San Jose State University SJSU ScholarWorks Faculty Publications Electrical Engineering 11-30-2010 Error Correcting Codes for Cooperative Broadcasting Robert H. Morelos-Zaragoza San Jose State University, robert.morelos-zaragoza@sjsu.edu Follow this and additional works at: http://scholarworks.sjsu.edu/ee_pub Part of the Electrical and Computer Engineering Commons Recommended Citation Robert H. Morelos-Zaragoza. "Error Correcting Codes for Cooperative Broadcasting" Faculty Publications (2010). This Presentation is brought to you for free and open access by the Electrical Engineering at SJSU ScholarWorks. It has been accepted for inclusion in Faculty Publications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.
Error Correcting Codes for Cooperative Broadcasting Robert Morelos Zaragoza Electrical Engineering Department San José State University Presented at the ISL Simposium Stanford University November 30, 2010
1. Motivation 2. Related work 3. Broadcast channels Outline 4. Multilevel codes and the u u+v construction Time sharing u v versus u u+v construction Decoding for u u+v construction: Two stages versus single state 5. Demapping and decoding performance BPSK and QPSK modulations 4 PAM modulation: Hierarchical mapping 6. Conclusions and future work November 30, 2010 2
1. Motivation Background on unequal error protection (UEP) codes: Ph.D. thesis onmultilevel error correcting codes UEP codes are based on the idea of superposition coding proposed in Cover s paper (1972) on broadcast channels The nodes of a wireless network (cooperative or not) always broadcast information i (i.e., every node, in principle, i receives this information) Cooperative broadcasting paper (Bergmans and Cover,1974) Superposition coding always outperforms orthogonal (time division orfrequency division) schemes 3November 30, 2010
Broadcasting in a Wireless Network Low SNR region (far) RN 2 S D High SNR region (close) RN 1 RN m 1. Relay RN 1 has higher SNR compared to RN 2 C 1 > C 2 2. Shortest path (smallest number of hops) not always most reliable November 30, 2010 4
2. Related work Cover (1972): Broadcast Channels Cloud structure of capacity achieving superposition codes Bergmans and Cover (1974): Cooperative Broadcasting Superposition codes always outperform orthogonal assignment Laneman, Tse and Wornell (2004): Cooperative Diversity in Wireless Networks: Efficient Protocols and Outage Behavior Amplify and forward, decode and forward, adaptive relaying, incremental relaying Stefanov and Erkip (2004): Cooperative Coding for Wireless Networks Propose superposition coding. Do not refer to Cover!!? Yi, Azimi, Kalyanaraman and Subramanian (2005): Error Control Code Combining Techniques in Cluster based Cooperative Wireless Networks Use Chase code combining with hybrid ARQ 5November 30, 2010
Related work (cont.) S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft,: XORs in The Air: Practical Wireless Network Coding, (2006) Opportunistic coding Chen and Ahmed (2008): Throughput Enhancement in Cooperative Diversity Wireless Networks using Adaptive Modulation Feedback CSI and adapt constellation accordingly Li, Ge, Tang and Xiong (2008): Cooperative Diversity Based on Alamouti Space Time Code Multiple access stage of relay nodes achieved with Alamouti s scheme L. Xiao, T.E. Fuja, J. Kliewer and D.J. Costello (2009): Error Performance Analysis of Signal Superposition Coded Cooperative Diversity Consider superposition coding performed at a single source, metrics H.J. Yang, Y. Choi and J. Chun (2010): Modified High Order PAMs for Binary Coded Physical Layer Network Coding Constellation design for superposition ii (physical ll layer) network coding 6November 30, 2010
3. Broadcast Channels Thomas Cover (1972) Gaussian broadcast channel with one source broadcasting information to two users: User 1 has a larger signal to noise i ratio (SNR) than User 2 (N 1 < N 2 ): Source sequence Sequence received by (low noise) user 1 Sequence received by (high noise) user 2 From: T. Cover, Broadcast Channels, IEEE Trans. Info. Theory, vol. IT 18, no. 1, pp. 2 14, Jan. 1972 7November 30, 2010
Coding for broadcast channels: Cloud concept Cover showed that a channel code achieving capacity has a cloud structure, shown below for a binary symmetric broadcast channel: From: T. Cover, Broadcast Channels, IEEE Trans. Info. Theory, vol. IT 18, no. 1, pp. 2 14, Jan. 1972 A cloud is a set of codewords (or sequences) that is selected with the information bits (most important or MSB) to be transmitted to the high noise user 8November 30, 2010
Cooperative Broadcasting (1974) Coordinated transmission (Need a side channel) S 1 D 1 (or RN 1 ) S 2 D 2 (or RN 2 ) Mixed-mode: OFDM with water-filling time sharing Naive time-sharing From: P. Bergmans and T. Cover, Cooperative Broadcasting, IEEE Trans. Info. Theory, vol. IT-20, no. 3, pp. 317-324, May 1974. November 30, 2010 9
4. Multilevel LUEP codes and u u+v construction An LUEP code has subcode partition chain C C with and d > d > > 1 2 L ' G1 G 2 ' ' G G 2 3 G =, G2 =, G L G L d., Practical two level LUEP codes can been constructed based on block, convolutional lor LDPC codes and Plotkin s (or u u+v ) ) construction: 0 G G = 1, ' ' G2 G2 ' where d = 2d < 1. 2 2 d G L 1, C L Cloud centers (coset leaders) Cloud (subcode) codewords 10 November 30, 2010
Time sharing ( u v ) versus Plotkin ( u u+v ) Time sharing: M 1 (MSB) C 1 Transmitter B 1 BPSK mapper m(b i ) AWGN channel Broadcast Receiver Y BPSK demapper L c (Y) Decoder 1 ^ M 1 M 2 (LSB) C 2 B 2 Decoder 2 ^ M 2 Plotkin: M 1 (MSB) M 2 C 1 C 2 11 01 B 1 Transmitter C B 2 BPSK mapper Broadcast Receiver m(c) AWGN Y BPSK L c (Y) Two-stage channel demapper decoder (LSB) M 2 ^ M 1 ^ November 30, 2010 11
Performance of u v vs. u u+v : short LDPC codes C 1, C 2 : LDPC (96,50) codes of degrees (3,6) and (4,8) November 30, 2010 12
A Plotkin u u+v coding scheme Multilevel codes always improve the throughput over any orthogonal (time or frequency division) approach Follow Bergmans and Cover s idea: Design an over theair u u+v (Plotkin) coding scheme: ( 0 ) m 1 v 1 S 1 D 1 (or RN 1 ) m ( v ) 2 2 v2 ( 0 v 1 ) + 21m 2 ( v 2 v 2 ) n 1 y + 1 = α 11m 1 α S 2 D 2 (or RN 2 ) (Subcode) (Coset) ( 0 v1 ) + 22 m2( v2 v2 ) n2 y + 2 = α12 m1 α November 30, 2010 13
Two sources with BPSK mapping BPSK Mapping m i from a bit to a signal set M i, i=1,2. Assume α i1 =α i2 M 1 ={s 11,s 12 } and M 2 ={s 21,s 22 }: s i1 (B i =0) E o s i2 (B i =1) E φ(t) At the receiver, the direct sum M 1 +M 2 is equal to a ternary signal set: (B 1 +B 2 =0) (B 1 +B 2 =1) (B 1 +B 2 =0) 2 E o 2 E y November 30, 2010 14
Single source and cooperative broadcasting Single source: M 1 (MSB) M 2 (LSB) C 1 C 2 B 1 B 2 Transmitter Cooperative (two sources): Broadcast Receiver C BPSK m(c) AWGN Y BPSK mapper channel demapper L c (Y) Decoder ^ M 1 ^ M 2 M 1 (MSB) C 1 Transmitter 1 B 1 BPSK m(b 1 ) mapper Transmitter 2 Σ Two-source Cooperative Broadcast Receiver AWGN channel Y superposition demapper L c (Y) Decoder ^ M 1 M 2 (LSB) C 2 B 2 BPSK mapper m(b 2 ) ^ M 2 November 30, 2010 15
Decoding for u u+v construction Two stages (Kumar Milenkovic, 2006) y BPSK Demapper 2 L v (y) Decoder H2 LSB BPSK mapper +/ 1 L u (y) BPSK Demapper 1 Decoder H1 MSB Single stage (SJSU, 2009) y Superposition Demapper L c (y) Decoder H = H1 0 H2 H2 u+v u LSB MSB Morelos Zaragoza Superposition Coding for Cooperative Broadcasting Slide 16
u u+v decoding: Simulation results C 1 : Regular (96,50) LDPC code with degrees (3,6); C 2 : Regular (96,49) LDPC codes with degrees (4,8) Uncoded BPSK Proposed Kumar Morelos Zaragoza Superposition Coding for Cooperative Broadcasting Slide 17
H matrix used in u u+v construction with LDPC codes of length th96 0 20 40 60 80 0 20 40 60 80 100 120 140 160 180 nz = 1056 November 30, 2010 18
H matrix used in u u+v construction with LDPC codes of length th204 November 30, 2010 19
Systematic encoding Use Gaussian elimination to produce systematic generator matrices and a permutation: Length 96: Length 204: November 30, 2010 20
Single source versus cooperative: BPSK Metrics Single source Cooperative (dual source) B 1 B 2 C=B 1 +B 2 y B 1 B 2 y 0 0 0-1 0 1 1 +1 1 0 1 +1 1 1 0 1 0 0-2 0 1 0 1 0 0 1 1 2 1 1 0-1 1 1-2 Log-likelihood ratio (LLR) metric: L c ( y ) = { c = y } { c= y} Pr 1 Pr 0 L c (y) L c (y) 0 1 0 1 0 y E E 2 E 0 2 E y November 30, 2010 21
LLR metrics for cooperative broadcasting and BPSK modulation (E/N 0 =10 db) 30 B m(b) 0 1 1 1 L c (r) 20 10 0-10 -20-30 -3-2 -1 0 1 2 3 r November 30, 2010 22
Cooperative u u+v : Lengths 96 and 204 AWGN channel November 30, 2010 23
Cooperative u u+v : Length 96 Flat Rayleigh hfdi fading channel 10 0 Uncoded LSB MSB 10-1 BER 10-2 10-3 10-4 0 2 4 6 8 10 12 E b /N 0 (db) November 30, 2010 24
Cooperative u u+v : Length 204 Flat Rayleigh hfdi fading channel 10 0 10-1 BER 10-2 10-3 10-4 0 2 4 6 8 10 12 E b /N 0 (db) November 30, 2010 25
Results are applicable to QPSK modulation November 30, 2010 26
QPSK = BPSK x BPSK: Subset mapping Two basis functions: φ 1 (t), φ 2 (t). Basic idea: Quadrature multiplexing Source i: BPSK mapping with φ i (t), i=1,2 Receiver processes two BPSK sequences in parallel branches M 1 (MSB) C 1 Transmitter 1 B 1 BPSK/I m(b 1 ) mapper Transmitter 2 B M2 2 BPSK/Q m(b 2 ) 2 C 2 (LSB) Σ Two-source Cooperative Broadcast Receiver AWGN channel BPSK/I demapper L c (Y) L c (Y) Decoder 1 BPSK/Q Decoder mapper demapper 2 ^ M 1 ^ M 2 November 30, 2010 27
QPSK subset vs Cooperative u u+v BPSK C 1, C 2 : LDPC (204,102) codes of node degrees (3,6) and (5,10) Uncoded BPSK QPSK BPSK November 30, 2010 28
Cooperative u u+v with 4 PAM and natural mapping 4 PAM Mapping m i from two bits to a signal set M i, i=1,2. Assume α i1 =α i2 M 1 ={s 11,ss 12,ss 13,s 14 } and M 2 ={s 21,ss 22,ss 23,s 24 } : s i1 (B i1b i2 =00) s i2 (01) s i3 (10) s i4 (11) 3 E /10 E /10 E /10 3 E /10 o φ(t) At the receiver, the direct sum M 1 +M 2 is equal to a 7 ary signal set: -6-4 -2 0 2 4 6 y / E/10 B 11 +B 12 = 0 0 1,0,1 1 1,0,1 0 0 B 21 +B 22 = 0 1 0,0,0 1 0,0,0 1 0 No dichotomy for bit B2 November 30, 2010 29
LLR metrics for cooperative broadcasting and 4 PAM modulation with natural mapping (E/N 0 =10) 30 20 10 L c1 (r) 0-10 -20-30 -8-6 -4-2 0 2 4 6 8 r 30 20 10 L c2 (r) 0-10 -20-30 -8-6 -4-2 0 2 4 6 8 r November 30, 2010 30
Metrics for cooperative broadcasting and 4 PAM modulation with Gray mapping (E/N 0 =10) 30 20 10 L c1 (r) 0-10 -20-30 -8-6 -4-2 0 2 4 6 8 r 30 20 10 L c2 (r) 0-10 -20-30 -8-6 -4-2 0 2 4 6 8 r November 30, 2010 31
Cooperative u u+v with 4 PAM and hierarchical mapping 4 PAM Mapping m i from two bits to a signal set M i, i=1,2. Assume α i1 =α i2 Again, two sets: M 1 ={s 11,s 12,s 13,s 14 } and M 2 ={s 21,s 22,s 23,s 24 } with two power levels At the receiver, the direct sum M 1 +M 2 is equal to a 28 ary signal set: -6-4 -2 0 2 4 6 y/ E/10 This idea was proposed by L. Xiao, T.E. Fuja, J. Kliewer and D.J. Costello (2009) Note: In their scheme, superposition takes place at the transmitter All dichotomies (i.e., metric L c =0) are removed, in exchange for decreased error performance Results are applicable to 16 QAM modulation November 30, 2010 32
Metrics for cooperative broadcasting with 4 PAM modulation and hierarchicalmapping 30 20 10 L c1 (r) 0-10 -20-30 -4-3 -2-1 0 1 2 3 4 r 30 20 10 L c2 (r) 0-10 -20-30 -4-3 -2-1 0 1 2 3 4 r November 30, 2010 33
Performance of 4 PAM with hierarchical mapping. Length 96 LDPC codes 10-1 LSB MSB 10-2 BER 10-3 10-4 18 19 20 21 22 23 24 25 26 E b /N 0 (db) November 30, 2010 34
Performance of 4 PAM with hierarchical mapping. Length 204 LDPC codes 10-1 MSB LSB 10-2 BER 10-3 10-4 18 19 20 21 22 23 24 25 26 E b /N 0 (db) November 30, 2010 35
Conclusions and future work Proposed a coding scheme for two user cooperative broadcasting ( over the air mixing ), based on Plotkin s u u+v construction using BPSK, QPSK, 4 PAM and 16 QAM modulations Cooperative broadcasting = Network coding over physical layer MILCOM 2010 presentation, comment from Matthew C. Valenti: i(his paper was on Receiver Design for Noncoherent Digital Network Coding ) Future directions Performance with longer LDPC codes (such as those used in WiMax) Design rules based on LDPC code parameters (minimum distance, node distributions) versus proportion of MSB and performance Use a software radio platform to study Synchronization techniques for over the air mixing Effect of channel estimation errors Error performance over realistic (frequency selective) wireless channels Noncoherent modulation: Differential encoding, FSK (as in Valenti s paper) 36
Supporting slides November 30, 2010 37
Performance of WiMax codes November 30, 2010 38
Combining cooperative u u+v with Alamouti: Length 96 codes November 30, 2010 39
Combining cooperative u u+v with Alamouti: Length 204 codes November 30, 2010 40