1
Log-Likelihood Algebra
Sum of two LLRs
L(d1) + L(d2) L (d1 d2 )
= log exp[L(d1)] + exp [L(d2)]
1 + exp[L(d1)].exp [L(d2)]
(-1) . sgn [L(d1)]. sgn [L(d2)] . Min ( |L(d1)| , |L(d2)| )
+
+
L (d)
L (d)
= - L (d)
0 = 0
2
Iterative decoding example
2D single parity code
di di = pij d1 = 1 d2 = 0 p12 = 1
d3 = 0 d4 = 1 d34 = 1
p13 = 1 p24 = 1 -
x1 = 0.75 x2 = 0.05 x12 = 1.25
x3 = 0.10 x4 = 0.15 x34 = 1.0
x13 = 3.0 x24 = 0.5 -
3
Iterative decoding example
Estimate Lc(xk)
= 2 xk /2
assuming 2 = 1
Lc(x1 )= 1.5 Lc(x2 )= 0.1 Lc(x12 )= 2.5
Lc(x3 )= 0.2 Lc(x4 )= 0.3 Lc(x34 )= 2.0
Lc(x13 )= 6.0 Lc(x24 )= 1.0 -
4
Iterative decoding example
Compute
Le( dj ) = [Lc ( x j) + L(dj ) ] Lc ( x ij)
+
Leh( d1) = [Lc ( x 2) + L(d2) ] Lc ( x 12 ) = new L( d1 )+
Leh( d2) = [Lc ( x 1) + L(d1) ] Lc ( x 12 ) = new L(d2)+
Leh( d3) = [Lc ( x 4) + L(d4) ] Lc ( x 34 ) = new L(d3) +
Leh( d4) = [Lc ( x 3) + L(d3) ] Lc ( x 34 ) = new L(d4)+
5
Iterative decoding example
Lev( d1) = [Lc ( x 3) + L(d3) ] Lc ( x 13 ) = new L( d1 )+
Lev( d2) = [Lc ( x 4) + L(d4) ] Lc ( x 24 ) = new L(d2)+
Lev( d3) = [Lc ( x 1) + L(d1) ] Lc ( x 13 ) = new L(d3) +
Lev( d4) = [Lc ( x 2) + L(d2) ] Lc ( x 24 ) = new L(d4)+
After many iterations the LLR is computed for decision making
L( di ) = Lc ( x i) + Leh(di) + Lev( dj )
^ ^^
6
First Pass output
Lc(x1 )= 1.5
Lc(x2 )= 0.1
Lc(x3 )= 0.2
Lc(x4 )= 0.3
Leh(d1 )= -0.1
Leh(d2 )= -1.5
Leh(d3 )= -0.3
Leh(d4 )= -0.2
Lev(d1 )= 0.1
Lev(d2 )= -0.1
Lev(d3 )= -1.4
Lev(d4 )= 1.0
L(d1 )= 1.5
L(d2 )= -1.5
L(d3 )= -1.5
L(d4 )= 1.1
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Parallel Concatenation Codes
Component codes are Convolutional codes Recursive Systematic Codes Should have maximum effective free
distance Large Eb/No maximizing minimum
weight codewords Small Eb/No optimizing weight
distribution of the codewords Interleaving to avoid low weight
codewords
8
Non - Systematic Codes - NSC
dkdk-1 dk-2
+
+
{dk}
{uk}
{vk}
uk = g1i dk-i (mod 2) ; i=1
L-1
vk = g2i dk-i (mod 2) ; i=1
L-1
G1 = [ 1 1 1 ]
G2 = [ 1 0 1 ]
9
Recursive Systematic Codes - RSC
akak-1 ak-2
+
{dk} {uk}
{vk}
+
ak = dk + gi’ ak-i (mod 2) ; gi’ = g1i if uk = dk
i=1 g2i if vk = dk
L-1
10
Trellis for NSC & RSC
NSC RSC
00
11
11
00
10
10
01
01
a = 00
b = 01
c = 10
d = 11
00
11
11
00
10
10
01
01
a = 00
b = 01
c = 10
d = 11
11
Concatenation of RSC Codes
{dk} {uk}
ak ak-1 ak-2
+{v1k}
+
ak ak-1 ak-2
+{v2k}
+
Interleaver
{vk}
{ 0 0 …. 0 1 1 1 0 0 …..0 0 }
{ 0 0 …. 0 0 1 0 0 1 0 … 0 0 } produce low weight codewords in component coders
12
Feedback Decoder
APP Joint Probability k i,m = P { dk = i, Sk = m / R1 N }
State at time k
Received sequenceFrom time 1 to N
APP P { dk = i / R1 N } = k i,m ; i = 0,1 for binary
m
Log Likelihood Ratio L( dk ) = Log k 1,m
m
k 0,m
m
Likelihood Ratio ( dk ) = k 1,m
m
k 0,m
m
13
MAP Rule dk =1 ; L(dk) > 0
dk =0 ; L(dk) < 0
Feedback Decoder
L( dk) = Lc ( x k) + L(dk) + Le( dk )
^ ^
L1( dk ) = [Lc ( x k) + Le1(dk ) ]L2( dk ) = [ f{L1 ( dn) }n k + Le2(dk ) ]
14
Feedback Decoder
DECODER 1
Interleaver
DECODER 2
De-Interleaver
De-Interleaver
yk
xk
dk
L1( dk )
L1( dn )
Le2( dk )
L2( dk )
y1k y2k
15
Modified MAP Vs. SOVA
SOVA Viterbi Algorithm acting on soft inputs over
forward path of the trellis for a block of bits Add BM to SM compare select ML path
Modified MAP Viterbi Algorithm acting on soft inputs over
forward and reverse paths of the trellis for a block of bits
Multiply BM & SM Sum in both directions best overall statistic
16
MAP Decoding Example
dk dk-1 dk-2
+
{dk}
{uk}
{vk}
00
11
11
00
10
10
01
01
a = 00
b = 10
c = 01
d = 11
17
MAP Decoding Example d = { 1, 0, 0 } u = { 1, 0, 0 } x = { 1., 0.5, -0.6 } v = { 1, 0, 1 } y = { 0.8, 0.2, 1.2 } Apriori probabilities 1 = 0 = 0.5
Branch Metric k i,m = P { dk = i, Sk = m , Rk }
= P { Rk / dk = i, Sk = m }
. P {Sk = m / dk = i } . P { dk = i }
P {Sk = m / dk = i } = 1 / 2 L = ¼ ; P { dk = i } = 1 / 2 ;
k i,m = P { xk / dk = i, Sk = m }. P { yk / dk = i, Sk = m } . { ki / 2L }
18
MAP Decoding Example k i,m = P { xk / dk = i, Sk = m }. P { yk / dk = i, Sk = m } . { k
i / 2L }
For AWGN channel :
k i,m = { ki / 2L } (1/2 ) exp { - (xk – uk
i )2 /(2 2 ) }dxk
. (1/2 ) exp { - (yk – vki,m )2 /(2 2 ) }dyk
k i,m = { Ak ki } exp { (xk . uk
i )+ (yk . Vki,m )/ 2 }
Assuming Ak = 1 2 =1 ;
k i,m = 0.5 exp { xk . uki + yk . vk
i,m }
19
Subsequent steps Calculate branch metric
k i,m = 0.5 exp { xk . uki + yk .
vki,m }
Calculate forward state metric
k+1m = k i,b(j,m) k
b(j,m) J=0
1
Calculate reverse state metric
km = k
j,m k+1f(j,m)
J=0
1
20
Subsequent steps Calculate LLR for all times
Log Likelihood Ratio L( dk ) = Log k
m k1,m k+1
f(1,m)
m
km k
0,m k+1f(0,m)
m
Hard decision based on LLR
21
Iterative decoding steps
LLR L( dk ) = L(dk) + { 2xk } + Log [ke ]
Likelihood Ratio ( dk )
=
=
=
{ k1} k
m exp { xk . uk1 + yk . Vk
1,m } k+1f(1,m)
m
{ k0} k
m exp { xk . uk0 + yk . Vk
0,m } k+1f(0,m)
m
km exp { yk . Vk
1,m } k+1f(1,m)
m
km exp { yk . Vk
0,m } k+1f(0,m)
m
{ k} exp { 2xk }
{ k} exp { 2xk } { k
e}
22
Iterative decoding
For the second iteration;
k i,m = ke i exp { xk . uk
i + yk . Vk
i,m } Calculate LLR for all times
Log Likelihood Ratio L( dk ) = Log k
m k1,m k+1
f(1,m)
m
km k
0,m k+1f(0,m)
m
Hard decision based on LLR after multiple iterations
