B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

860 ERROR CORRECTING CODES
Step 2: At every variable node (indexed by £), update
Q (m +l) (O)
_ (m+l) . (0)
u
- a u
Pe
Q (m +l) (l)
_ (m+l) .
- a i,e (l)
i,t
Pe
TT
TT
R (m ) (O)
ml
R (m l (1) ml
(14.81a)
(14.81b)
where the normalization factor a (r: i+ 1 )
is selected such that
l,L
Step 3: At the variable nodes, also estimate the a posteriori probabilities
p ( m+ l )
[v e = 0lr] = ai m+ ! ) . Pe (0) TT Rt1 co)
mEµe
p ( m+ I )
[v e = llr] = at + l )
·p e (l) TT R 1 (1)
mEµe
(14.82a)
(14.82b)
where the normalization factor afn + l )
is selected such that
Step 4: Make hard decisions of each code bit
p ( m+ l )
[ve = 0lr] + p ( m+ l )
[v e = llr] = 1
A
• { p(m+l) [v e = l lr] l
v e = sign log p (m+l) [ve = 0lr]
If the decode codeword satisfies all parity checks, stop decoding. Otherwise, go back
to step 1 for another iteration.
Notice that external input signals {r;} are involved only during the estimation of a priori
probabilities Pt (1) and Pe (0). SPA uses the a priori probabilities as follows:
p(rlv e = 1)
/Jf (1) = --c---------,---
p(r lvc = 1) + p(rlvc = 0)
and
p(rlv e = 0)
/J£ (0) = -------­
p(rlvc = 1) + p(rlv e = 0)
For a more concrete example, consider the example of an AWGN channel with BPSK
modulation. For v_e, the received signal sample is
re = je;,(2ve - 1) + we
where we is Gaussian with zero mean and variance N /2. Because {rd are independent, when
we receive re = r e , we can simply use
p e (l) = ------
1 + exp (-4!}-re)
and Pe(O) = ------
1 + exp ( 4!}-r e )
1
This completes the introduction of sum-product algorithm for the decoding of LDPC codes.