1 / 25

MAP decoding: The BCJR algorithm

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 L a ( u l ) = ln( P ( u l =1)/ P ( u l =0) ) Decoder outputs

loweryr
Download Presentation

MAP decoding: The BCJR algorithm

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 0,44 3,06 0,45 1,60 3,02 -0,75 +1,45 +0,25 -0,35 +0,75 -1,45 +0,45 1,60 0 1,59 1.34 -0,45 3,44 0 -0,45 +0,35 -0,25 +0.48 +0.62 -1,02    0 0 • Assume Es/N0=1/4 = - 6.02 dB • R=3/8 so Eb/N0=2/3 = - 1.76dB

  14. Example: Max-log-MAP 0,45 2,34 3,05 1,60 -0,20 -0,75 +1,45 -0,35 +0,25 +0,75 -1,45 1,60 +0,45 0 1.20 3,31 -0,45 1,25 0 -0,45 +0,35 -0,25 -0,07 +0,10 -0,40    0 0 • Assume Es/N0=1/4 = - 6.02 dB • R=3/8 so Eb/N0=2/3 = - 1.76dB

  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 7 5 10 1 !4 2 5 3 6 6 27 7 17

  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 • 12.27-12.39

More Related