
in lies in
the m
iddle o
f a rising part o
f the staircase‐
like g
raph
5/18/20
1029
The Q
uantizatio
n Process (co
ntd.)
Unifo
rm quan
tization: th
e represen
tation levels are u
nifo
rmly sp
aced
Nonunifo
rm quan
tization: n
onunifo
rmly sp
aced step
‐sizes
5/18/20
1030
W. S
tallings, D
ata and Computer C
ommunica
tions, 8
th Ed., P
rentice H
all, 2007.
unifo
rmnonunifo
rm
What In
form
ation Is Lo
st?Sam
plin
g
Use sam
ples g
(nTs ), T
s =1/fs =
1/(2W), to
represen
t g(t)
The valu
es of g
(t)betw
een [(n
‐1)Ts , n
Ts ] are lo
st, for all n
Is th
is a problem
?
Quan
tization
Use d
iscrete amplitu
de v(t)
to rep
resent co
ntin
uous
amplitu
de m
(t)
Intro
duce d
istortio
n (o
r, noise) to
the in
form
ation sig
nal
Is th
is a problem
?
Human percep
tion can
tolerate certain
disto
rtions
Need
to quan
tify the level o
f noise ad
ded by q
uan
tization
5/18/20
1031
Quantizatio
n Noise
Quan
tization noise: q
= m ‐v
5/18/20
1032
Erro
r qis
bounded
by ±Δ/2
m
v
Quantizatio
n Noise M
odel
Input: zero
‐mean
random variab
le M
Quan
tizer g(·): m
aps M
of co
ntin
uous am
plitu
de in
to
a discrete ran
dom variab
le output V
V: also
zero m
ean
Q=M‐V
: quan
tization erro
r, also zero
mean
Unifo
rm quan
tizer, input am
plitu
de is in
[‐mmax ,
mmax ], L
represen
tation levels an
d is su
fficiently larg
eStep
‐size: Δ= 2m
max /L
Q
is bounded by: ‐Δ
/2 ≤q ≤Δ/2
Q
is unifo
rmly d
istributed
with probab
ility den
sity functio
n as:
5/18/20
1033
otherw
ise ,0
2/2/
- ,
/1
)(
fQ
Quantizatio
n Noise M
odel (co
ntd.)
Since Q
is zero m
ean, its p
ower (o
r, variance) is
If each
sample is rep
resented
with R
bits, th
en
R=log2 L an
dL=2R
The n
oise p
ower is:
If averag
e signal p
ower is P
S , the sig
nal‐to
‐noise ra
tio
is:
5/18/20
1034
122/
-
2/
3
11
)(
][
E2
22/2/
-
22/2/
-
22
qdq
qdq
qf
PQ
Q
RQ
mP
22m
ax 23 1
RS
Q SO
m
P
P P2
2max
23
)S
NR
(
Quantizatio
n Noise: Exam
ple
A fu
ll‐load
sinusoidal
m(t)
of am
plitu
de A
m, averag
e power is:
Full‐lo
ad
Am=m
max
Averag
e noise p
ower is:
The q
uan
tizer’s output sig
nal‐to
‐noise ratio
is:
In dB, it is:
5/18/20
1035
2/2m
SA
P
Rm
RQ
Am
P2
22
2max
23 1
23 1
R
Rm
mO
A
A2
22
2
22 3
3/2
2/
power
noise
Average
power
signal
Average
)S
NR
(
RO
dB6
8.1
)S
NR
(log
10)
SN
R(
10
Quantizatio
n Noise Exam
ple
(contd.)
5/18/20
1036
RdB
68.
1)
SN
R(
Nonunifo
rm Quantizatio
nExam
ple 7.2:
Assu
mes fu
ll‐load sig
nal, i.e., A
m=m
max
W
hat if A
m<m
max ?
If A
m=m
max /3, , 9
.6 dB lo
wer
For p
ractical systems
Fixed
L, m
maxan
d R
But sig
nal m
ay not alw
ays be fu
ll‐load ones
Voice: lo
ud talk
to w
eak talk
: 1000 to 1
60 dB variatio
n
Need
nonunifo
rm quan
tization fo
r constan
t SNR
Larg
e step‐size fo
r high am
plitu
de reg
ion, sm
all step‐size fo
r sm
all amplitu
de reg
ion
Equivalen
t to usin
g a co
mpresso
r
5/18/20
1037
RdB
68.
7)
SN
R(
Solutio
n I: N
onunifo
rm Quant.
Unifo
rm quan
tization: th
e represen
tation levels are u
nifo
rmly sp
aced
Nonunifo
rm quan
tization: n
onunifo
rmly sp
aced step
‐sizes
5/18/20
1038
W. S
tallings, D
ata and Computer C
ommunica
tions, 8
th Ed., P
rentice H
all, 2007.
unifo
rmnonunifo
rm
Solutio
n II: C
ompander
Compresso
r compressio
n law
, the μ
–law:
μis a p
ositive co
nstan
t
Slope is
5/18/20
1039
)1
log(
|)|
1log(
||
mv
|)|
1()
1log(
||
||
mv
d
md
Approxim
atelylin
ear for lo
w in
put
levels: μ|m
|<<1
Approxim
atelylogarith
m fo
r high
input levels: μ
|m|>>1
Compander (co
ntd.)
Compresso
r compressio
n law
, the A
–law:
Ais a p
ositive co
nstan
t ~ 10
0
Slope is
5/18/20
1040
1|
|1
,log
1
|)|
log(1
1|
|0
,
log1
||
||
mA
A mA
Am
A
mA
v
1|
|1/A
|,
|)
1log(
/1
||
0 ,
/)
1log(
||
||
mm
A
Am
AA
vd
md
Linear fo
r low in
put levels
Logarith
m fo
r high in
put levels
Compander (co
ntd.)
Receiver sid
e
An exp
ander: w
ith a ch
aracteristic complem
entary to
the co
mpresso
r
Compresso
r + exp
ander: co
mpan
der
5/18/20
1041
compresso
runifo
rm
quan
tizer
expan
der
Tran
smitter:
Receiver:
chan
nel
Encoding
Tran
slate the d
iscrete set of sam
ple valu
es (after quan
tization) to
a more ap
propriate fo
rm of sig
nal
Code: a p
lan fo
r such tran
slation
Codeelem
ent o
rsym
bol: a d
iscrete event in
a code
Codew
ord
or ch
aracter: a particu
lar arrangem
ent o
f sym
bols u
sed to rep
resent a sin
gle sam
ple valu
e
A binary co
de
Each
symbol m
ay be eith
er of tw
odistin
ct values
A tern
ary code
Each
symbol m
ay be eith
er of th
reedistin
ct values
Each
code w
ord co
nsists o
f Rbits, L
=2Rlevels
5/18/20
1042
Encoding (co
ntd.)
Enco
ding: estab
lish a o
ne‐to
‐one m
apping betw
een
represen
tation levels an
d co
de w
ords
For exam
ple: exp
ress the o
rdinal n
umber o
f the
represen
tation level as a b
inary n
umber
Tab
le 7.2, R=4 bits/sam
ple
5/18/20
1043
Level in
dex
0 ~ (L‐1)
Binary co
de o
flen
gth R=log2 (L
)One‐to
‐one
map
ping
Encoding
(contd.)
5/18/20
1044
Data rate =
R ×
fs
Line C
odes
Line co
de: a b
inary stream
of d
ata takes o
n an
electrical represen
tation
for tran
smissio
nRetu
rn‐to‐zero
: the p
ulse sh
ape u
sed to rep
resent th
e bit alw
ays returns to
the 0 vo
lts or th
e neu
tral level befo
re the en
d of th
e bit
Nonretu
rn‐to‐zero
: …. d
oes n
ot …
.
Unipolar
& polar
(or b
ipolar)
Unipolar N
onretu
rn‐to‐Zero
(NRZ) sig
nalin
gSym
bol 1: rep
resented
by a p
ulse o
f amplitu
de A
for th
e duratio
n of th
e symbol
Sym
bol 0
: represen
ted by sw
itching off th
e pulse
5/18/20
1045
Line C
odes (co
ntd.)
Polar N
onretu
rn‐to‐Zero
(NRZ) sig
nalin
gSym
bols 1 an
d 0 are rep
resented
by p
ulses o
f amplitu
de +
Aan
d –A, resp
ectivelyEasier to
gen
erate and m
ore p
ower‐efficien
t
Unipolar R
eturn‐to‐Zero
(RZ) sig
nalin
gSym
bol 1: p
ulse w
ith am
plitu
de A
for h
alf‐symbol w
idth
Sym
bol 0
: switch
ing off
Bipolar R
eturn‐to‐Zero
(BRZ) sig
nalin
gSym
bol 1: p
ositive an
d neg
ative pulses w
ith +Aan
d –Afor
half‐sym
bol w
idth
Sym
bol 0
: switch
ing off
Also called
Altern
ative Mark
Inversio
n (A
MI) sig
nalin
g
5/18/20
1046
Mark
symbol o
ne; S
pace
sym
bol zero
Line C
odes (co
ntd.)
Split‐P
hase sig
nalin
g (M
anch
ester Code)
Sym
bol 1: p
ositive p
ulse w
ith +Afollo
wed by a n
egative p
ulse w
ith –A,
both half‐sym
bol w
ide
Sym
bol 0
: polarities o
f the tw
o pulses are reversed
Differen
tial Enco
ding
A tran
sition to desig
nate sym
bol 0
No tran
sition to desig
nate sym
bol 1
5/18/20
1047
Line C
odes
(contd.)
Unipolar N
RZ
Polar N
RZ
Unipolar R
Z
Bipolar R
Z or A
MI
Split‐p
hase o
r Man
chester co
de
5/18/20
1048
Which
Code to
Choose
Signal sp
ectrum
high freq
uen
cies: ban
dwidth req
uirem
ents
low freq
uen
cies: allows co
uplin
g via tran
sform
er, isolatio
n
Self‐syn
chronizatio
n
with an extern
al clock or clo
ck reco
very from th
e signal
Coding/deco
ding co
mplexity
Baselin
e wan
derin
g
drift o
f average received
power fo
r differen
t signal
DC co
mponen
t:
low freq
uen
cy componen
ts, high req
uirem
ent o
n ch
annel
Built‐in
error d
etection/co
rrection
Im
munity to
noise an
d in
terference
…
5/18/20
1049
Which
Code to
Choose (co
ntd.)
5/18/20
1050
From D
ata Communica
tions a
nd Netw
orking, B
ehrouz A
. Forouzan
Example 7
.3: Tim
e Divisio
n M
ultip
lexing
5/18/20
1051
TDM (co
ntd.)
5/18/20
1052
Low sp
eed lin
ks
High sp
eed lin
k
Why n
eed bufferin
g?
Sch
edulin
g, sig
nalin
g, …
T1 System
: Transm
ission Bandwidth
Voice sig
nal:
30
0 H
z to 310
0 H
z
Low‐pass filter: W
=3.1 k
Hz
Sam
plin
g rate: 8
kHz > N
yquist rate 6
.2 kHz
Piece‐lin
ear characteristic (15 lin
ear segmen
ts) to
approxim
ate the lo
garith
mic μ
‐law, μ
=255
PCM: R
=8 bits/sam
ple
T1 systemAcco
mmodate 24
voice ch
annels
Fram
e: one b
yte per ch
annel, 24
chan
nels, 1 b
it for
synch
ronizatio
n
24×8+1=193 b
its
Sam
plin
g rate 8
kHz
125 μs fram
e perio
d
1.544 M
bps
5/18/20
1053
T1 System
5/18/20
1054
Delta M
odulatio
n
5/18/20
1055
W. S
tallings, D
ata and Computer C
ommunica
tions, 8
th Ed., P
rentice H
all, 2007.
Data ra
te?‐R=1/T
s
Quantiza
tion
error?
Delta M
odulatio
n (co
ntd.)
Provid
es a staircase approxim
ation fo
r the in
form
ation w
ave
The erro
r betw
een th
e staircase approxim
ation an
d th
e orig
inal sig
nal
is quan
tized an
d tran
smitted
5/18/20
1056
Delta M
odulatio
n (co
ntd.)
5/18/20
1057
nks
sq
ss
qs
sq
sq
ss
qs
qkT
em
nTe
TnT
eT
nTm
nTe
TnT
mnT
m1
)(
)0(
)(
)(
)2
()
()
()
(
Delta M
odulatio
n (co
ntd.)
In order fo
r the staircase ap
proxim
ation to in
crease as fast as the o
riginal sig
nal, w
e require th
e conditio
n:
Two typ
es of d
istortio
n:
W
hen th
e conditio
n
Does n
ot h
old tru
e slo
pe o
verload
slop overlo
ad disto
rtion
Gran
ular n
oise:
Occu
rs at a relatively flat
region of m
(t)
Analo
gous to
quan
tization
noise in
PCM
Conflictin
g desig
n
choices
5/18/20
1058
Them
e Example: D
igitization of
Video
and M
PEG
Video
Can be rep
resented
by th
ree dim
ensio
ns
Two dim
ensio
ns are sp
atial still im
age
Third dim
ensio
n is tem
poral
imag
e evolves o
ver time
Still im
age is rep
resented
by th
ree dim
ensio
ns
Luminan
ce (brig
htness)
Two ch
rominan
ce (color) co
mponen
ts
Spatial an
d tem
poral co
rrelation
Video co
ding
Rem
ove sp
atial and tem
poral red
undan
cy
5/18/20
1059
Them
e Example: D
igitization of
Video
and M
PEG
(contd.)
Video co
der: ap
plied
to each
of th
e three co
mponen
ts
5/18/20
1060
Them
e Example: D
igitization of
Video
and M
PEG
(contd.)
Sam
plin
g
A sam
ple o
f a video sig
nal is an
N×M
matrix
W
ith pictu
re elemen
ts (pels) p
er unit tim
e co
rresponding to a co
mplete stillim
age
Video Fram
e
Human eyes is m
ore sen
sitive to lu
minan
ce than
chrominan
ce
Lower fram
e rate (a quarter) fo
r chrominan
ce signals
MPEG co
ding
First fram
e: intrafram
e coding m
ode (I‐p
icture)
Subseq
uen
t frames: in
terframe p
redictio
n (P‐pictu
re)
5/18/20
1061
Them
e Example: D
igitization of
Video
and M
PEG
(contd.)
Discrete co
sine tran
sform (D
CT)
Closely related
to discrete F
ourier tran
sform
Iden
tifies spatial co
rrelation of th
e pels
5/18/20
1062
Σ∫
Cos
exp{}
Them
e Example: D
igitization of
Video
and M
PEG
(contd.)
Quan
tization
The co
efficients fo
r each D
CT block are q
uan
tizedOnly n
on‐zero
coefficien
ts (and th
eir positio
n) are
transm
itted
Enco
ding
Non‐zero
coefficien
ts and positio
ns are co
ded w
ith an
advan
ced data‐co
mpressio
n tech
nique, ru
n‐len
gth en
tropy
enco
ding
Motio
n‐co
mpen
sation fo
r P‐pictu
resTem
poral co
rrelation co
rresponds to
motio
n of a p
el or a
block in th
e same d
irection
Only th
e motio
n vecto
rs and a sm
all number o
f non‐zero
DCT co
efficients h
ave to be tran
smitted
lower rate
5/18/20
1063
Summary
W
hy an
alog to digital
Three step
sSam
plin
g: th
e link betw
een an
alog w
aveform an
d its
discrete‐tim
e represen
tation
Quan
tization: th
e link betw
een an
alog w
aveform an
d its
discrete am
plitu
de rep
resentatio
nEnco
ding: tran
slate discrete am
plitu
des to
a more
appropriate fo
rm of sig
nal
Sam
plin
g th
eorem
PAM, P
DM, P
PM, D
MQuan
tization noise
Nonunifo
rm quan
tization/co
mpan
ding
Tim
e divisio
n m
ultip
lexing
5/18/20
1064
top related