MAP decoding, BCJR algorithm

8/4/2019 MAP decoding, BCJR algorithm

1/25

1

MAP decoding: The BCJR algorithm

Maximum a posteriori probability (MAP) decoding

Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm (1974)

Baum-Welch algorithm (1963?)*

Decoder inputs:

Received sequence r (soft or hard) A prioriL-valuesLa(ul) = ln(P(ul= 1)/P(ul= -1))

Decoder outputs:

A posteriori probability (APP)L-valuesL(ul) = ln(P(ul= 1|

r)/P(ul= -1|r)) > 0: ul is most likely to be 1

< 0: ul is most likely to be -1


2/25

2

BCJR (cont.)


3/25

3

BCJR (cont.)


4/25

4

BCJR (cont.)


5/25

5

BCJR (cont.)

AWGN


6/25

6

MAP algorithm

Initialize forward and backward recursions 0(s) and 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 APPL-values

Complexity: Approximately 3xViterbi

Requires detailed knowledge of SNR Viterbi just maximizes rv, and does not require exact

knowledge of SNR


7/25

7

BCJR (cont.)

Information bits

Termination bits


8/25

8

BCJR (cont.)


9/25

9

BCJR (cont.)


10/25

10

Log-MAP algorithm

Initialize forward and backward recursions 0*(s

) and 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 APPL-values

Advantages over MAP algorithm:

Easier to implement

Numerically more stable


11/25

11

Max-log-MAP algorithm

Replace max* by max, i.e., remove table look-up correction term

Advantage: Simpler and much faster Forward and backward passes are equivalent to a Viterbi decoder

Disadvantage: Less accurate, but the correction term is limited in sizeby ln(2)

Can improve accuracy by scaling with an SNR-(in)dependent scalingfactor*


12/25

12

Example: log-MAP


13/25

13

Example: log-MAP AssumeEs/N0 = 1/4 = -6.02 dB

R = 3/8, soEb/N0 = 2/3 = -1.76dB

-0,45

+0,45

-0,25

+0,25

-0,75

+0,75

+0,35

-0,35

+1,45

-1,45

0

1.60

0 0

1,60

01,59

3,06

3,44

3,02

-0,45

0,45

1.34

0,44


14/25

14

Example: log-MAP AssumeEs/N0 = 1/4 = -6.02 dB

R = 3/8, soEb/N0 = 2/3 = -1.76dB

-0,45

+0,45

-0,25

+0,25

-0,75

+0,75

+0,35

-0,35

+1,45

-1,45

0

1.60

0 0

1,60

01,59

3,06

3,44

3,02

-0,45

0,45

1.34

0,44

+0.48 +0.62 -1,02


15/25

15

Example: Max-log-MAP AssumeEs/N0 = 1/4 = -6.02 dB

R = 3/8, soEb/N0 = 2/3 = -1.76dB

-0,45

+0,45

-0,25

+0,25

-0,75

+0,75

+0,35

-0,35

+1,45

-1,45

0

1,60

0 0

1,60

01,25

3,05

3,31

2,34

-0,45

0,45

1.20

-0,20


16/25

16

Example: Max-log-MAP AssumeEs/N0 = 1/4 = -6.02 dB

R = 3/8, soEb/N0 = 2/3 = -1.76dB

-0,45

+0,45

-0,25

+0,25

-0,75

+0,75

+0,35

-0,35

+1,45

-1,45

0

1,60

0 0

1,60

01,25

3,05

3,31

2,34

-0,45

0,45

1.20

-0,20

-0,07 +0,10 -0,40


17/25

17

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:

Bit-level encoders

Syndrome trellis decoding (Riedel)* Punctured codes

Start with low-rate convolutional mothercode (rate 1/n?)

Puncture (delete) some code bits according to apredetermined pattern

Punctured bits are not transmitted. Hence, the code rate isincreased, but the free distance of the code could be reduced

Decoder inserts dummy bits with neutral metrics contribution


18/25

18

Example: Rate 2/3 punctured from rate 1/2

The punctured code is also a convolutional code dfree = 3


19/25

19

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

dfree = 3


20/25

20

More on punctured convolutional codes

Rate-compatible punctured convolutional (RCPC) codes: Used for applications that need to support several code rates,

e.g., adaptive coding or hybrid ARQ

Sequence of codes is obtained by repeated puncturing

Advantage: One decoder can decode all codes in the family Disadvantage: Resulting codes may be sub-optimum

Puncturing patterns:

Usually periodic puncturing patterns

Found by computer search Care must be exercised to avoid catastrophic encoders


21/25

21

Best punctured codes

1

5 3

7

717

10

4 2

6 27

75


22/25

22

Tailbiting convolutional codes Purpose: Avoid the terminating tail (rate loss) and maintain a

uniform level of protection

Note: Cannot avoid distance loss completely unless the length isnot too short. When the length gets larger, the minimum distanceapproaches the free distance of the convolutional code

Codewords can start in any state

This gives 2as many codewords

However, each codeword must end in the same state that it

started from. This gives 2-as many codewords

Thus, the code rate is equal to the encoder rate

Tailbiting codes are increasingly popular for moderate lengthpurposes

Some of the best known linear block codes are tailbiting codes

Tables of optimum tailbiting codes are given in the book

DVB: Turbo codes with tailbiting component codes


23/25

23

Example: Feedforward encoder

Feedforward encoder: Always possible to find an information vector

that ends in the proper state (inspect the last mk-bit input tuples)


24/25

24

Example: Feedback encoder Feedback encoder:

Not always

possible, for everylength, to constructa tailbiting code

For each u: Must

find uniquestarting state

L* = 6 not OK

L* = 5 OK

In general,L*should not have thelength of a zeroinput-weight cycleas a divisor


25/25

25

Circular trellis

Decoding of tailbiting codes: Try all possible starting states (multiplies complexity by 2), i.e., run the

Viterbi algorithm for each of the 2 subcodes and compare the best paths fromeach subcode

Suboptimum Viterbi: Initialize an arbitrary state at time 0 with zero metricand find the best ending state. Continue one round from there with the best

subcode MAP: Similar

MAP decoding, BCJR algorithm

Documents