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 christop@myralink.com Christopher Jones, MyraLink

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 3.6dB @ BER = 10-4 Christopher Jones, MyraLink

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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