Outline Transmitters (Chapters 3 and 4, Source Coding and Modulation) (week 1 and 2) Receivers (Chapter 5) (week 3 and 4) Received Signal Synchronization.
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Outline• Transmitters (Chapters 3 and 4, Source Coding and
Modulation) (week 1 and 2)• Receivers (Chapter 5) (week 3 and 4) • Received Signal Synchronization
(Chapter 6) (week 5)
• Channel Capacity (Chapter 7) (week 6)• Error Correction Codes (Chapter 8) (week 7 and 8)• Equalization (Bandwidth Constrained Channels) (Chapter
10) (week 9)• Adaptive Equalization (Chapter 11) (week 10 and 11)• Spread Spectrum (Chapter 13) (week 12)• Fading and multi path (Chapter 14) (week 12)
Channel Capacity (Chapter 7) (week 6)
• Discrete Memoryless Channels
• Random Codes
• Block Codes
• Trellis Codes
Channel Models
• Discrete Memoryless Channel– Discrete-discrete
• Binary channel, M-ary channel
– Discrete-continuous• M-ary channel with soft-decision (analog)
– Continuous-continuous• Modulated waveform channels (QAM)
Discrete Memoryless Channel
• Discrete-discrete– Binary channel, M-ary channel
11
11
...
....
.)|(..
...)|(
jiji
p
pxyP
xXyYP
P
Probability transition matrix
Discrete Memoryless Channel
• Discrete-continuous– M-ary channel with soft-decision (analog)
outputx0
x1
x2
.
.
.xq-1
y
)|(
)|( 1
kxXyp
xXyp
P
22 2/)(
2
1)|(
kxy
k exyp
AWGN
Discrete Memoryless Channel
• Continuous-continuous– Modulated waveform channels (QAM)– Assume Band limited waveforms, bandwidth = W
• Sampling at Nyquist = 2W sample/s
– Then over interval of N = 2WT samples use an orthogonal function expansion:
)(
)(
1
tfx
txN
iii
)(
)(
1
tfn
tnN
iii
N
iii tfy
ty
1
)(
)(
Discrete Memoryless Channel
• Continuous-continuous– Using orthogonal function expansion:
)(
)(
1
tfx
txN
iii
)(
)(
1
tfn
tnN
iii
N
iiii
N
ii
T
i
N
ii
T
i
N
iii
tfnx
tfdttftntx
tfdttfty
tfy
ty
1
10
*
10
*
1
)(
)()()()(
)()()(
)(
)(
Discrete Memoryless Channel
• Continuous-continuous– Using orthogonal function expansion get an
equivalent discrete time channel:
Nx
x
.
.
.
.1
Ny
y
y
.
.
.2
1
22 2/)(
2
1)|( iii xy
i
ii exyp
111 nxy Gaussian noise
Capacity of binary symmetric channel
• BSC
pp
pp
1
1P
}1,0{ }1,0{ YX
0 0
11
X Yp1
p1
p p
Capacity of binary symmetric channel
• Average Mutual Information 0 0
11
X Yp1
p1
p p
)1()1()0(
1log)1)(1(
)1()0()1(log)1(
)1()1()0(log)0(
)1()0()1(
1log)1)(0(
)1(
)1|1(log)1|1()1(
)0(
)1|0(log)1|0()1(
)1(
)0|1(log)0|1()0(
)0(
)0|0(log)0|0()0();(
XPpXpP
ppXP
XpPXPp
ppXP
XPpXpP
ppXP
XpPXPp
ppXP
YP
XYPXYPXP
YP
XYPXYPXP
YP
XYPXYPXP
YP
XYPXYPXPYXI
Capacity of binary symmetric channel
• Channel Capacity is Maximum Information– earlier showed:
0 0
11
X Yp1
p1
p p
pppp
XPpXpP
ppXP
XpPXPp
ppXP
XPpXpP
ppXP
XpPXPp
ppXPYXI
XPXP
2log)1(2log)1(
)1()1()0(
1log)1)(1(
)1()0()1(log)1(
)1()1()0(log)0(
)1()0()1(
1log)1)(0());(max(C
2
1)0()1(
2
1)0()1());(max( XPXPYXI
Capacity of binary symmetric channel• Channel Capacity
– When p=1 bits are inverted but information is perfect if invert them back!
0 0
11
X Yp1
p1
p p
pppp 2log)1(2log)1(C 22
Capacity of binary symmetric channel• Effect of SNR on Capacity
– Binary PAM signal (digital signal amplitude 2A)
0 0
11
X Yp1
p1
p p
)(tg
)(tg)(tg )(tg
)(tgA
A
02
02
/)(
0
2
/)(
0
1
1)|(
1)|(
Nr
Nr
b
b
eN
srp
eN
srp
AGWN
Capacity of binary symmetric channel• Effect of SNR
– Binary PAM signal (digital signal amplitude 2A)
0 0
11
X Yp1
p1
p p
)(tg
)(tg)(tg )(tg
)(tgA
A
)|(2
1)|(
20
0 /)(
0
10
2
sePN
Q
dreN
seP
b
Nr b
0
1s
Capacity of binary symmetric channel• Effect of SNR
– Binary PAM signal (digital signal amplitude 2A)
0 0
11
X Yp1
p1
p p
0
0
22
112
1
2
12
22
2
)|()|(
NAQ
QSNRQ
NQ
sePsePP
bb
b
b
Capacity of binary symmetric channel• Effect of SNR
– Binary PAM signal (digital signal amplitude 2A)
0 0
11
X Yp1
p1
p p
noise rms
22
4
22
2
12
2 noise rms
2
1
2
1
0
0
0
AQ
AQ
NAQ
NAQP
N
b
Not sure about thisDoes it depend on bandwidth?
Capacity of binary symmetric channel• Effect of SNR
– Binary PAM signal (digital signal amplitude 2A)
0 0
11
X Yp1
p1
p p
noise rms 2
Amplitudeerfc
2
1
noise rms
Amplitude
2
1
2
1QpPb
pppp 2log)1(2log)1(C
Capacity of binary symmetric channel• Effect of SNR
– Binary PAM signal (digital signal amplitude 2A)
0 0
11
X Yp1
p1
p p
SNR Pb1 0.3085382 0.1586553 0.0668074 0.022755 0.006216 0.001357 0.0002338 3.17E-059 3.4E-06
10 2.87E-0711 1.9E-0812 9.87E-1013 4.02E-1114 1.28E-1215 3.19E-1416 6.11E-16
Pb (BER) vs SNR for binary channel
1.40E+01, 1.28E-12
1.20E+01, 9.87E-10
1E-16
1E-15
1E-14
1E-13
1E-12
1E-11
1E-10
1E-09
1E-08
1E-07
1E-06
1E-05
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
SNR = A/rms noise
BE
R
noise rms 2
Amplitudeerfc
2
12
1bP
Capacity of binary symmetric channel• Effect of SNR
– Binary PAM signal (digital signal amplitude 2A)
0 0
11
X Yp1
p1
p p
pppp 2log)1(2log)1(C 22 SNR Pb C
0 0.5 01 0.308538 0.1085222 0.158655 0.3689173 0.066807 0.6461064 0.02275 0.8433855 0.00621 0.9455446 0.00135 0.9851857 0.000233 0.9968578 3.17E-05 0.9994819 3.4E-06 0.999933
10 2.87E-07 0.99999311 1.9E-08 0.99999912 9.87E-10 113 4.02E-11 114 1.28E-12 115 3.19E-14 116 6.11E-16 1
Capacity © and BER vs SNR for binary channel
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
SNR = A/rms noise
BE
R
noise rms 2
Amplitudeerfc
2
12
1bP
At capacity SNR = 7, so waste lots of SNR to get low BER!!!
Capacity of binary symmetric channel• Effect of SNRb
– Binary PAM signal (digital signal amplitude 2A)
0 0
11
X Yp1
p1
p p
pppp 2log)1(2log)1(C 22 SNRb (dB)Pb C
-20 0.443769 0.009143-18 0.429346 0.014452-16 0.411325 0.022809-14 0.388906 0.03591-12 0.361207 0.056319-10 0.32736 0.087793-8 0.286715 0.135561-6 0.239229 0.206245-4 0.186114 0.306729-2 0.130645 0.4407970 0.07865 0.6025972 0.037506 0.7692614 0.012501 0.903056 0.002388 0.9757578 0.000191 0.997366
10 3.87E-06 0.99992512 9.01E-09 114 6.81E-13 1
Capacity C and BER vs SNR for binary channel
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-20 -12 -4 4 12
SNR per bit (dB)
Ca
pa
cit
y C
an
d B
ER
b
bb QpP
erfc2
1
2
Channel Capacity of Discrete Memoryless Channel
• Discrete-discrete– Binary channel, M-ary channel
11
11
...
....
.)|(..
...)|(
jiji
p
pxyP
xXyYP
P
Probability transition matrix
Channel Capacity of Discrete Memoryless Channel
Average Mutual Information
1
0
1
1
1
0
1
1
)(
)|(log)|()(
)(
)|(log)|()();(
q
j
Q
i j
jijij
q
j
Q
i j
jijij
yP
xyPxyPxP
yYP
xXyYPxXyYPxXPYXI
Channel Capacity of Discrete Memoryless Channel
Channel Capacity is Maximum InformationOccurs for only if Otherwise must work out max
1)( ,0)(
1
0
1
1)(
1
0
1
1)()(
1
0
)(
)|(log)|()(max
)(
)|(log)|()(max);(max
q
jjj
j
jj
xPxP
q
j
Q
i j
jijij
xP
q
j
Q
i j
jijij
xPxP
yP
xyPxyPxP
yYP
xXyYPxXyYPxXPYXIC
jpxP j allfor ,)( symmetric P
)( jxP
Channel Capacity Discrete Memoryless Channel
• Discrete-continuous
• Channel Capacity
x0
x1
x2
.
.
.xq-1
y
)|(
)|( 1
kxXyp
xXyp
P
1
0
1
0)()(
)|()()(
where
)(
)|(log)|()(max);(max
q
iii
q
i
iii
xPxP
xXyYpxXPyYp
dyyYp
xXyYpxXyYpxXPYXIC
ii
Channel Capacity Discrete Memoryless Channel
• Discrete-continuous
• Channel Capacity with AWGN
x0
x1
x2
.
.
.xq-1
y
22 2/)(
2
1)|(
kxy
k exyp
1
01
0
2/)(
2/)(
2/)(
)( 22
22
22
2
1)(
2
1
log2
1)(max
q
iq
i
xyi
xy
xyi
xPdy
exXP
eexXPC
i
i
i
i
Channel Capacity Discrete Memoryless Channel
• Binary Symmetric PAM-continuous
• Maximum Information when:
x0
x1
x2
.
.
.xq-1
y
2
1)()( AXPAXP
dyeee
e
dyeee
eC
y
AA
A
y
AA
A
22
2222
22
22
2222
22
2/
2/2/
2/2
2/
2/2/
2/2
2
1
2log
2log
2
1
Channel Capacity Discrete Memoryless Channel
• Binary Symmetric PAM-continuous
• Maximum Information when:
dyeee
e
dyeee
eC
y
AA
A
y
AA
A
22
2222
22
22
2222
22
2/
2/2/
2/2
2/
2/2/
2/2
2
1
2log
2log
2
1
Channel Capacity Discrete Memoryless Channel
• Binary Symmetric PAM-continuous
• Versus Binary Symmetric discrete SNRb (dB)Pb C-20 0.443769 0.009143-18 0.429346 0.014452-16 0.411325 0.022809-14 0.388906 0.03591-12 0.361207 0.056319-10 0.32736 0.087793-8 0.286715 0.135561-6 0.239229 0.206245-4 0.186114 0.306729-2 0.130645 0.4407970 0.07865 0.6025972 0.037506 0.7692614 0.012501 0.903056 0.002388 0.9757578 0.000191 0.997366
10 3.87E-06 0.99992512 9.01E-09 114 6.81E-13 1
Capacity C and BER vs SNR for binary channel
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-20 -12 -4 4 12
SNR per bit (dB)
Ca
pa
cit
y C
an
d B
ER
Discrete Memoryless Channel
• Continuous-continuous– Modulated waveform channels (QAM)– Assume Band limited waveforms, bandwidth = W
• Sampling at Nyquist = 2W sample/s
– Then over interval of N = 2WT samples use an orthogonal function expansion:
)(
)(
1
tfx
txN
iii
)(
)(
1
tfn
tnN
iii
N
iii tfy
ty
1
)(
)(
Discrete Memoryless Channel
• Continuous-continuous– Using orthogonal function expansion get an
equivalent discrete time channel:
Nx
x
.
.
.
.1
Ny
y
y
.
.
.2
1
22 2/)(
2
1)|( iii xy
i
ii exyp
111 nxy Gaussian noise
Discrete Memoryless Channel
• Continuous-continuous
• Capacity is (Shannon))(
)(
1
tfx
txM
iii
)(
)(
1
tfn
tnM
iii
M
iii tfy
ty
1
)(
)(
);(1
maxlim)(
YXIT
CxpT
ii
N
i i
iijii
NNN
NNNNNNN
dxydyp
xypxpxyp
ddp
pppI
WTN
NN
1 )(
)|(log)()|(
)(
)|(log)()|();(
2
XY
yxy
xyxxyYX
22 2/)(
2
1)|( iii xy
i
ii exyp
Discrete Memoryless Channel
• Continuous-continuous
• Maximum Information when:
0
2
0
2
1 0
2
2
1
)(
21log
21log
2
1
21log);(max
NWT
NN
NI
x
x
N
i
xNN
xp
YX
22 2/
2
1)( xix
x
i exp
Statistically independent
zero mean Gaussian inputs
then
Discrete Memoryless Channel
• Continuous-continuous
• Constrain average power in x(t):
22
1
2
0
2
2
)(2
1
)]([1
xx
N
ii
T
av
WT
N
xE
dttxET
P
Discrete Memoryless Channel
• Continuous-continuous
• Thus Capacity is:)(
)(
1
tfx
txM
iii
)(
)(
1
tfn
tnM
iii
M
iii tfy
ty
1
)(
)(
0
0
2
)(
1log
21loglim
);(1
maxlim
WN
PW
NW
IT
C
av
x
T
NNxpT
YX
22 2/)(
2
1)|( iii xy
i
ii exyp
Discrete Memoryless Channel
• Continuous-continuous
• Thus Normalized Capacity is:)(
)(
1
tfx
txM
iii
)(
)(
1
tfn
tnM
iii
M
iii tfy
ty
1
)(
)(
WCN
WN
C
CPWN
P
W
C
WCb
b
bavav
/
12
1log
but ,1log
/
0
02
02
22 2/)(
2
1)|( iii xy
i
ii exyp
etab/No (dB)C/W-1.44036 0.1-1.36402 0.15-1.24869 0.225-1.07386 0.3375-0.8075 0.50625
-0.39875 0.7593750.234937 1.1390631.230848 1.7085942.822545 2.5628915.41099 3.844336
9.669259 5.76650416.65749 8.64975627.92605 12.9746345.69444 19.4619573.22669 29.19293115.4055 43.78939179.5542 65.68408 0.1
1
10
-10 0 10 20 30
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