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July 2003 doc.: IEEE 802.11-03/0532-00-htsg
The Robustness of
Low-Density Parity-Check Codes
In Quasi-Static and Fast Rayleigh
Fading MIMO Channels
Chris Jones
Cenk Kose
Tao Tian
Rick Wesel
MyraLink Electrical Engineering
Consulting UCLA
Submission Slide 1 christop@myralink.com Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Linear Gaussian Channels
X1 Y1
X2 Y2
X3 Y3
Y ΗX W
MI(H) log det I HH
†
Submission Slide 2 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Shannon proved that a code
exists for each H
• Shannon proved that for each channel H
there is a code that can reliably transmit at
rate R as long as R < MI, where
MI(H) log det I HH †
MI H log 2 1 1 1 2
Submission Slide 3 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Universal Channel Codes
[Root & Varaiya 68]:There exists a
single code that supports rate R for
the entire family of linear Gaussian
vector channels Y=HX+W with
MI(H) > R.
Submission Slide 4 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
The full range of 2x2 H’s
1 0 cos sin e j
H 1 j
0 sin e cos
2 [0, ] [0,2 ]
1
MI ((H ) log 2det I 1 1 2
MI H ) 1 N HH
2 t 2
Mutual information depends only on the eigenvalues,
Or, on the `effective’ SNR and the eigenskew.
Submission Slide 5 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Performance on Sampling of
Channels
32-state Trellis Codes
1.8
Excess MI per antenna
Universal, 2x2 8-PSK
1.6
Yan-Blum, 2x2 4-PSK
1.4 Siwag-Fitz, 2x2 4-PSK
1.2
1
0.8
0.6
0 0.2 0.5 1
Eigenvalue skew
Submission Slide 6 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
LDPC on Sampling of Channels
rate 1/3 length 15,000 irregular LDPC code on 2x2 with QPSK => 4/3 bps
BER = 10-5
Loss of
one TX Channel
Submission Slide 7 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Conclusions for 2x2
• A 32-state universal space-time trellis code
consistently requires 1.06 bits of excess mutual
information per-antenna or less.
• A blocklength 15,000 universal space-time
LDPC code requires 0.24 bits of excess mutual
information per-antenna or less.
• Universal design guarantees good performance
under any quasistatic distribution.
Submission Slide 8 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Diagonal H yields a periodic SNR
• Root and Varaiya result implies that a single
code can support rate R per p dimensions
over all channels
yi = i mod p xi + wi
that satisfy
i Es
p
log 1 2
i 1
R
• In other words, any periodic SNR variation
that maintains mutual information should be
fine.
Submission Slide 9 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
OFDM creates a periodic channel
a1 a2
a0 ap-1
ai
0 P-1
xi {1,1} yi
~
a [a0 , a1 , , a p 1 ] n (0, 2 )
ai 0
The mutual information (capacity) of this channel is given by :
1 p 1
~ N
MI (a) I X ; X
p i 0 ai
Submission Slide 10 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Four OFDM-256 Channel Profiles
4 ISI Taps
8 ISI Taps
16 ISI Taps
16 ISI Taps
125 SubChannels
erased
Submission Slide 11 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
How does the performance on
each of these channels compare ?
- Measured in terms of SNR, it’s hard to tell.
- Instead, we measure the channel Mutual Information
and plot versus this quantity instead of in terms of SNR.
- Channel Mutual Information provides an Absolute measure
with which to compare performance.
Submission Slide 12 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
LDPC Robustness Over OFDM-256 Channel Profiles
Rate 1/3 length 15,000
irregular LDPC
SNR Performance
On Channels a,b,c,d
MI Performance
On Channels a,b,c,d
(Tightly Clustered)
Submission Slide 13 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Rate Vs. Diversity for Bit Multiplexed MIMO
S/P
LDPC Code &
Map
Rate ≤ 1/2
Full Diversity (loosely) ≡ System can operate when all but one TX trans. is lost
Full Rate ≡ The upper bound on achievable rate when all but one TX trans. is lost
In the above, Full Rate equals 2 bps. The code rate which supports this is 1/2
However, for the eigenskew 0 channel (half of all symbols are punctured) the
code can not be guaranteed to operate – rate ½ code under 50% erasure
System design ranges from Full Rate (Rate > ½ code)
to Full Diversity (Rate < ½ code)
Submission Slide 14 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Diversity in systems with more than 2 trans. streams
Assume N r Nt
S/P
LDPC Code &
Map
Rate = ?
Q: Should the rate of this system be low enough to support loss of all but
one transmit channel ? e.g. Rate ≤ 1/Nt
A: From channel data, the answer is no. More than one transmit channel
(eigenvalue) is very unlikely to be lost.
Nt 1
A possible max rate rule : Code Rate
Nt
Submission Slide 15 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Connecting Code Rate, Diversity
and Throughput
log2(M)*Nt
System
Throughput
log2(M)*(Nt-1)
“Full Rate” log2(M)
Practical Full
Diversity
1/ Nt (Nt-1)/Nt 1
Full Diversity
Code Rate
Submission Slide 16 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Code Rate, Diversity and
Throughput – 16QAM 4Tx Antenna
System 16 bits
Throughput
12 bits
“Full Rate” 4 bits
Practical Full
Diversity
1/ 4 3/4 1
Full Diversity
Code Rate
Submission Slide 17 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Code Rate, Diversity and
Throughput – QPSK 2Tx Antenna
System
Throughput 4 bits
“Full Rate” 2 bits
Practical Full
Diversity
1/ 2 1
Full Diversity
Code Rate
Submission Slide 18 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
SNR Performance in Fast Rayleigh Fading
2bps
4bps
rate 1/2 length 15,000
rate 1/3 length 15,000
0.5dB 3.2dB
Submission Slide 19 Christopher Jones, MyraLink
Length 4096 Rate ½ 3.6dB @ BER = 10-4
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
MI Performance in Fast Rayleigh Fading
In Blue, 1x1 to 4x4 Gauss Sig Cap QPSK 4x4 Cap
Rate 1/2 op points
Rate 1/3 op points
(BER = 10-5) QPSK 3x3 Cap
QPSK 2x2 Cap
QPSK 1x1 Cap
BPSK 1x1 Cap
Submission Slide 20 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
MI Per Real Dim. in Fast Rayleigh Fading
Submission Slide 21 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Conclusion
• Bit Multiplexed LDPC Coding provides :
– Scalability (in antenna dimension &
modulation cardinality)
– Robustness (via consistency of mutual
information performance across
a broad range of channel
realizations)
– Rate flexibility (via code puncturing or
shortening – not shown here)
– Low complexity kernel decoding operations are
available (not shown here)
Submission Slide 22 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Appendix
LDPC Background
Submission Slide 23 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
What is a low-density parity check
(LDPC) code?
• It is simply a binary linear block code in which
the parity matrix has a low density of ones.
• A Regular LDPC code has the same number
of ones in each column and the same number
of ones in each row.
• Back in the 60’s Gallager showed that the
class of regular LDPC codes was a capacity-
achieving class.
• That means that as the blocklength goes to
infinity, certain codes of this type can have a
block error rate that goes to zero while
maintaining any rate below channel capacity.
Submission Slide 24 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Design of Irregular LDPC Codes
• Irregular LDPC codes tend to begin to
work at lower SNRs.
• However, they have so-called “error floors”
• Irregular LDPC codes are designed in two
steps
– Obtain a degree distribution through density
evolution
– Design a particular parity matrix that has that
degree distribution. (affects error floor).
Submission Slide 25 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Decoding LDPC Codes
• It has been known that these are “good” codes
for forty years.
• Gallager even described a message –passing
decoder.
• However, with the advent of turbo codes, LDPC
codes were rediscovered.
• The LDPC message-passing decoder has been
refined in light of what we know from turbo
decoding.
• We will now construct the bi-partite graph on
which decoding takes place.
Submission Slide 26 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
An Irregular Parity-Check
Code
1 1 1 0 1 0 0
H 1 1 0 1 0 1 0 n k 3 rows
1 0 1 1 0 0 1
n7 columns
Submission Slide 27 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Variable Nodes
A B C D E F G A
1 1 1 0 1 0 0 1
1 1 0 1 0 1 0 2
1 0 1 1 0 0 1 3 B
C
Variable Nodes v D
E
F
G
Submission Slide 28 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Constraint Nodes
A B C D E F G A 1
1 1 1 0 1 0 0 1
1 1 0 1 0 1 0 2
1 0 1 1 0 0 1 3 B
Constraint
C 2
Nodes u
D
E 3
F
G
Submission Slide 29 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Column identifies edges from a variable node.
A B C D E F G A 1
1 1 1 0 1 0 0 1
1 1 0 1 0 1 0 2
1 0 1 1 0 0 1 3 B
C 2
D
E 3
F
G
Submission Slide 30 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Column identifies edges from a variable node.
A B C D E F G A 1
1 1 1 0 1 0 0 1
1 1 0 1 0 1 0 2
1 0 1 1 0 0 1 3 B
C 2
D
E 3
F
G
Submission Slide 31 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Row identifies edges into a constraint node.
A B C D E F G A 1
1 1 1 0 1 0 0 1
1 1 0 1 0 1 0 2
1 0 1 1 0 0 1 3 B
C 2
Each constraint node
represents a D
parity check equation
E 3
x A xB xC xE 0
F
G
Submission Slide 32 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Bi-Partite Graph Representation
A B C D E F G A +
1 1 1 0 1 0 0 1
1 1 0 1 0 1 0 2
1 0 1 1 0 0 1 3 B
C +
D
E +
F
G
Submission Slide 33 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Message-Passing Decoder
A 1
• On each iteration, each B
constraint node
provides a probability 2
C
for each variable with
which it shares an
D
edge.
• These probabilities are 3
E
then combined for the
computation of the new
variable probability. F
G
Submission Slide 34 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
The equation implemented by constraint node.
x A xB xC xE 0 A 1
B
C 2
D
E 3
F
G
Submission Slide 35 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Computing a variable node probability from the constraint node probability.
A 1
Let pA P(vA 1)
p A u1 P (vB vC vE 1) B
pB 1 pC 1 pE C 2
pC 1 pB 1 pE
pE 1 pB 1 pC
D
pB pC pE E 3
1 1
,E} 1 2 pi
2 2 i{ B ,C F
G
Submission Slide 36 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Computing a variable node probability from the constraint node probability .
A 1
B
C 2
D
E 3
F
G
Submission Slide 37 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Computing an extrinsic probability from the variable node probabilities.
A 1
For u1 : pA pA u2 pA u3
B
C 2
D
E 3
F
G
Submission Slide 38 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Degree-Distribution Definition
(Applicable to the design of
Irregular LDPC Codes)
Submission Slide 39 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Left Degree of an edge
A 1
E3 3 B
C 2
Number of edges that arrive
at degree-3 nodes
D
E 3
F
G
Submission Slide 40 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Left Degree of an edge
A 1
B
E3 3 C 2
E2 6 D
E 3
F
G
Submission Slide 41 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Left Degree of an edge
A 1
E3 3 B
E2 6
C 2
E1 3
D
E 3
F
G
Submission Slide 42 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Left Degree of an edge
E3 3 A 1
E2 6 B
E1 3
E Ei 12
C 2
i D
Ei i 1
( x) x E 3
i E F
x x 1
4
1
2
1
4
2
G
Submission Slide 43 Christopher Jones, MyraLink
July 2003 doc.: IEEE 802.11-03/0532-00-htsg
Right Degree of an edge
A 1 E4 12
E Ei 12
B
C 2 i
D Ei i 1
( x) x
i E
E 3
F x 3
G
Submission Slide 44 Christopher Jones, MyraLink
