14
Notes on a posteriori probability (APP) metrics for LDPC IEEE 802.3an Task Force November, 2004 Raju Hormis, Xiaodong Wang Columbia University, NY e-mail: [email protected]
Notes on a posteriori probability (APP) metrics for LDPC
IEEE 802.3an Task Force
November, 2004
Raju Hormis, Xiaodong Wang
Columbia University, NY
e-mail: [email protected]
Outline
Section 55.3.11.3 of draft D1.1 discusses decoding of LDPC code groups. This presentation outlines a
method for calculation of exact LDPC metrics, with examples.
• General definition of APP metrics for iterative decoders.
• A simple example: PAM-2.
• APP metric for 1-D lattices.
• APP metric for 2-D and N-D lattices.
• Conclusions.
2
APP calculation for iterative decoders
Iterative decoders for LDPC and turbo-codes require a posteriori probabilities (APP’s) as metrics,
rather than direct channel observations.
• Usually expressed as a log ratio, and often referred to as LLR (log likelihood ratio).
• Let {b0, b1, b2, ..., bn} be the set of bits mapped to a symbol ak in constellation A. We will denote
the APP for, say bit b0, as Lb0.
• Consider the AWGN channel model: y = ak + υ , where ak is the transmitted symbol, y is the
channel observation, and υ is AWGN.
• We can write the definition of Lb0 as
Lb0 , loge
[p(b0 = 0/y)
p(b0 = 1/y)
], and applying Bayes’ rule, we get
= loge
[p(y/b0 = 0)p(b0 = 0)
p(y)
p(y)
p(y/b0 = 1)p(b0 = 1)
],
= loge
[p(y/b0 = 0)
p(y/b0 = 1)
].
3
APP computation for PAM-2 in AWGN
• Consider the case of PAM-2 which takes a0 = +1 when b0 = 0, a1 = −1 when b0 = 1.
• We will make use of the conditional Gaussian pdf of a received symbol y in AWGN:
p(y/a) =1√
2πσ2e− (y−a)2
2σ2 ,
where σ2 is the noise variance.
• We can now derive the APP value for b0 as
Lb0 = loge
p(y/b0 = 0)
p(y/b0 = 1), as derived earlier,
= loge
e−(y−a0)2/2σ2
e−(y−a1)2/2σ2 ,
= loge
e−(y−1)2/2σ2
e−(y+1)2/2σ2 ,
Lb0 =2
σ2y .
• The APP value is a simpe scaled version of channel observation y. Notice that scaling is inversely
proportional to noise variance.
4
APP for 1-D lattice: PAM-8 constellation 1
b2b1b0 PAM level
. . . . . .
100 +9
010 +7
011 +5
001 +3
000 +1
110 -1
111 -3
101 -5
100 -7
010 -9
. . . . . .
To compute the APP’s for PAM-8, notice that
p(b0 = 1/y) =∑
k
p(ak/y), where ak ∈ A, s.t. b0 = 1
p(b0 = 0/y) =∑
j
p(aj/y), where aj ∈ A, s.t. b0 = 0
Using the above, we can now define the LLR for b0 as
Lb0 , loge
[p(b0 = 0/y)
p(b0 = 1/y)
],
= loge
[∑ak∈A, b0=0 p(ak/y)∑aj∈A, b0=1 p(aj/y)
]. Now, applying Bayes rule we get,
Lb0 = loge
∑ak∈{···,+7,+1,−1,−7,···} e
− (y−ak)2
2σ2
∑aj∈{···,+5,+3,−3,−5,···} e
− (y−aj)2
2σ2
.
1S. Rao, R. Hormis, and E. Krouk, “The 4-D PAM-8 proposal for 10G-Base-T”, http://www.ieee802.org/3/10GBT/public/nov03/rao 1 1103.pdf, Nov. 2003
5
APP for 1-D lattice: PAM-8 constellation
−8 −6 −4 −2 0 2 4 6 8−500
−400
−300
−200
−100
0
100
200
300
400
500
channel observation y
AP
P
Lb
0 L
b1
PAM-8: APP for bits b0 and b1, noise σ = 0.15
6
APP calculation for 2-D and N-D lattices
• In this case, the received symbol y and the constellation symbols, ak ∈ A, can be viewed as vectors.
Let us denote the components of vector ak as [ak0 ak1 · · ·]. Similarly, y = [y0 y1 · · ·].
• We can write the APP for bit b0 as
Lb0 = loge
[∑ak∈A,b0=0 p(ak/y)∑aj∈A,b0=1 p(aj/y)
],
= loge
[∑ak∈A, b0=0 p(y/ak)∑aj∈A, b0=1 p(y/aj)
], assuming that p(an) constant ∀n.
• Notice that p(y/ak) = p(y0/ak0) . p(y1/ak1), since y0, y1, · · · are conditionally independent given
ak0, ak1, · · · respectively.
• We can then simplify the APP for b0 as
Lb0 = loge
[∑ak∈A, b0=0 p(y0/ak0) p(y1/ak1) · · ·∑aj∈A, b0=1 p(y0/aj0) p(y1/aj1) · · ·
],
which involves products of 1-D conditional Gaussian pdf’s .
7
APP for 2D-lattice: 2D-PAM12 Key-eye proposal 2
8/27/20044
128 Points - Cross Constellation
1001
000
1011
000
1100
000
1010
000
0000
011
0011
011
0010
011
0110
000
0111
000
0100
000
1110
000
1111
000
1101
000
0000
000
0001
000
0011
000
0010
000
0001
011
1000
110
1001
000
1011
000
1010
000
1000
110
1001
110
1011
110
1100
110
1010
110
0101
110
0011
011
0010
011
0110
110
0111
110
0100
110
1111
110
1101
111
0001
011
0001
110
0011
111
0010
110
0000
001
0011
011
0010
011
0001
011
1000 1001
000
1011
000
1010
000
1000 1001
011
1011
011
1100
011
1010
011
0000
110
0101
001
0011
011
0010
011
0110
011
0111
011
1110
110
1101
011
0001
011
0001
011
0011
011
0010
011
1000
100
1011
100
1100
100
1010
100
0101
100
0110
100
0100
100
1110
100
1111
100
1101
100
0000
100
0001
100
0011
100
0010
100
1000
010
1001
010
1011
010
1100
001
0101
010
0110
010
0111
010
0100
011
1110
011
1111
011
1101
010
0000
011
0011
010
0010
001
1000
101
1001
101
1011
101
1100
101
1010
101
0101
101
0110
101
0111
101
0100
101
1110
101
1111
101
1101
101
0000
101
0001
101
0011
101
0010
101
1000
000
1001
100
1100
010
1010
010
0101
000
0111
100
0100
010
1110
010
1111
010
1101
110
0000
010
0001
010
0011
110
0010
010
1000
111
1001
111
1011
111
1100
111
1010
111
0101
111
0110
111
0111
111
0100
111
1110
111
1111
111
0000
111
0001
111
0010
111
1000
011
1001
001
1011
001
1010
001
0101
011
0110
001
0111
001
0100
001
1110
001
1111
001
1101
001
0000
001
0001
001
0011
001
-11 -9 -7 -5 -3 -1 1 3 5 7 9 11
-11
-9-7
-5-3
-11
35
79
11
co-set
uncoded
2Weizhuang Xin, “Modified 128 point cross constellation mapping”, http://www.ieee802.org/3/10GBT/email/msg01026.html , Aug. 2004
8
APP for 2D-lattice: 2D-PAM12 Key-eye proposal
2-D APP function of b0, noise σ = 0.15.
9
APP for 2D-lattice: 2D-PAM12 Key-eye proposal
−15 −10 −5 0 5 10 15−200
−100
0
100
200
cross−section of LLR(b0), y
1 = 0
horiz. 1−D PAM−12 symbol y0
AP
P(b
0)
−15 −10 −5 0 5 10 15−400
−300
−200
−100
0
100
200
cross−section of LLR(b0), y
1 = +11
horiz. 1−D PAM−12 symbol y0
AP
P(b
0)
Cross-sections of 2-D APP function of b0, at y0 = +11 and y0 = 0, noise σ = 0.15.
10
APP for 2D-lattices: 2D-PAM12 Teranetics proposal 3
One Dimensional Map
-15 -10 -5 0 5 10 15-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
00
L
10
L
11
L
01
L
00
C
10
C
11
C
01
C
00
R
10
R
11
R
01
R
One Dimens ional Map for 12 PAM
Coset Labels
Point Labels
-10 -5 0 5 10
-10
-5
0
5
10
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
000
001
011
100
010
101
111
110
The Mapping of the Uncoded Bits in Two Dimensions
3D. Dabiri and J. Tellado, “Modifications to LDPC proposal offering lower symbol rate and lower latency”,
http://www.ieee802.org/3/an/public/mar04/dabiri 1 0304.pdf , Mar. 2004
11
APP for 2D-lattices: 2D-PAM12 Teranetics proposal
2-D APP function for bit b0, noise σ = 0.15.
12
APP for 2D-lattices: 2D-PAM12 Teranetics proposal
−15 −10 −5 0 5 10 15−600
−400
−200
0
200
400
600
cross−section of LLR(b0), y
1 = 0
horiz. 1−D PAM−12 symbol y0
AP
P(b
0)
−15 −10 −5 0 5 10 15−200
−100
0
100
200
cross−section of LLR(b0), y
1 = +11
horiz. 1−D PAM−12 symbol y0
AP
P(b
0)
Cross-section of APP function of b0, y0 = +11 and y0 = 0, noise σ = 0.15.
13
Conclusions
• From first principles, we derived the generalized APP metric for multi-dimensional constellations.
• Non-rectangular constellations lead to APP’s that are not always separable into 1-D functions
. . . the 1-D functions would be approximations.
Recommendations:
• Use of 2-D (N-D) APP metrics, depending on the dimensionality of the constellation.
• The APP metrics tend to be approximately piece-wise linear (planar) – perhaps easy to map using
LUT’s or digital gates.
• The piece-wise linear APP metric surface should be documented.
14
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