A code construction for the block-fading cooperative channel by yfv54841

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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

								
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