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A code construction for the block-fading cooperative channel presented by Joseph J. Boutros Texas A&M University at Qatar In Collaboration with Dieter Duyck and Marc Moeneclaey Ghent University, Belgium Seminar at TELECOM ParisTech Paris, France 18 December 2008 Presentation outline 1 Introduction 2 System model and notations 3 Outage probability analysis 4 Coded cooperation via full-diversity LDPC 5 Density evolution on the block-fading relay channel 6 Numerical results Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Introduction (1) Diversity (Proakis 2000, Tse & Viswanath 2005): transmitting information over diﬀerent paths in time, frequency, space, etc. Diversity: it saves power and improves error rate performance on fading channels. Cooperative Communications (Hunter & Nosratinia 2002-2004, Laneman 2002, Sendonaris, Erkip & Aazhang 2003): let two or more single-antenna devices cooperate together in order to create space (multi-antenna) diversity. Relay channel (van der Meulen 1971, Cover & El Gamal 1979): the simplest example of cooperation with one source, one relay, and one destination. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 1 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Introduction (1) Diversity (Proakis 2000, Tse & Viswanath 2005): transmitting information over diﬀerent paths in time, frequency, space, etc. Diversity: it saves power and improves error rate performance on fading channels. Cooperative Communications (Hunter & Nosratinia 2002-2004, Laneman 2002, Sendonaris, Erkip & Aazhang 2003): let two or more single-antenna devices cooperate together in order to create space (multi-antenna) diversity. Relay channel (van der Meulen 1971, Cover & El Gamal 1979): the simplest example of cooperation with one source, one relay, and one destination. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 1 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Introduction (1) Diversity (Proakis 2000, Tse & Viswanath 2005): transmitting information over diﬀerent paths in time, frequency, space, etc. Diversity: it saves power and improves error rate performance on fading channels. Cooperative Communications (Hunter & Nosratinia 2002-2004, Laneman 2002, Sendonaris, Erkip & Aazhang 2003): let two or more single-antenna devices cooperate together in order to create space (multi-antenna) diversity. Relay channel (van der Meulen 1971, Cover & El Gamal 1979): the simplest example of cooperation with one source, one relay, and one destination. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 1 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Introduction (2) A Cooperative Multiple Access Channel (MAC). User 1 1111 0000 0000 1111 0000 1111 Destination 1111 0000 0000 1111 0000 1111 0000 1111 1111 0000 1111 0000 0000 1111 1111 0000 1111 00002 User The speciﬁc task of the relay is determined by the protocol. Amplify and Forward: the relay ampliﬁes the received signal and noise. Decode and Forward: the relay decodes, re-encodes the message, and sends it to the destination. Coded Cooperation: the relay decodes the message and transmits additional parity bits. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 2 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Introduction (2) A Cooperative Multiple Access Channel (MAC). User 1 1111 0000 0000 1111 0000 1111 Destination 1111 0000 0000 1111 0000 1111 0000 1111 1111 0000 1111 0000 0000 1111 1111 0000 1111 00002 User The speciﬁc task of the relay is determined by the protocol. Amplify and Forward: the relay ampliﬁes the received signal and noise. Decode and Forward: the relay decodes, re-encodes the message, and sends it to the destination. Coded Cooperation: the relay decodes the message and transmits additional parity bits. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 2 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results System model and notations (1) Block fading on all links, independent from one codeword to another. Channel state information at the decoder side. BPSK modulation and a rate -R binary [N, K] code. Transmission on orthogonal slots (orthogonal protocol). Half-duplex devices (simultaneous reception and transmission too complicated). D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 3 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results System model and notations (2.a) Case 1: Frame 1 Frame 2 User 1 X1 X2p User 2 X2 X1p Both interuser channels are successfully decoded. Each user cooperates in the second frame. Deﬁnition: Cooperation Level We deﬁne the level of cooperation, β, as the ratio of the length of frame 2 by the total length of both frames. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 4 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results System model and notations (2.a) Case 1: Frame 1 Frame 2 User 1 X1 X2p User 2 X2 X1p Both interuser channels are successfully decoded. Each user cooperates in the second frame. Deﬁnition: Cooperation Level We deﬁne the level of cooperation, β, as the ratio of the length of frame 2 by the total length of both frames. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 4 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results System model and notations (2.b) Case 2: Frame 1 Frame 2 User 1 X1 X1p User 2 X2 X2p Both interuser channels are in outage. Each user sends its own parity bits in the second frame. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 5 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results System model and notations (2.c) Case 3: Frame 1 Frame 2 User 1 X1 X1p User 2 X2 X1p User2-to-User1 channel is in outage. User 1 sends its own parity bits in second frame. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 6 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results System model and notations (2.d) Case 4: Frame 1 Frame 2 User 1 X1 X2p User 2 X2 X2p User1-to-User2 channel is in outage. User 2 sends its own parity bits in second frame. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 7 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Outage probability analysis (1) In coded cooperation for a two-user MAC with BPSK signaling, the outage event Eo related to user 1 is: Proposition: outage of the two-user cooperative MAC with BPSK Eo = [(I12 > R/(1 − β)) ∩ (I21 > R/(1 − β)) ∩ (I1d (case 1) < R)] ∪ [(I12 < R/(1 − β)) ∩ (I21 < R/(1 − β)) ∩ (I1d (case 2) < R)] ∪ [(I12 > R/(1 − β)) ∩ (I21 < R/(1 − β)) ∩ (I1d (case 3) < R)] ∪ [(I12 < R/(1 − β)) ∩ (I21 > R/(1 − β)) ∩ (I1d (case 4) < R/(1 − β))] −2y12α12 I12 = 1 − Ey log2 1 + exp 2 σ12 −2y21α21 I21 = 1 − Ey log2 1 + exp 2 σ21 D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 8 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Outage probability analysis (2) The average mutual information I1d can be easily determined: Case 1: −2y1dα1 −2y2dα2 1−(1−β) Ey log2 1 + exp 2 −β Ey log2 1 + exp 2 σ1d σ2d Case 2: −2y1dα1 1 − Ey log2 1 + exp 2 σ1d Case 3: −2y1dα1 1 − (1 − β) Ey log2 1 + exp 2 σ1d −2(y ′ )(α2 + α2 )3/2 1 2 −β Ey′ log2 1 + exp σ1d α2 + σ2d α2 2 1 2 2 Case 4: −2y1dα1 1 − Ey log2 1 + exp 2 σ1d D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 9 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Outage probability analysis (3) Take the extremal channel where fading is 0 and +∞, i.e., the block erasure channel. After a simple analysis of I1d , we obtain Corollary: Rate limitation due to cooperation In coded cooperation for the 2-user MAC with a cooperation level β, transmitting at a coding rate higher than min(β, 1 − β) will always restrict the diversity order of the outage probability to one. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 10 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Full-Diversity Cooperative LDPC (1) Rate-compatible convolutional codes (Hagenauer 1988) can be used for coded cooperation. But the WER will vary as log(N )d (Boutros & Fabregas 2005). Special LDPC codes for the relay channel without fading has been studied (Razaghi & Yu 2007). Randomly structured LDPC for the block-fading relay channel has been proposed (Hu & Duman 2007). No guaranty for full-diversity and limitation in the highest possible rate. Punctured LDPC codes suﬀer from weak performance and a random structure does not guaranty full-diversity. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 11 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Full-Diversity Cooperative LDPC (2) Initially, we adopt the following structure for LDPC rate compatibility (equivalent to a serial concatenation). For example, rate(H1 )=2/3 and rate(H1 )=1/2. SOURCE RELAY i p1 p2 H1 0 0 H2 Then, H2 will be replaced by a root-LDPC and H1 will be split into a direct sum to balance the coding gain. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 12 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Full-Diversity Cooperative LDPC (3) Compact Tanner graph for a root-LDPC (full-diversity code) with rate 1/2 that is MDS on a 2-state channel (Boutros, Zemor, Fabregas & Biglieri 2007). S 1111 0000 0000 1111 0000 1111 O 1111 0000 1i 0000 1111 0000 1111 1111 0000 0000 1111 1 U R 0000 1111 0000 1111 0000 1111 2 3c C 1111 0000 1p 0000 1111 0000 1111 1111 0000 0000 1111 E 3 3 1111 0000 R 0000 1111 1111 0000 0000 1111 2p 0000 1111 4c E 1111 0000 1111 0000 2 0000 1111 1 L 0000 1111 A 1111 0000 0000 1111 0000 1111 2i 0000 1111 0000 1111 Y 1111 0000 1111 0000 D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 13 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Full-Diversity Cooperative LDPC (4) We obtain a symmetric structure for a cooperative root-LDPC SOURCE RELAY 1i 1p p′1 2i 2p p′2 H1 0 0 0 1c 0 0 0 H1 2c 1 1 1 0 0 H2i H2p 0 3c 1 1 1 H1i H1p 0 1 0 0 4c 1 It is possible to extend to more than 2 relays (i.e. higher diversity orders). D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 14 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Full-Diversity Cooperative LDPC (5) The general Tanner graph for an irregular cooperative root-LDPC is 1111 0000 1111 0000 λ1(x) ρ1(x) N 0000 1111 1111 0000 S p′1 0000 1111 6 0000 1111 1111 0000 1c N O 1111 0000 6 U 0000 1 (x) 1111λ 1111 0000 1111 0000 N 0000 1111 R 1i 1111 6 0000 0000 11111111 0000 C ˜ 1111 0000 λ 2 (x) 1 3c N 1111 0000 E 0000 1111 1111 N 0000 λ1(x) 6 0000 1111 1p 1111 6 0000 1111 0000 1111 0000 ρ2(x) 1111 0000 λ2(x) ρ2(x) 0000 2 (x) 1111 0000 1111 λ N 0000 1111 1111 0000 2p 0000 1111 λ1(x) 4c N 6 1111 0000 1111 0000 6 R 0000 1111 ˜ λ2(x) E 0000 1111 1 0000 1111 L 1111 N 0000 0000 1111 2i 2i 1111 6 0000 0000 1111 A 1111 0000 1 (x) λ 0000 1111 2c N 6 Y 0000 1111 1111 0000 1111 N 0000 0000 1111 ρ1(x) p′2 6 0000 1111 0000 1111 λ1(x) 1111 0000 1111 0000 D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 15 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Full-Diversity Cooperative LDPC (6) Proposition: Cooperative (rate-compatible) Root-LDPC The rate-compatible root-LDPC code deﬁned by the parity-check matrix (or equivalently the Tanner graph) shown in the previous slide achieves diversity 2 on a block-fading relay channel. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 16 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Density Evolution 12 density evolution equations are needed to study DE for our cooperative root-LDPC code. For the sake of simplicity, we do not show those details. Dr Boutros would like to thank his student, Dieter Duyck, for the hard work done in establishing the DE equations and writing its numerical implementation. Results are shown in the next slides. D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 17 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Numerical Results for a cooperative full-diversity root-LDPC code (1) R = 1/3, Interuser=5dB, β = 0.5. 10+0 BPSK Outage Probability Irregular LDPC Code Regular LDPC Code 10-1 10-2 WER 10-3 10-4 10-5 10-6 3 6 9 12 15 18 21 24 EsN0[dB] D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 18 / 19 Introduction System Model Outage Analysis Full-Diversity Cooperative Coding Density Evolution Numerical Results Numerical Results for a cooperative full-diversity root-LDPC code (2) R = 0.45, Interuser=12dB, Relay-Destination=4dB, β = 0.5. 10+0 BPSK Outage probability Irregular LDPC code 10-1 10-2 WER 10-3 10-4 10-5 10-6 3 6 9 12 15 18 21 24 EsN0[dB] D. Duyck, J.J. Boutros, and M. Moeneclaey MIR Seminar at INFRES December 2008 19 / 19