CNU Dept. of Electronics D. J. Kim 1 Lecture on Communication Theory Chapter 8. Digital Passband Transmission 8.1 Introduction Analog Digital Pas sband Amplitude Mod. ASK ( Amplitude Shift Keying ) Freq. Mod FSK ( Frequency Shift Keying ) Phase Mod PSK ( Phase Shift Keying ) keying means switching Coherent RX : phase locked Noncoherent RX : Phase unlocked 8.2 Passband Transmission Model
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Chapter 8. Digital Passband Transmission 8.1 Introduction
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CNU Dept. of Electronics
D. J. Kim1
Lecture on Communication Theory
Chapter 8. Digital Passband Transmission
8.1 Introduction
Analog Digital Passband
Amplitude Mod. ASK ( Amplitude Shift Keying )Freq. Mod FSK ( Frequency Shift
8.3 Gram-Schmidt Orthogonalization Procedure: Representation of any set of M energy signals { si(t) } as a linear combinations of N orthogonal basis functions where N M.
- Real-valued energy signals s1(t), s2(t),, sM(t) is
in the form
where
- Real valued basic functions are orthonormali.e.
Mi
Tttsts
N
jjiji ,,2,1
0 )()(
1
NjMi
dtttssT
jiij ,,2,1,,2,1
)()(0
jiji
dtttT
ji if 0 if 1
)()(0
CNU Dept. of Electronics
D. J. Kim5
Lecture on Communication Theory
Gram-Schmidt orthogonalization procedure ; Basis functions 을 구하는 방법
Input s1(t), s2(t),, sM(t)
Let
Def
Then
Def
so
1
11
)()(
Ets
t
111111111 e wher )()()( EststEts
T
dtttss0 1221 )()(
)()()( 12122 tststg
orthogonal ; 0)()( 21210 12 ssdtttgT
normalize ;
Tdttg
tgt
0
22
22
)(
)()(
221
22
1212 )()(
sE
tsts
TT
dtttdtt0 210
22 0)()( ,1)(
CNU Dept. of Electronics
D. J. Kim6
Lecture on Communication Theory
In general form
where
Def
-
Examples
- Fourier series expansion of a periodic signal
- Representation of a band-limited signal in terms of its
samples taken at the Nyquist rate
1
1
)()()(i
jjijii tststg
1,,2,1,1)()(0
ijdtttssT
jiij
Ndttg
tgtT
i
ii ,1,2, i
0
2 )(
)()(
for and tindependenlinearly not areThey
set tindependenlinearly a form signal The
NitgMN
MNtststs
i
Mi
0)(
)(,),(),( 2
CNU Dept. of Electronics
D. J. Kim7
Lecture on Communication Theory
Example
Ex1) Gram-Schmidt Orthogonality Procedure
CNU Dept. of Electronics
D. J. Kim8
Lecture on Communication Theory
and
T TdttsE0
211 3
)(
otherwise 0, 0 3,3)()(
1
11
TtTEtst
T T TdtT
dtttss0
3
01221 33)1()()(
T TdttsE0
222 3
2)(
otherwise
,03233)()()(
221
22
12122
TtTT
sE
tstst
031 s
32
332 33)1(
T
T
TdtT
s
)()()()( 232131313 tststst
otherwise 0
1 TtTtstststg
32)()()()( 23213133
otherwise ,
,0323
)(
)()(
0
23
33
TtTT
dttg
tgtT
CNU Dept. of Electronics
D. J. Kim9
Lecture on Communication Theory
for 0)(g ,4 4 ti
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D. J. Kim10
Lecture on Communication Theory
8.4 Geometric Interpretation of signalsIn a vector form of signal
where
Mi
s
ss
iN
i
i
i ,,2,12
1
s
N
jjiji tsts
1
)()(
CNU Dept. of Electronics
D. J. Kim11
Lecture on Communication Theory
- :signal vector
- Signal space : N-dimensional Euclidean space
- Length or absolute value or norm :
Squared-length
- Cosine of angle between vector and
- Euclidean distance
is
is
N
jiji
Tii s
1
22 sss
is js
ji
jTi
ij ss
sscos
jiikd ss
N
jkjijkiik ssd
1
222 ss
T
ki dttsts0
2)()(
CNU Dept. of Electronics
D. J. Kim12
Lecture on Communication Theory
8.5 Response of bank of correlators to noisy input
Received signal
Output of correlator j
11 wsi
)(tx22 wsi
NiN ws
,M,,i
Tttwtstx i 21
0 )()()(
T
jj
T
jiij
T
jijjj
dtttww
dtttss
Njwsdtttxx
0
0
0
)()(
)()( where
,,2,1 )()(
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Lecture on Communication Theory
Statistical Characterization of the Correlator Output
Let
Mean
Covariance
, of fct sample with x(t)prtX ..:)(
jj wvrW value sample with..:
jijjX WsEXEj
μ
ijjij sWEs
222 varσ jijjjX WEsXEXj
jN
duuuWdtttWET
j
T
j
all for 2
)()()()(
0
00
)]μ)(μ[(]cov[kj XkXjkj XXEXX
kj
WWE kj
for 0
][
)(2
0 tWN process noise the ofdensity spectral power : where
CNU Dept. of Electronics
D. J. Kim14
Lecture on Communication Theory
NX
XX
2
1
X
NXXX ,,, 21
x
Mimxfmf ij
N
jXi j
,,2,1 , 1
xX
yprobabilit transition channel
fcts likelihood
element nobservatio vector nobservatio
:
::
i
j
mf
x
x
x
X
2
00
1exp1ijjijX sx
NNmxf
j
MisxN
NmfN
jijj
Ni ,,2,11exp
1
2
0
20
xX
Define the vector of N random variables
are statistically independent
Conditional probability density fct of
위 식을 만족하는 channel : memoryless channel
여기서
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Lecture on Communication Theory
8.6 Coherent Detection of Signals in Noise
- M possible signals :
- Received signal
- Receiver’s job : “best estimate” of TXed signal
- Represent as appropriate set of N orthonormal basis functions
represent by a point in N-dimensional Euclidean space
i.e. , transmitted signal point or message point
- Signal Constellation : Set of message points
- Decision making process : Given observation vector x,
Pe 를 최소화 하도록 x 를 mi 의 estimate 으로
mapping
)(,),(),( 21 tststs M
Mi
Tttwtstx i ,,2,1
0)()()(
2)( 0Ntw psd where 의
)(tsi
)(tsi
)(tsi
Miwi ,,2,1 , sx
m
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Lecture on Communication Theory
1. Maximum Likelihood Decoder
1) MAP (Maximum a Posteriori) rule
Optimal decision
Select which minimize
Select which maximize
In other words, select i
: prior probability of occurrence of symbol mi
: likelihood fct that results when symbol mk
is transmitted
imm ˆ
xx sentnot , iie mPmP
xsent 1 imP
im
)(x
xx
X
X
fmfP
mP iii sent
)(
maxmaxxx
xX
X
fmfp
mP ii
iiisent
ip
kmf xX
im
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Lecture on Communication Theory
MAP 의 예
2) ML Decoder
From MAP
is independent of the transmitted signal,
Assume with equal probability
Then
xXf im
ik pp
Decoder ML select
iimf
ixXmax
21
11 PP if
31
32
1
1
P
P
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D. J. Kim18
Lecture on Communication Theory
In other wordsSet if is maximum for k=i
Likelihood fct
So, is a monotone increasing fct.
Called the metric
ML rule
3) Graphical interpretation of ML decision rule
Let Z denote the N-dimensional space of all possible
observation vector x
i.e. Z : observation space
Z is partitioned into M decision regions
Decision rule :
observation vector x lies in region Z i
if is maximum for k=i
imm ˆ kmf xX
0imf xX
imf xXln
iimf
ixXln
selectmax
MZZZ ,,, 21
kmf xXln
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Lecture on Communication Theory
Consider AWGN Channel
Metric
Decision rule :
Observation vector x lies in region Zi
if is maximum for k=i
or if is minimum for k=i
or if is minimum for k=i
Conclusion : ML decision rule is simply to choose
the message point closest to the received
signal point
MksxN
NmfN
jkjj
Nk ,,2,11expπ
1
2
0
20
xX
MksxN
NNmfN
jkjjk ,,2,1 1ln
2ln
1
2
00
xX
N
jkjj sx
N 1
2
0
1
N
jkjj sx
1
2
ksx
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Lecture on Communication Theory
Examples
boundary Decision
)( axf X
aa
)( axf X
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Lecture on Communication Theory
8.A Constellation ( 성상도 )
1. Example of constellations
The alphabet is the set of symbols that are available for transmission
- Binary antipodal (BPSK)
- a a
}A{I km
}A{R ke
Figure 1. Two popular constellations for passband PAM transmission. The constants b and c affect the power of the transmitted signal
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Lecture on Communication Theory
2. Slicer
- makes the decision about the intended symbol.
- selects that minimizes
- ML : maximum likelihood MAP : maximum a - posteriori detector
3. Minimum Distance
- Euclidean distance 두점 사이의 거리
- dmin 성상도에서 가장 가까운 두점 사이의 거리
8PSK 4VSB
kA2ˆ kk A-Q
-3 -1 1 3
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Lecture on Communication Theory
4. Decision Regions
Figure 2 . Received samples perturbed by additive Gaussian noise form a Gaussian cloud around each of the points in the signal constellation
Figure 3. The ML detectors for the constellations in Figure 1 have the decision regions shown.
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Lecture on Communication Theory
5. Power constraintsex)
Telephone channel : average power is constrained by regulation P(voiceband data signal) P(voice signal) to limit crosstalk interference.
Radio channel : Power regulation to avoid interference with other radio SVC to avoid nonlinearity in the RF circuitry
Transmitted power
where T : symbol interval
power constraint ;
sets an upper bound on the minimum distance dmin
for any given constellation design.
221gAX T
P
whiteis sequence symbol assume )(σ 2 jwTAA eS
dwjwGg
22 )(21σ
PσσT
P gAX 221
22
gA
PT
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Lecture on Communication Theory
6. Constellation Design
1) objective : maximize the distance btw symbols
while not exceeding the power constraint
2) 방법
zero mean
모든 점들간의 거리가 똑같이
모든 점들이 동일 원 내에 분포
3) QAM
Figure 4. Some QAM constellations.
Figure 5. Cross constellations.
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Lecture on Communication Theory
4) AM-PM( 복잡 )
5) Hexagonal constellations ( 복잡 )
6) Higher Dimensions ; 3rd, 4th-order
Figure 6. Constellations using phase-shift keying and amplitude modulation.
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Lecture on Communication Theory
8.7 Probability of Error
- Observation space Z is partitioned, in accordance with ML
decision rule, into a set of M regions
- Suppose symbol mi (signal vector Si) is TXed.
- Observation vector : x
Average probability of symbol error Pe
< Problem > Numerical computation of integral is impractical
MiZ i ,,2,1
sent sent in lie not does i
M
iiie mPmZPP
1
x
M
iii mZP
M 1
1 sent in lie not does x
M
iii mZP
M 1
11 sent in lies x
M
iZ i
i
dmxfM 1
11 xx
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Lecture on Communication Theory
1. Union Bound on the Probability of Error Example of QPSK
In General
확률가까울에보다가
는
vector nobservatio
where
k
k
e
ss
ssPssPssPssPsP
1
12
4123122121
,,,,
x
MiPmPM
ikk
kiie ,,2,1,1
2
ss
sent is when, than to closer is iikki PP sssxss ,2
2 0
2
0
exp1ikd du
Nu
N
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Lecture on Communication Theory
where
Thus
Avg. Probability of symbol error
For symmetric geometry
Consider
Thus
kiikd ss where
0221
Ndikerfc
u
dzzu 2exp2)
erfc(
MiN
dmP ikie ,,2,1
221
0
erfc
erfc1
M
i
M
ikk
ikM
iiee N
dM
mPM
P1 01 22
11
all for erfc iN
dmPPM
ikk
ikiee
1 0221
-exp1erfc
0
2
0 22 Nd
Nd ikik
all for iN
dPM
ikk
ike
1 0
2
2exp
π21
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Lecture on Communication Theory
If TXed signal power >> noise spectral density No
Approximation of the bound
Mmin : # of TXed signals that attain the minimum Euclidean distance for each mi
0
2
,
min
2minexp
π2 NdMP ik
kikie
2min M 4min M
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Lecture on Communication Theory
2. Bit Versus Symbol Error Probabilities
BER ( Bit Error Rate ) or Probability of bit errorm bits 1 symbol
1) Case 1Gray code : 옆 심볼과는 1 bit 만 차이 나게 design
옆 심볼과 error 가 날 경우 1 bit error 만 나게 함
In general
Mm 2log
MPe
2log BER
ee PM
P BER 2log
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Lecture on Communication Theory
2) Case 2
Let M=2K, where K is an integer
Assume that all symbol errors are equally likely and occur with
probability
where Pe is avg. probability of symbol error
Error 가 있는 symbol 에서 번째 i bit 가 error 일 확률 (2K-1
경우의 수 )
121
Kee P
MP
eeK
K-
PM
MP
12
122BER
1
2BER lim e
M
P
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Lecture on Communication Theory
8.8 Correlation Receiver
- ML Decision rule
Observation vector x lies in region Zi
if is minimum for k=i
- Consider
- Other form of ML Decoder
Observation vector x lies in region Zi
if is maximum for k = i
where ; energy
ksx
N
jkj
N
jkjj
N
jj
N
jkjj ssxxsx
1
2
11
2
1
2 2
k
N
jkjjkkk
Esx21maxmin
1
sx
k
N
jkjj Esx
21
1
N
jkjk sE
1
2
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Lecture on Communication Theory
- Implementation
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Lecture on Communication Theory
8.9 Detection of signals with unknown phase
noiseUncertaninty
phase
Synchronization with the phase of the carrier : costly