Maximum likelihood decoding
v
u
ECC
encoder
v’,u’
r = v+ n
Channel
ECC
decoder
Decoder: Must determine v’ to minimize P(E|r)=P(v’v|r)
P(E) = r P(E|r)P(r)
P(r) is independent of decodingoptimum decoding must
•
minimize P(v’v|r)
•
maximize P(v’=v|r)
•
choose v’ as codeword v that maximizes
P(v|r) = P(r|v)P(v) /P(r)
•
i. e. (if P(v) the same for all v) that maximizes P(r|v)
1
ML decoding (cont.)
DMC:
•
P(r|v) = j P(rj|vj)….
•
ML decoder: Choose v to maximize this expression
•
…or log P(r|v) = j log P(rj|vj)….
•
Is an ML decoder optimum?
•
Only if all v are equally probable as input vectors
2
ML decoding on BSC
BSC:
•
P(rj|vj) = 1-p if rj=vj, , p otherwise
•
log P(r|v) = j log P(rj|vj)
•
Hamming distance: Let r and v differ in d(r,v) positions
•
j log P(rj|vj) = d(r,v) log p + (n-d(r,v))log(1-p)
= d(r,v) log (p/(1-p)) + nlog(1-p)
•
log (p/(1-p)) < 0 for p<0.5, so an ML decoder for a BSC
must choose v to minimize d(r,v)
3
Channel capacity
Shannon (1948)
•
Every channel has a capacity C (determined by noise
and available bandwidth)
•
Eb(R) is a positive function of R for R<C
•
There exists a block code of length n such that with ML
decoding
–nEb(R)
P(E) < 2
•
Similar for convolutional codes
•
In fact the average code performs like this.
Nonconstructive proof
•
But ML decoding for long random codes is infeasible!
4
Performance measures
Tradeoff of main parameters:
•
Code rate
•
Error probability
•
Word error rate (WER, FER, BLER)
•
Bit error rate (BER)
at a given channel and channel quality
•
Decoding complexity
Performance is often displayed as a curve of an error rate as
a function of channel quality
5
Error rate curves
Log ER
Coding threshold
Coding gain
SNR : Eb/N0 (dB)
Eb= Es/R
6
Asymptotic coding gain
Log ER
Coding gain
Asymptotic coding gain
SNR : Eb/N0 (dB)
7
Asymptotic coding gain
puncoded  e
1
2
 Eb / N 0
High SNR, coded with soft-decision decoding:
pcoded,SD  Kcodee dmin REb / N0
Asymptotic coding gain:
Eb / N 0 uncoded
Eb / N 0 coded
 d min R
...or 10 log10 (Rd min). For HD decoding: 10 log10 (Rd min/2)
Thus SD gives 10 log10 2 = 3dB better ACG than HD
8
Performance close to the Shannon bound
Log ER
SNR : Eb/N0 (dB)
9
Coded modulation
•
Encoding + modulation:
•
Need a distance preserving modulation mapping,
preserving distance between different codewords
•
Thus we can view codes also in the modulation domain
•
Combined coding and modulation: Design codes
specifically to increase distance
•
Exploit that in a large signal constellation some points
are further apart
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Coded modulation
•
Some schemes work on a signal constellation by:
1. Encode some input bits by an ECC. Let the output
bits determine a subconstellation
2. Let the remaining input bits determine a point in the
subconstellation
•
•
TCM – coded modulation based on a convolutional
ECC
•
BCM – coded modulation based on block ECC
Also: coded modulation with turbo codes and LDPC
codes
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Trellises of linear block codes (CH 9)
A representation that facilitates soft-decision decoding
Recall:
•
•
A linear block code is the row space of a generator
matrix
000
Example:
011
101
110
Sender
Receiver
E
E
E
O
O
E
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Trellises
A trellis is a directed graph:
•
A set  of depths or time instances, ordered (usually)
from 0 to n
•
At each time instant, a set of nodes, vertices,
representing the (code) state at that time instant. Usually
(in an ordinary block code) one initial state s0 at time
instant 0 and one final state sf at time n
•
Edges can go from a state at time i to a state at time i+1
•
Each edge is labeled by one (or more) symbol(s) from
the code alphabet (usually binary)
•
A sequence of edge labels obtained by traversing the
trellis from s0 to sf is a codeword
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Linear trellises
•
Necessary (but not sufficient) conditions for a trellis (or
corresponding code) to be linear:
•
•
Oi= fi(si, Ii)
•
Output block Oi
•
Input block Ii
•
State si at time instant i
•
Depends on time instant
si+1= gi(si, Ii)
Oi
si+1
si
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More properties of linear trellises
•
In the trellis of a linear code, the set of states i at time i is
called the state space
•
A trellis is time invariant iff 
•
a finite period initial delay 
•
An output function f and a state transition function g
•
a ”template”state space 
such that
•
i   for i < , and i =  for i  
•
fi=f and gi=g for all i 
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Bit-level trellises of linear block codes
•
[n,k] code C
•
Bit-level trellis: n+1 time instants and n trellis sections
•
One initial state s0 at time 0, one final state sf at time n
•
For each time i >0, there is a fixed number Incoming(i) of
incoming branches. For all i, Incoming(i) is 1 or 2. Two
branches going to the same state have different labels.
•
For each time i <n, there is a fixed number Outgoing(i) of
outgoing branches. For all i, outgoing(i) is 1 or 2. Two
branches coming from the same state have different labels.
•
Each codeword corresponds to a distinct path from s0 to sf
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Bit-level trellises of linear block codes
•
The number |i| is (sometimes) called the state space
complexity at time instant i. For a linear code, we will
show that |i| for all i , |i| is a power of 2
•
Thus we can define the state space dimension
i = log2|i|
•
The sequence
( 0=0, 1 ,..., i,..., n=0)
is called the state space dimension profile, and determines
the complexity of a maximum likelihood (soft-decision)
decoder for the code.
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Generator matrix: TO form
1
0
G  
0

0
1
1
1
1
1
1
1
0
1
0
1
0
0
1
1
0
0
1
0
0
0
1
1
1
1
0
G'  
0

0
1
1
1
0
0
0
1
0
1
1
0
1
0
1
1
1
1
0
0
0
0
1
1
1
1
1

1

1
0
0

0

1
18