MAP decoding: The BCJR algorithm
•
•
•
•
Maximum A posteriori Probability
Bahl-Cocke-Jelinek-Raviv (1974)
• Baum-Welch algorithm (1963?)*
Decoder inputs:
• Received sequence r (soft or hard)
• A priori L-values La(ul) = ln( P(ul=1)/ P(ul=0) )
Decoder outputs
• A posteriori L-values L(ul) = ln( P(ul=1 | r)/ P(ul=0 | r) )
• > 0: ul is most likely to be 1
• < 0: ul is most likely to be 0
1
BCJR (Continued)
2
BCJR (Continued)
3
BCJR (Continued)
4
BCJR (Continued)
AWGN
5
MAP algorithm
• Initialize Forward recursion 0(s)
and backward recursion N(s)
• Compute branch metrics {l(s’,s)}
• Carry out forward recursion {l+1(s)} based on {l(s)}
• Carry out backward recursion {l-1(s)} based on {l(s)}
• Compute APP L-values
• Complexity: approximately 3xViterbi
• Requires detailed knowledge of SNR
• Viterbi just maximizes rv, does not require exact SNR
6
BCJR (Continued)
Information bits
Termination bits
7
BCJR (Continued)
8
BCJR (Continued)
9
Log-MAP algorithm
•
•
•
•
•
•
Initialize Forward recursion 0*(s)
and backward recursion N*(s)
Compute branch metrics {l*(s,s’)}
Carry out forward recursion {l+1*(s)} based on {l*(s)}
Carry out backward recursion {l-1*(s)} based on {l*(s)}
Compute APP L-values
Advantages over MAP algorithm:
• Easier to implement
• Numerically stable
10
Max-log-MAP algorithm
•
•
•
•
Replace max* by max: delete table lookup correction term
Advantage: Simpler, faster (?)
Forward and backward pass equivalent to Viterbi decoder
Disadvantage: Less accurate (but the correction term is limited in size
by ln 2)
11
Example: log-MAP
12
Example: log-MAP
•
•

Assume Es/N0=1/4 = - 6.02 dB
+0.48
R=3/8
so Eb/N0=2/3 = -+0.62
1.76dB
+0,45
0
+0,25
-0,35
-1,45
1,60
1.34 1,59
-0,45 3,44
-0,45
1,60
+1,45
-0,75
+0,75

-1,02
0,44 3,06
0,45 3,02

-0,25
0
+0,35
0
0
13
Example: Max-log-MAP
•
•

Assume Es/N0=1/4 = - 6.02 dB
-0,07
R=3/8
so Eb/N0=2/3 = -+0,10
1.76dB
+0,45
0
+0,25
-0,35
-1,45
1,60
1.20 1,25
-0,45 3,31
-0,45
1,60
+1,45
-0,75
+0,75

-0,40
-0,20 3,05
0,45 2,34

-0,25
0
+0,35
0
0
14
Punctured convolutional codes
•
•
•
Recall that an (n,k) convolutional code has a decoder trellis with 2k
branches going into each state
More complex decoding
Solutions
• Syndrome decoding
• Bit-oriented trellis
• Punctured codes
• Start with low-rate convolutional mother code (rate 1/n?)
• Puncture (delete) some code bits according to predetermined
pattern
• Punctured bits are not transmitted. Hence the code rate is
increased, but the distance between codewords is reduced
• Decoder inserts dummy bits with neutral metrics contribution
15
Example: Rate 2/3 punctured from rate 1/2
•
•
The punctured code is also a convolutional code
Note the bit-level trellis
dfree = 3
16
Example: Rate 3/4 punctured from rate 1/2
dfree = 3
17
More on punctured convolutional codes
•
Rate compatible punctured convolutional codes
•
•
For applications that needs to support several code rates
• For example adaptive coding
• Sequence of codes obtained by repeated puncturing
• Advantage: One decoder can decode all codes in family
• Disadvantage: Resulting codes may be sub-optimum
Puncturing patterns
• Usually periodic puncturing maps
• Found by computer search
• Care must be exercised to avoid catastrophic encoders
18
Best punctured codes
10
1
5
7
3
!4
2
6
5
7
17
6
27
19
Tailbiting convolutional codes
•
•
•
•
Purpose: Avoid the terminating tail (rate loss)
• without distance loss?
• Cannot avoid distance loss completely
Tailbiting codes: Paths through trellis
• Codewords can start in any state
• Gives 2 as many codewords
• But each codeword must end in the same state that it started from
• …Gives 2- as many codewords
Tailbiting codes increasingly popular for moderate length purposes
DVB: Turbo codes with tailbiting component codes
20
”Circular” trellis
•
Decoding
•
Try all possible starting states (multiplies complexity by 2 )
•
Suboptimum Viterbi: Let decoding proceed until best ending state found,
continue ”one round” from there
•
MAP: Similar
21
Example: Feedforward encoder
Feedforward: always possible to find an information vector that ends in
22
the proper state-> can start in any state and end in any state
Example: Feedback encoder
•
•
•
•
Feedback
encoder: Not
always possible,
for every length,
to construct a
tailbiting
systematic
recursive encoder
For each u: must
find unique
starting state
L*=6 not OK
L*=5 OK
23
Tailbiting codes as block codes
•
Tailbiting codes are block codes
24
Suggested exercises
•
12.27-12.39
25