1. Chris Jones Cenk Kose Tao Tian Rick Wesel
July 2003
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
Electrical Engineering
UCLA
MyraLink
Consulting
Christopher Jones, MyraLink

2. Linear Gaussian Channels
July 2003
Linear Gaussian Channels
Christopher Jones, MyraLink

3. Shannon proved that a code exists for each H
July 2003
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
Christopher Jones, MyraLink

4. Universal Channel Codes
July 2003
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.
Christopher Jones, MyraLink

5. July 2003
The full range of 2x2 H’s
Mutual information depends only on the eigenvalues,
Or, on the `effective’ SNR and the eigenskew.
Christopher Jones, MyraLink

6. Performance on Sampling of Channels
July 2003
Performance on Sampling of Channels
32-state Trellis Codes
1.8
Universal, 2x2 8-PSK
1.6
Yan-Blum, 2x2 4-PSK
1.4
Siwag-Fitz, 2x2 4-PSK
Excess MI per antenna
1.2 1
0.8
0.6
0.2
0.5 1
Eigenvalue skew
Christopher Jones, MyraLink

7. LDPC on Sampling of Channels
July 2003
rate 1/3 length 15,000 irregular LDPC code on 2x2 with QPSK => 4/3 bps
BER = 10-5
Loss of
one TX Channel
Christopher Jones, MyraLink

8. July 2003
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.
Christopher Jones, MyraLink

9. Diagonal H yields a periodic SNR
July 2003
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
In other words, any periodic SNR variation that maintains mutual information should be fine.
that satisfy
Christopher Jones, MyraLink

10. OFDM creates a periodic channel
July 2003
OFDM creates a periodic channel a2 a1 a0
ap-1 ai
P-1
The mutual information (capacity) of this channel is given by :
Christopher Jones, MyraLink

11. Four OFDM-256 Channel Profiles
July 2003
Four OFDM-256 Channel Profiles
4 ISI Taps
8 ISI Taps
16 ISI Taps
16 ISI Taps
125 SubChannels
erased
Christopher Jones, MyraLink

12. How does the performance on each of these channels compare ?
July 2003
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.
Christopher Jones, MyraLink

13. LDPC Robustness Over OFDM-256 Channel Profiles
July 2003
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)
Christopher Jones, MyraLink

14. Rate Vs. Diversity for Bit Multiplexed MIMO
July 2003
Rate Vs. Diversity for Bit Multiplexed MIMO
S/P
&
Map
LDPC Code
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)
Christopher Jones, MyraLink

15. Diversity in systems with more than 2 trans. streams
July 2003
Diversity in systems with more than 2 trans. streams
Assume
S/P
&
Map
LDPC Code
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.
A possible max rate rule :
Christopher Jones, MyraLink

16. Connecting Code Rate, Diversity and Throughput
July 2003
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
Christopher Jones, MyraLink

17. Code Rate, Diversity and Throughput – 16QAM 4Tx Antenna
July 2003
Code Rate, Diversity and Throughput – 16QAM 4Tx Antenna
System Throughput
16 bits
12 bits
“Full Rate”
4 bits
Practical Full Diversity
1/ 4
3/4 1
Full Diversity
Code Rate
Christopher Jones, MyraLink

18. Code Rate, Diversity and Throughput – QPSK 2Tx Antenna
July 2003
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
Christopher Jones, MyraLink

19. SNR Performance in Fast Rayleigh Fading
July 2003
SNR Performance in Fast Rayleigh Fading
2bps
4bps
rate 1/2 length 15,000
rate 1/3 length 15,000
0.5dB
3.2dB
Length 4096 Rate ½ BER = 10-4
Christopher Jones, MyraLink

20. MI Performance in Fast Rayleigh Fading
July 2003
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
Christopher Jones, MyraLink

21. Per Real Dim. in Fast Rayleigh Fading
July 2003
Per Real Dim. in Fast Rayleigh Fading
Christopher Jones, MyraLink

22. Conclusion Bit Multiplexed LDPC Coding provides :
July 2003
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)
Christopher Jones, MyraLink

23. Appendix LDPC Background
July 2003
Appendix LDPC Background
Christopher Jones, MyraLink

24. What is a low-density parity check (LDPC) code?
July 2003
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.
Christopher Jones, MyraLink

25. Design of Irregular LDPC Codes
July 2003
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).
Christopher Jones, MyraLink

26. July 2003
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.
Christopher Jones, MyraLink

27. An Irregular Parity-Check Code
July 2003
An Irregular Parity-Check Code
Christopher Jones, MyraLink

28. Variable Nodes Variable Nodes v A B C D E F G July 2003
Christopher Jones, MyraLink

29. Constraint Nodes Constraint Nodes u A 1 B C 2 D E 3 F G July 2003
Christopher Jones, MyraLink

30. Column identifies edges from a variable node.
July 2003
Column identifies edges from a variable node. A 1 B C 2 D E 3 F G
Christopher Jones, MyraLink

31. Column identifies edges from a variable node.
July 2003
Column identifies edges from a variable node. A 1 B C 2 D E 3 F G
Christopher Jones, MyraLink

32. Row identifies edges into a constraint node.
July 2003
Row identifies edges into a constraint node. A 1 B C 2
Each constraint node
represents a
parity check equation D E 3 F G
Christopher Jones, MyraLink

33. Bi-Partite Graph Representation
July 2003
Bi-Partite Graph Representation A + B C + D E + F G
Christopher Jones, MyraLink

34. Message-Passing Decoder
July 2003
Message-Passing Decoder A 1
On each iteration, each constraint node provides a probability for each variable with which it shares an edge.
These probabilities are then combined for the computation of the new variable probability. B C 2 D E 3 F G
Christopher Jones, MyraLink

35. The equation implemented by constraint node.
July 2003
The equation implemented by constraint node. A 1 B C 2 D E 3 F G
Christopher Jones, MyraLink

36. July 2003
Computing a variable node probability from the constraint node probability. A 1 B C 2 D E 3 F G
Christopher Jones, MyraLink

37. July 2003
Computing a variable node probability from the constraint node probability. A 1 B C 2 D E 3 F G
Christopher Jones, MyraLink

38. July 2003
Computing an extrinsic probability from the variable node probabilities. A 1 B C 2 D E 3 F G
Christopher Jones, MyraLink

39. July 2003
Degree-Distribution Definition (Applicable to the design of Irregular LDPC Codes)
Christopher Jones, MyraLink

40. Left Degree of an edge A 1 B C 2 Number of edges that arrive
July 2003
Left Degree of an edge A 1 B C 2
Number of edges that arrive
at degree-3 nodes D E 3 F G
Christopher Jones, MyraLink

41. Left Degree of an edge A 1 B C 2 D E 3 F G July 2003
Christopher Jones, MyraLink

42. Left Degree of an edge A 1 B C 2 D E 3 F G July 2003
Christopher Jones, MyraLink

43. Left Degree of an edge A 1 B C 2 D E 3 F G July 2003
Christopher Jones, MyraLink

44. Right Degree of an edge A 1 B C 2 D E 3 F G July 2003
Christopher Jones, MyraLink