1. 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

2. BCJR (Continued)

3. BCJR (Continued)

4. BCJR (Continued)

5. BCJR (Continued)
AWGN

6. 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

7. BCJR (Continued)
Information bits
Termination bits

8. BCJR (Continued)

9. BCJR (Continued)

10. 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

11. 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)

12. Example: log-MAP

13. Example: log-MAP +0.48 +0.62 -1,02    Assume Es/N0=1/4 = - 6.02 dB
R=3/8 so Eb/N0=2/3 = dB
-0,45
0,45
3,44
3,02
1.34
0,44
1,59
3,06
1,60
-0,25
+0,25
-0,75
+0,75
+0,35
-0,35
+1,45
-1,45
-0,45
+0,45
1,60

14. Example: Max-log-MAP -0,07 +0,10 -0,40   
Assume Es/N0=1/4 = dB
R=3/8 so Eb/N0=2/3 = dB
-0,45
0,45
3,31
2,34
1.20
-0,20
1,25
3,05
1,60
-0,25
+0,25
-0,75
+0,75
+0,35
-0,35
+1,45
-1,45
-0,45
+0,45
1,60

15. 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

16. Example: Rate 2/3 punctured from rate 1/2
dfree = 3
The punctured code is also a convolutional code
Note the bit-level trellis

17. Example: Rate 3/4 punctured from rate 1/2
dfree = 3

18. 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

19. Best punctured codes 1 10 5 3 !4 2 6 7 5 7 17 6 27

20. 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

21. ”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

22. Example: Feedforward encoder
Feedforward: always possible to find an information vector that ends in the proper state-> can start in any state and end in any state

23. 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

24. Tailbiting codes as block codes
Tailbiting codes are block codes

25. Suggested exercises