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

BCJR (Continued)

BCJR (Continued)

BCJR (Continued)

BCJR (Continued) AWGN

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

BCJR (Continued) Information bits Termination bits

BCJR (Continued)

BCJR (Continued)

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

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)

Example: log-MAP

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 = - 1.76dB -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

Example: Max-log-MAP -0,07 +0,10 -0,40    Assume Es/N0=1/4 = - 6.02 dB R=3/8 so Eb/N0=2/3 = - 1.76dB -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

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

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

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

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

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

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

”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

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

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

Tailbiting codes as block codes Tailbiting codes are block codes

Suggested exercises 12.27-12.39