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 10 and 11)
•
•
•
•
Equalization (Bandwidth Constrained Channels) (Chapter 10)
Adaptive Equalization (Chapter 11)
Spread Spectrum (Chapter 13)
Fading and multi path (Chapter 14)
Error Correction Codes (Chapter 8)
• Trellis/Convolution Codes
• Trellis Codes and QAM
NRZI code with memory L=1
Trellis for NRZI
Convolution Code Generator
Viterbi Algorithm
QAM set partition via convolution code
8 state Trellis for rectangular QAM
Trellis/Convolution Codes
• Modulation with Memory
– Example: NRZI
• Trellis diagram
• Maximum likelihood sequence
detectors
NRZI code with memory L=1
Trellis for NRZI
– Viterbi Algorithm
Viterbi Algorithm
Modulation with Memory
• Example: NRZI (non return to zero invert)
– Binary PAM
NRZI code with memory L=1
Modulation with Memory
• Example: NRZI (non return to zero invert)
– Encoded by:
ak
Modulo 2 addition

bk 1
bk
Has one memory and 1/1 code rate
(No error correction)
Trellis diagram
• Indicates state transitions for various inputs
ak

bk 1
bk
Trellis Diagram for NRZI
Maximum likelihood sequence
detectors
• Consider NRZI matched filter decoder
output:
rk    b  nk
p (rk | s1 ) 
p (rk | s2 ) 
1
2 n
1
2 n
e
 ( rk   b ) 2 / 2 n2
e
 ( rk   b ) 2 / 2 n2
Maximum likelihood sequence
detectors
• Consider joint pdf of a matched filter
decoder output sequence and a transmitted
(m )
sequence s : m = 1,2,…M (M symbols)
K
p (r1 , r2 ,  rK | s
(m)
1
 ( rk 
e
2 n
)
k 1
 1

 2
n

K
K
 
 e k 1



 b ) 2 / 2 n2
( rk  s k( m ) ) 2 / 2 n2
Maximum likelihood sequence
detectors
• Now find maximum likely hood:
max( p (r1 , r2 ,  rK | s
(m)
))

K
min( D(r, s
(m)
)   (rk  s
k 1
(m) 2
k
)
Minimizes Euclidean distance
Maximum likelihood sequence
detectors
•
Viterbi algorithm
–
Eliminates sequences as data is collected
1. compute distances as you go
2. Keep only the smallest distances for each
trellis state at each symbol time
3. After some number of symbols K decide
what the first symbol was by:
a) Consensus or b) Minimum D path
4. Do this for each time step
Maximum likelihood sequence
detectors
ak
• Viterbi algorithm for NRZI
s1
s r s0
k
D(r, s
b
1
2
3
4
5
6
7
s
1
1
1
1
0
0
1
r
shat
0 0.00
0
0.00
1 1.00
0 -0.01
1 1.01
0 -0.01
0 0.01
0
0
0
0.99
0
0.00
0
1.02
0
0.00
0
0.00
0
0.00
s=
s=
s=
s=
s=
s=
bhat
0
0
0.99
0
0
0.99
0
0
1.02
0
0
1.02
0
0
0.00
0
0
0.00
bhat
1
1
0.00
1
1
2.01
1
1
0.00
1
1
2.03
1
1
0.98
1
1
1.00
shat
1
0.99
s=
1
0.00
s=
1
1.02
s=
1
0.00
s=
1
1.01
s=
1
0.98
s=
1
0.99
bhat
0
1
0.99
0
1
1.02
0
1
1.02
0
1
1.01
0
1
2.00
0
1
1.98
bhat
1
0
1.98
1
0
0.00
1
0
2.04
1
0
0.00
1
0
1.01
1
0
0.98

K
(m)
)   (rk  sk( m ) ) 2
k 1
The example
shows how
the Viterbi
algorithm by
trying all
possible
states and
digital inputs
can correctly
estimate the
digital inputs
(green)
bk 1
bk
Maximum likelihood sequence
detectors
ak
• Viterbi algorithm for NRZI
s1
s r s0
k
D(r, s
b
1
2
3
4
5
6
7
s
0
1
1
0
0
1
1
r
shat
1 1.01
0
1.02
1 1.00
0 0.00
1
1
1 0.99
1 1.01
0 0.01
0
1.00
0
0.00
0
1.00
0
0.98
0
1.02
0
0.00
s=
s=
s=
s=
s=
s=
bhat
0
0
2.02
0
0
1.00
0
0
1.00
0
0
1.98
0
0
2.00
0
0
1.02
bhat
1
1
1.02
1
1
2.01
1
1
0.00
1
1
1.00
1
1
0.98
1
1
2.01
shat
1
0.00
s=
1
0.00
s=
1
1.01
s=
1
0.00
s=
1
0.00
s=
1
0.00
s=
1
0.98
bhat
0
1
0.00
0
1
1.01
0
1
1.01
0
1
0.00
0
1
0.00
0
1
0.99
bhat
1
0
1.00
1
0
0.00
1
0
2.00
1
0
0.98
1
0
1.02
1
0
0.00

K
(m)
)   (rk  sk( m ) ) 2
k 1
Try it again
with different
initial state
and data.
The trellis to
the right
matches the
Excel
spreadsheet
bk 1
bk
Trellis/Convolution Codes
• The general Convolution Code
K=number of stages
k=number of input bits per stage
n=number of output bits
The code rate
Rc  k n
Represents how much faster data must be sent
Trellis/Convolution Codes
• The general Convolution Code
– The code is described by the “generators”
• i.e., the weights on each mod 2 adder
– e.g., for K = 3, k = 1, n = 3, Rc  k n  1 3
g1  1 0 0  4 (octal)
1
g 2  1 0 1  5 (octal)
g 3  1 1 1  7 (octal)
0
1
1
1
1
Trellis/Convolution Codes
• The general Convolution Code
– There are better codes
– K = 3, k = 1, n = 3, Rc  k n  1 3
– With generator 5 7 7 is optimal
g1  1 0 1  5 (octal)
g 2  1 1 1  7 (octal)
g 3  1 1 1  7 (octal)
Trellis/Convolution Codes
• The general Convolution Code
– Trellis and Transfer function
• Tell us dfree = minimum hamming distance
between paths through the Trellis
– Coding gain:
Coding gain  10 log 10 ( Rc d free )
e.g., Rc  1 3 , d free  6
Coding gain  10 log 10 (1 3  6)  3dB
e.g., Rc  1 3 , d free  8
Coding gain  10 log 10 (1 3  8)  4.3dB
Maximum likelihood sequence
detector
• Viterbi algorithm for K = 3, k = 1, n = 3,with generator 5 7 7
1
binsbin
1
g1 1
g2 1
g3 1
1
0
1
1
2 1
g1
g2
g3
1
1
1
1
1
0
1
1
3 0
g1
g2
g3
0
1
1
1
1
0
1
1
4 0
g1
g2
g3
0
1
1
1
0
0
1
1
boutsigma=
0.01
r
1
1 0
-0.01
1 1
0.99
1 1
1.00
metric=
next state=
1
1 0
0.03
1 1
1.00
1 1
1.00
metric =
next state=
1
1 1
0.99
1 0
-0.01
1 0
-0.01
metric =
next state=
1
1 1
1.01
1 1
1.01
1 1
1.00
metric =
next state=
shat
0 0
1 0
1 1
1 1
0
1
1
1
0
1
1
1
0
0
0
1
1
0
0
1
1
1
0
1
1
1
0
0
0
1
1
0
0
1
1
1
0
1
1
1
0
0
0
1
1
0
0
1
1
1
0 0
shat
1 0
0 1 0
0 1 1
0 1 1
1.98
1
1 0
0 1 0
0 1 1
0 1 1
3.01
1
1 0
0 1 0
0 1 1
0 1 1
1.92
1
1 0
0 1 0
0 1 1
0 1 1
4.95
1
0
1
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
0
shat
0 0
1 1 0
1 1 1
1 1 1
1.01
0
0 0
1 1 0
1 1 1
1 1 1
1.95
0
0 0
1 1 0
1 1 1
1 1 1
2.99
0
0 0
1 1 0
1 1 1
1 1 1
1.92
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
shat
1 0
1 1 0
1 1 1
1 1 1
1.01
1
1 0
1 1 0
1 1 1
1 1 1
0.94
1
1 0
1 1 0
1 1 1
1 1 1
3.06
1
1 0
1 1 0
1 1 1
1 1 1
0.00
1
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
shat
0 1
0 1 0
0 1 1
0 1 1
1.98
0
0 1
0 1 0
0 1 1
0 1 1
2.00
0
0 1
0 1 0
0 1 1
0 1 1
1.99
0
0 1
0 1 0
0 1 1
0 1 1
3.03
0
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
shat
1 1
0 1 0
1 1 1
1 1 1
0.00
1
1 1
0 1 0
1 1 1
1 1 1
1.01
1
1 1
0 1 0
1 1 1
1 1 1
4.98
1
1 1
0 1 0
1 1 1
1 1 1
3.01
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
shat
0 1
1 1 0
0 1 1
0 1 1
2.99
0
0 1
1 1 0
0 1 1
0 1 1
3.95
0
0 1
1 1 0
0 1 1
0 1 1
1.95
0
0 1
1 1 0
0 1 1
0 1 1
4.01
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
shat
1 1
1 1 0
0 1 1
0 1 1
2.99
1
1 1
1 1 0
0 1 1
0 1 1
2.94
1
1 1
1 1 0
0 1 1
0 1 1
0.00
1
1 1
1 1 0
0 1 1
0 1 1
3.97
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0.00
0
1
1
0.00
0
1
1
3.03
0
1
1
2.97
Maximum likelihood sequence
detector
• Viterbi algorithm for K = 3, k = 1, n = 3,with generator 5 7 7
• Example of error correction? sigma = 0.5
• Decision at step 11 corrects bit error at step 2
1
binsbin
1
g1 1
g2 1
g3 1
1
0
1
1
2 0
g1
g2
g3
0
1
1
1
1
0
1
1
3 0
g1
g2
g3
0
1
1
1
0
0
1
1
4 1
g1
g2
g3
1
1
1
1
0
0
1
1
boutsigma=0.5 r
1
1 0
0.17
1 1
1.06
1 1
0.85
metric=
next state=
1
1 1
0.88
1 0
0.41
1 0
0.74
metric =
next state=
1
1 1
1.65
1 1
1.58
1 1
0.98
metric =
next state=
0
1 1
0.94
1 1
1.67
1 1
1.38
metric =
next state=
shat
0 0
1 0
1 1
1 1
0
1
1
1
0
1
1
1
0
0
0
1
1
0
0
1
1
1
0
1
1
1
0
0
0
1
1
0
0
1
1
1
0
1
1
1
0
0
0
1
1
0
0
1
1
1
0 0
shat
1 0
0 1 0
0 1 1
0 1 1
1.86
1
1 0
0 1 0
0 1 1
0 1 1
2.22
1
1 0
0 1 0
0 1 1
0 1 1
6.68
1
1 0
0 1 0
0 1 1
0 1 1
7.13
1
0
1
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
0
shat
0 0
1 1 0
1 1 1
1 1 1
0.72
0
0 0
1 1 0
1 1 1
1 1 1
1.15
0
0 0
1 1 0
1 1 1
1 1 1
1.24
0
0 0
1 1 0
1 1 1
1 1 1
2.16
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
shat
1 0
1 1 0
1 1 1
1 1 1
0.72
1
1 0
1 1 0
1 1 1
1 1 1
0.48
1
1 0
1 1 0
1 1 1
1 1 1
1.56
1
1 0
1 1 0
1 1 1
1 1 1
4.81
1
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
shat
0 1
0 1 0
0 1 1
0 1 1
1.86
0
0 1
0 1 0
0 1 1
0 1 1
1.55
0
0 1
0 1 0
0 1 1
0 1 1
6.99
0
0 1
0 1 0
0 1 1
0 1 1
9.79
0
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
shat
1 1
0 1 0
1 1 1
1 1 1
0.05
1
1 1
0 1 0
1 1 1
1 1 1
1.90
1
1 1
0 1 0
1 1 1
1 1 1
4.22
1
1 1
0 1 0
1 1 1
1 1 1
2.72
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
shat
0 1
1 1 0
0 1 1
0 1 1
2.53
0
0 1
1 1 0
0 1 1
0 1 1
1.46
0
0 1
1 1 0
0 1 1
0 1 1
5.04
0
0 1
1 1 0
0 1 1
0 1 1
5.94
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
shat
1 1
1 1 0
0 1 1
0 1 1
2.53
1
1 1
1 1 0
0 1 1
0 1 1
0.79
1
1 1
1 1 0
0 1 1
0 1 1
5.13
1
1 1
1 1 0
0 1 1
0 1 1
9.00
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0.05
0
1
1
1.23
0
1
1
4.30
0
1
1
5.78
Trellis Codes and QAM
• Bandwidth Constrained Channels
– Use set partitioning to combine code with M-ary bandwidth constrained
signaling Ungerboeck (1982)
– Three rules:
• Use all subsets with equal frequency in trellis
• Use Transitions that join use maximum distance subsets
• Parallel transitions are assigned max Euclidean distance (unencoded bit transitions)
QAM set partition via convolution code
8 state Trellis for rectangular QAM
Set Partitioning
• Chop up the Constellations to increase Euclidean distance
between points
• In rectangular QAM case each partition increases d by 2
64 QAM Set Partitioning
• Each 4 x 4 block of 64 QAM constellation is
sm2
replaced by 16 QAM partition
sm1
64 QAM Set Partitioning
• Examples:
Decide between these in Real decoder
D0
D2
Trellis Codes and QAM
• Eight state trellis for QAM
Contains no overlapping locations
Trellis Codes and QAM
• The code generator
Trellis Codes and QAM
• The code generator
– Alternate = essentially a kind of parity bit
Trellis Codes and QAM
• The code generator
– Alternate = essentially a kind of parity bit
– The weights = “parity matrix” not generator
matrix
– Ungerboeck IEEE Communications Magazine February 1987-Vol. 25. No. 2
Trellis Codes and QAM
• The code generator
g1  0 0 1 0 0 0
g 2  0 1 0 0 1 0
g 3  1 0 0 1 0 0
Trellis Codes and QAM
• The code generator
Memory = 3, thus 8 states


g1  0 0 1 0 0 0
g 2  0 1 0 0 1 0
g 3  1 0 0 1 0 0
Trellis Codes and QAM
• The code generator
g1  0 0 1 0 0 0
g 2  0 1 0 0 1 0
g 3  1 0 0 1 0 0
bin
sbin
1 1
g1 0 0
g2 0 1
g3 1 0
1
1
0
0
1
0
0
1
1
0
1
0
2 1
0
g1
g2
g3
0
0
0
1
0
0
1
0
1
1
0
0
1
0
0
1
1
0
1
0
3 1
1
g1
g2
g3
0
0
0
1
1
0
1
0
0
1
0
0
0
0
0
1
1
0
1
0
1


0
0
0
0
1
0
0
1
2
0
1
0
3
0
1
1
4
1
0
0
bout
0.01 r
1
sigma=
0 1
1.00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0 0
0.01
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
0 0
0.00
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
0
0
1
1
metric=
1.00 1.97 2.00 2.97 1.97 1.00 2.97 2.00 2.00 2.97 1.00 1.97 2.97 2.00 1.97 1.00 0.00 0.97 1.00 1.98 0.
next state=
1
0 1
1.01
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0 1
1.02
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
0 1
1.01
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
0
0
1
1
metric =
4.05 3.02 3.04 2.01 2.03 3.06 1.01 2.04 3.04 2.01 4.05 3.02 1.01 2.04 2.03 3.06 3.04 2.01 2.03 1.00 1.
next state=
1
0 0
0.00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0 0
-0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
0 0
0
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
0
0
1
1
Trellis Codes and 64 QAM
• Send 64 QAM but only 5 bits per symbol
– 6 bits = 3 data bits + 3 encoded bits
– The 3 encoded bits come from 2 data bits
One of eight subsets – 3 bits
Select the levels for each dimension
2 bits in, 3 bits out


One of eight symbols – 3 bits
4 levels per dimension
Trellis Codes and QAM
• Trellis seems not the same?
– Maybe books trellis is parity realization
0
0
0
0
1
0
0
1
2
0
1
0
3
0
1
1
4
1
0
0
5
1
0
1
6
1
0.01
r
sigma=
1
1.00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
-0.01
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
0.00
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
0
0
1
1
0
0
1
1
1
1
metric=
1.00 2.03 2.00 3.03 2.03 1.00 3.03 2.00 2.00 3.03 1.00 2.03 3.03 2.00 2.03 1.00 0.00 1.03 1.00 2.03 1.03 0.00 2.03 1.00 1.00 2.03
next state=
bout
1
1
1
0.99
1.00
0.99
metric =
next state=
0
0
0
0.00
0.01
-0
metric =
next state=
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
0
0
1
1
0
0
1
1
1
1
3.98 2.97 2.99 1.99 1.97 2.98 0.98 1.99 2.99 1.99 3.98 2.97 0.98 1.99 1.97 2.98 3.00 1.99 2.01 1.00 0.99 1.99 0.00 1.01 2.01 1.00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
0
0
1
1
0
0
1
1
1
1
0.98 1.97 2.01 3.00 0.99 0.00 2.01 1.02 3.01 4.00 1.99 2.98 3.02 2.03 1.99 1.00 1.99 2.98 3.01 4.00 1.99 1.01 3.02 2.03 4.01 5.00