Hybrid Coding. Young-Han Kim, UCSD. Living Information Theory Workshop Happy<70.42> th birthday,professorberger!
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1 Hybrid Coding Young-Han Kim, UCSD Living Information Theory Workshop Happy<70.42> th birthday,professorberger! 1/21
2 Hybrid Coding Young-Han Kim, UCSD Living Information Theory Workshop Happy<70.42> th birthday,professorberger! SungHoonLim,PaoloMinero,andAbbasElGamal 1/21
3 Gratitude#1 2/21
4 Gratitude # /
5 Gratitude # /
6 Gratitude#2 3/21
7 Gratitude#2 Strong typicality (Extended) Markov lemma 3/21
8 Gratitude#3 4/21
9 Gratitude#3 4/21
10 Gratitude#3 But life is short and information endless: nobody has time for everything. In practice we are generally forced to choose between an unduly brief exposition and no exposition at all. Abbreviation is a necessary evil and the abbreviator s businessistomakethebestofajobwhich, though intrinsically bad, is still better than nothing. He must learn to simplify, but not to the point of falsification. He must learn to concentrate upon the essentials of a situation, but without ignoring too many of reality s qualifying side issues. Aldous Huxley Brave New World Revisited 4/21
11 Point-to-Point Communication System S n Source channel X n p(y x) Source channel Ŝ n encoder decoder Question Find the sufficient and necessary condition to achieve lim sup n 1 n n i=1 d(s i,ŝ i ) D 5/21
12 Point-to-Point Communication System S n Source M Channel X n Channel ˆM p(y x) Source Ŝ n encoder encoder encoder encoder Question Find the sufficient and necessary condition to achieve lim sup n 1 n n i=1 d(s i,ŝ i ) D Answer: Source channel separation(shannon 1948, 1959) Sufficient and necessary condition: R(D) = min I(S;Ŝ)< max I(X;Y)=C p(ŝ s) E(d(S,Ŝ)) D p(x) 5/21
13 Point-to-Point Communication System S n Source M Channel X n Channel ˆM p(y x) Source Ŝ n encoder encoder encoder encoder Question Find the sufficient and necessary condition to achieve lim sup n 1 n n i=1 d(s i,ŝ i ) D Answer: Source channel separation(shannon 1948, 1959) Sufficient and necessary condition: R(D) = min I(S;Ŝ)< max I(X;Y)=C p(ŝ s) E(d(S,Ŝ)) D p(x) 5/21
14 Point-to-Point Communication System S n Source M Channel X n Channel ˆM p(y x) Source Ŝ n encoder encoder encoder encoder Question Find the sufficient and necessary condition to achieve lim sup n 1 n n i=1 d(s i,ŝ i ) D Answer: Source channel separation(shannon 1948, 1959) Sufficient and necessary condition: R(D) = min I(S;Ŝ)< max I(X;Y)=C p(ŝ s) E(d(S,Ŝ)) D p(x) Digital source channel interface 5/21
15 Uncoded Transmission S n Source channel X n p(y x) Source channel Ŝ n encoder decoder 6/21
16 Uncoded Transmission S n X n Ŝ n x(s) p(y x) ŝ(y) 6/21
17 Uncoded Transmission S n X n Ŝ n x(s) p(y x) ŝ(y) Optimal for quadratic Gaussian source coding over the Gaussian channel with average power constraint(goblick 1965) 6/21
18 Uncoded Transmission S n X n Ŝ n x(s) p(y x) ŝ(y) Optimal for quadratic Gaussian source coding over the Gaussian channel with average power constraint(goblick 1965) R(D)=C 6/21
19 Uncoded Transmission S n X n Ŝ n x(s) p(y x) ŝ(y) Optimal for quadratic Gaussian source coding over the Gaussian channel with average power constraint(goblick 1965) R(D)=C Tocodeornottocode (Gastpar,Rimoldi,andVetterli2003) 6/21
20 Uncoded Transmission S n X n Ŝ n x(s) p(y x) ŝ(y) Optimal for quadratic Gaussian source coding over the Gaussian channel with average power constraint(goblick 1965) R(D)=C Tocodeornottocode (Gastpar,Rimoldi,andVetterli2003) Analog source channel interface 6/21
21 Hybrid Coding S n Source channel X n p(y x) Source channel Ŝ n encoder decoder 7/21
22 Hybrid Coding S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder 7/21
23 Hybrid Coding S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Sufficient condition: I(U;S)<I(U;Y) forsomep(u s), x(u,s),ŝ(u,y)suchthate(d(s,ŝ)) D 7/21
24 Hybrid Coding S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Sufficient condition: I(U;S)<I(U;Y) forsomep(u s), x(u,s),ŝ(u,y)suchthate(d(s,ŝ)) D Analog/digital hybrid source channel interface 7/21
25 Hybrid Coding S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Sufficient condition: I(U;S)<I(U;Y) forsomep(u s), x(u,s),ŝ(u,y)suchthate(d(s,ŝ)) D Analog/digital hybrid source channel interface U= :uncodedtransmission 7/21
26 Hybrid Coding S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Sufficient condition: I(U;S)<I(U;Y) forsomep(u s), x(u,s),ŝ(u,y)suchthate(d(s,ŝ)) D Analog/digital hybrid source channel interface U= :uncodedtransmission U=(Ŝ,X) p(ŝ s)p(x):separatesourceandchannelcoding 7/21
27 Hybrid Coding S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Sufficient condition: I(U;S)<I(U;Y) forsomep(u s), x(u,s),ŝ(u,y)suchthate(d(s,ŝ)) D Analog/digital hybrid source channel interface U= :uncodedtransmission U=(Ŝ,X) p(ŝ s)p(x):separatesourceandchannelcoding Beaten? (Shamai, Verdú, and Zamir 1998, Mittal and Phamdo 2002, Tinguely and Lapidoth 2006, Caire and Narayanan 2007, Kochman, Khina, Erez, and Zamir 2009, Tian, Diggavi, and Shamai 2010) 7/21
28 Hybrid Coding S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Sufficient condition: I(U;S)<I(U;Y) forsomep(u s), x(u,s),ŝ(u,y)suchthate(d(s,ŝ)) D Analog/digital hybrid source channel interface U= :uncodedtransmission U=(Ŝ,X) p(ŝ s)p(x):separatesourceandchannelcoding Beaten? (Shamai, Verdú, and Zamir 1998, Mittal and Phamdo 2002, Tinguely and Lapidoth 2006, Caire and Narayanan 2007, Kochman, Khina, Erez, and Zamir 2009, Tian, Diggavi, and Shamai 2010) 7/21
29 Proof of Achievability S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder 8/21
30 Proof of Achievability S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Random codebook generation 2 nr sequences u n (m) n i=1 p U(u i ) form [1 2 nr ] 8/21
31 Proof of Achievability S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Random codebook generation 2 nr sequences u n (m) n i=1 p U(u i ) form [1 2 nr ] Joint typicality encoding Findanindexmsuchthat(u n (m),s n ) T (n) є andtransmit x i = x(u i (m),s i ): 8/21
32 Proof of Achievability S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Random codebook generation 2 nr sequences u n (m) n i=1 p U(u i ) form [1 2 nr ] Joint typicality encoding Findanindexmsuchthat(u n (m),s n ) T (n) є andtransmit x i = x(u i (m),s i ): successfulw.h.p.ifr>i(u;s) 8/21
33 Proof of Achievability S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Random codebook generation 2 nr sequences u n (m) n i=1 p U(u i ) form [1 2 nr ] Joint typicality encoding Findanindexmsuchthat(u n (m),s n ) T (n) є andtransmit x i = x(u i (m),s i ): successfulw.h.p.ifr>i(u;s) Joint typicality decoding Findtheuniqueindex ˆmsuchthat(u n ( ˆm),y n ) T (n) є ŝ i = ŝ(u i ( ˆm),y i ): and reconstruct 8/21
34 Proof of Achievability S n Source U n (M) encoder X n x(u, s) p(y x) Channel U n ( ˆM) Ŝ n ŝ(u, y) decoder Random codebook generation 2 nr sequences u n (m) n i=1 p U(u i ) form [1 2 nr ] Joint typicality encoding Findanindexmsuchthat(u n (m),s n ) T (n) є andtransmit x i = x(u i (m),s i ): successfulw.h.p.ifr>i(u;s) Joint typicality decoding Findtheuniqueindex ˆmsuchthat(u n ( ˆm),y n ) T (n) є ŝ i = ŝ(u i ( ˆm),y i ): successfulw.h.p.ifr<i(u;y) and reconstruct 8/21
35 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding 9/21
36 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding R 1 > I(U 1 ;S 1 ), R 2 > I(U 2 ;S 2 ) 9/21
37 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding R R 1 > I(U 1 ;S 1 ), 1 < I(U 1 ;Y,U 2 ), R R 2 > I(U 2 ;S 2 ) 2 < I(U 2 ;Y,U 1 ), R 1 +R 2 < I(U 1,U 2 ;Y)+I(U 1 ;U 2 ) 9/21
38 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding I(U 1 ;S 1 Q)<I(U 1 ;Y,U 2 Q), I(U 2 ;S 2 Q)<I(U 2 ;Y,U 1 Q), I(U 1 ;S 1 Q)+I(U 2 ;S 2 Q)<I(U 1,U 2 ;Y Q)+I(U 1 ;U 2 Q) forsomep(q)p(u 1 s 1,q)p(u 2 s 2,q), x j (u j,s j,q),ŝ j (u 1,u 2,y,q)such thate(d j (S j,ŝ j )) D j, j=1,2 10/21
39 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding 11/21
40 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding Special cases (Ŝ 1,Ŝ 2 )=(S 1,S 2 ):losslesscoding(cover,elgamal,andsalehi1980) 11/21
41 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding Special cases (Ŝ 1,Ŝ 2 )=(S 1,S 2 ):losslesscoding(cover,elgamal,andsalehi1980) Y=(X 1,X 2 ): noiselesschannel(berger1978,tung1978) 11/21
42 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding Special cases (Ŝ 1,Ŝ 2 )=(S 1,S 2 ):losslesscoding(cover,elgamal,andsalehi1980) Y=(X 1,X 2 ): noiselesschannel(berger1978,tung1978) Gaussian(S 1,S 2 )andgaussianmac(tinguelyandlapidoth2006) 11/21
43 Correlated Sources over a MAC S1 n S2 n Source encoder 1 Source encoder 2 U1(M n 1) x 1(u 1,s 1) U2(M n 2) x 2(u 2,s 2) X n 1 X n 2 p(y x 1,x 2) Channel decoder U1( n ˆM 1) ŝ 1(u 1,u 2,y) U2( n ˆM 2) ŝ 2(u 1,u 2,y) Ŝ n 1 Ŝ n 2 Hybrid coding Berger Tung source coding + MAC coding Special cases (Ŝ 1,Ŝ 2 )=(S 1,S 2 ):losslesscoding(cover,elgamal,andsalehi1980) Y=(X 1,X 2 ): noiselesschannel(berger1978,tung1978) Gaussian(S 1,S 2 )andgaussianmac(tinguelyandlapidoth2006) Common part: Kaspi Berger coding + Slepian Wolf MAC coding 11/21
44 2-Sender 2-Receiver Channel S1 n S2 n Encoder 1 Encoder 2 X1 n X2 n p(y 1,y 2 x 1,x 2) 1 2 Decoder 1 Decoder 2 Ŝ n 11,Ŝ n 21 Ŝ n 12,Ŝ n 22 12/21
45 2-Sender 2-Receiver Channel S1 n S2 n Encoder 1 Encoder 2 X1 n X2 n p(y 1,y 2 x 1,x 2) 1 2 Decoder 1 Decoder 2 Ŝ n 11,Ŝ n 21 Ŝ n 12,Ŝ n 22 Hybrid coding 12/21
46 2-Sender 2-Receiver Channel S n 1 S n 2 Encoder 1 Encoder 2 X n 1 X n 2 p(y 1,y 2 x 1,x 2) 1 2 Decoder 1 Decoder 2 Ŝ n 11,Ŝ n 21 Ŝ n 12,Ŝ n 22 Hybrid coding Conceptually simple yet performs very well 12/21
47 2-Sender 2-Receiver Channel S n 1 S n 2 Encoder 1 Encoder 2 X n 1 X n 2 p(y 1,y 2 x 1,x 2) 1 2 Decoder 1 Decoder 2 Ŝ n 11,Ŝ n 21 Ŝ n 12,Ŝ n 22 Hybrid coding Conceptually simple yet performs very well PaoloMinero stalkonmonday10:05am(rightbeforedavidtse) 12/21
48 Diamond Network M Encoder X n 1 2 p(y 2,y 3 x 1) p(y 4 x 2,x 3) Yn 4 3 Relay 2 Relay 3 X n 2 X n 3 Decoder ˆM Existing coding schemes Decode forward(digital-to-digital interface) 13/21
49 Diamond Network M Encoder X n 1 2 p(y 2,y 3 x 1) p(y 4 x 2,x 3) Yn 4 3 Relay 2 Relay 3 X n 2 X n 3 Decoder ˆM Existing coding schemes Decode forward(digital-to-digital interface) Amplify forward(analog-to-analog interface) 13/21
50 Diamond Network M Encoder X n 1 2 p(y 2,y 3 x 1) p(y 4 x 2,x 3) Yn 4 3 Relay 2 Relay 3 X n 2 X n 3 Decoder ˆM Existing coding schemes Decode forward(digital-to-digital interface) Amplify forward(analog-to-analog interface) Noisy network coding(analog-to-digital interface) 13/21
51 Diamond Network M Encoder X n 1 2 p(y 2,y 3 x 1) p(y 4 x 2,x 3) Yn 4 3 Relay 2 Relay 3 X n 2 X n 3 Decoder ˆM Existing coding schemes Decode forward(digital-to-digital interface) Amplify forward(analog-to-analog interface) Noisy network coding(analog-to-digital interface) Question Can we do better than these coding schemes? 13/21
52 Diamond Network M Encoder X n 1 2 p(y 2,y 3 x 1) p(y 4 x 2,x 3) Yn 4 3 Relay 2 Relay 3 X n 2 X n 3 Decoder ˆM Answer Hybrid coding(analog-to-[analog/digital] interface) 14/21
53 Diamond Network M Encoder X n 1 2 p(y 2,y 3 x 1) p(y 4 x 2,x 3) Yn 4 3 Relay 2 Relay 3 X n 2 X n 3 Decoder ˆM Answer Hybrid coding(analog-to-[analog/digital] interface) Yj n Source U n j(m j ) encoder x j (u j,y j ) X n j 14/21
54 Diamond Network M Encoder X n 1 2 p(y 2,y 3 x 1) p(y 4 x 2,x 3) Yn 4 3 Relay 2 Relay 3 X n 2 X n 3 Decoder ˆM Answer Hybrid coding(analog-to-[analog/digital] interface) Yj n Source U n j(m j ) encoder x j (u j,y j ) X n j Naturally combines noisy network coding and amplify forward 14/21
55 Example 1: Deterministic Diamond Network M Encoder X n 1 2 (y 2,y 3)(x 1) y 4(x 2,x 3) 3 Relay 2 Relay 3 X n 2 X n 3 4 Decoder ˆM 15/21
56 Example 1: Deterministic Diamond Network M Encoder X n 1 2 (y 2,y 3)(x 1) y 4(x 2,x 3) 3 Relay 2 Relay 3 X n 2 X n 3 4 Decoder ˆM Cutset bound C max p(x 1,x 2,x 3 ) min{h(y 2,Y 3 ), H(Y 2 )+H(Y 4 X 3 ), H(Y 3 )+H(Y 4 X 2 ), H(Y 4 )} 15/21
57 Example 1: Deterministic Diamond Network M Encoder X n 1 2 (y 2,y 3)(x 1) y 4(x 2,x 3) 3 Relay 2 Relay 3 X n 2 X n 3 4 Decoder ˆM Cutset bound C max p(x 1,x 2,x 3 ) min{h(y 2,Y 3 ), H(Y 2 )+H(Y 4 X 3 ), H(Y 3 )+H(Y 4 X 2 ), H(Y 4 )} General lower bound(avestimehr, Diggavi, and Tse 2007) C max p(x 1 )p(x 2 )p(x 3 ) min{h(y 2,Y 3 ), H(Y 2 )+H(Y 4 X 3 ), H(Y 3 )+H(Y 4 X 2 ), H(Y 4 )} 15/21
58 Example 1: Deterministic Diamond Network M Encoder X n 1 2 (y 2,y 3)(x 1) y 4(x 2,x 3) 3 Relay 2 Relay 3 X n 2 X n 3 4 Decoder ˆM Cutset bound C max p(x 1,x 2,x 3 ) min{h(y 2,Y 3 ), H(Y 2 )+H(Y 4 X 3 ), H(Y 3 )+H(Y 4 X 2 ), H(Y 4 )} General lower bound(avestimehr, Diggavi, and Tse 2007) C max p(x 1 )p(x 2 )p(x 3 ) min{h(y 2,Y 3 ), H(Y 2 )+H(Y 4 X 3 ), H(Y 3 )+H(Y 4 X 2 ), H(Y 4 )} Hybrid coding lower bound C max p(x 1 )p(x 2 y 2 )p(x 3 y 3 ) min{h(y 2,Y 3 ), H(Y 2 )+H(Y 4 X 3 ), H(Y 3 )+H(Y 4 X 2 ), H(Y 4 )} 15/21
59 Example 1: Deterministic Diamond Network M Encoder X n 1 2 (y 2,y 3)(x 1) y 4(x 2,x 3) 3 Relay 2 Relay 3 X n 2 X n 3 4 Decoder ˆM BlackwellBC(y 2 (x 1 ),y 3 (x 1 )) BinaryerasureMAC y 4 (x 2,x 3 ) X Y 2 Y 3 Y 4 = X 2 +X 3 X 2,X 3 {0,1}, Y 4 {0,1,2} 16/21
60 Example 1: Deterministic Diamond Network M Encoder X n 1 2 (y 2,y 3)(x 1) y 4(x 2,x 3) 3 Relay 2 Relay 3 X n 2 X n 3 4 Decoder ˆM BlackwellBC(y 2 (x 1 ),y 3 (x 1 )) BinaryerasureMAC y 4 (x 2,x 3 ) X Y 2 Y 3 Y 4 = X 2 +X 3 X 2,X 3 {0,1}, Y 4 {0,1,2} Generallowerbound=1.5<log3=Hybridcodinglowerbound 16/21
61 Example 2: Gaussian Diamond Network Z 2 X 1 g 21 Y 2 X 2 g 42 Z 4 Y 4 g 31 Y 3 X 3 g 43 Z 3 17/21
62 Example 2: Gaussian Diamond Network Z 2 g Y 2 X 2 1 Z 4 X 1 Y 4 1 Y 3 X 3 g Z 3 17/21
63 AFvs.NNCvs.HybridCoding g 18/21
64 AFvs.NNCvs.HybridCoding g 18/21
65 Example 3: Gaussian Two-Way Relay Channel Z 3 g 31 g 32 X 1 X 2 Y 3 X 3 Y 1 Y g 2 13 g 23 Z 1 Z 2 19/21
66 Example 3: Gaussian Two-Way Relay Channel Z 3 r 3/2 (1 r) 3/2 X 1 X 2 Y 3 X 3 Y 1 Y 2 r 3/2 (1 r) 3/2 Z 1 Z 2 r /21
67 Hybrid Coding vs. Other Coding Schemes R CS R AF R NNC R DF r 20/21
68 Hybrid Coding vs. Other Coding Schemes R CS R HC R AF R NNC R DF r 20/21
69 Concluding Remarks Hybrid coding 21/21
70 Concluding Remarks Hybrid coding Versatile interface for source channel coding and relaying 21/21
71 Concluding Remarks Hybrid coding Versatile interface for source channel coding and relaying Joint source channel coding can be simple(and fun too!) 21/21
72 Concluding Remarks Hybrid coding Versatile interface for source channel coding and relaying Joint source channel coding can be simple(and fun too!) Hybrid coding > amplify forward + noisy network coding 21/21
73 Concluding Remarks Hybrid coding Versatile interface for source channel coding and relaying Joint source channel coding can be simple(and fun too!) Hybrid coding > amplify forward + noisy network coding Hybrid coding + structured coding > decode forward 21/21
74 Concluding Remarks Hybrid coding Versatile interface for source channel coding and relaying Joint source channel coding can be simple(and fun too!) Hybrid coding > amplify forward + noisy network coding Hybrid coding + structured coding > decode forward Back to reality 21/21
75 Concluding Remarks Hybrid coding Versatile interface for source channel coding and relaying Joint source channel coding can be simple(and fun too!) Hybrid coding > amplify forward + noisy network coding Hybrid coding + structured coding > decode forward Back to reality Barron and Oppenheim(2002): US Patent 6,441,764 21/21
76 Concluding Remarks Hybrid coding Versatile interface for source channel coding and relaying Joint source channel coding can be simple(and fun too!) Hybrid coding > amplify forward + noisy network coding Hybrid coding + structured coding > decode forward Back to reality Barron and Oppenheim(2002): US Patent 6,441,764 Meir Feder: Amimon high-definition wireless audio-video modem 21/21
77 Concluding Remarks Hybrid coding Versatile interface for source channel coding and relaying Joint source channel coding can be simple(and fun too!) Hybrid coding > amplify forward + noisy network coding Hybrid coding + structured coding > decode forward Back to reality Barron and Oppenheim(2002): US Patent 6,441,764 Meir Feder: Amimon high-definition wireless audio-video modem Any good code for both(source) encoding and(channel) decoding? 21/21
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