Second-Order Asymptotics for the Gaussian MAC with Degraded Message Sets

Size: px

Start display at page:

Download "Second-Order Asymptotics for the Gaussian MAC with Degraded Message Sets"

Robyn Stevenson
6 years ago
Views:

Tan Department of Engineering, University of Cambridge Electrical and Computer Engineering,

1 Second-Order Asymptotics for the Gaussian MAC with Degraded Message Sets Jonathan Scarlett and Vincent Y. F. Tan Department of Engineering, University of Cambridge Electrical and Computer Engineering, National University of Singapore Information Theory and Applications Workshop (Feb 2013) Full version: Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

2 Second-Order Asymptotics for Point-to-Point Channels Second-order asymptotics for the DMC and the AWGN channel are well-understood Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

3 Second-Order Asymptotics for Point-to-Point Channels Second-order asymptotics for the DMC and the AWGN channel are well-understood M (n, ε, S): maximum codebook size for n uses of the AWGN channel with SNR S and average error probability ε Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

4 Second-Order Asymptotics for Point-to-Point Channels Second-order asymptotics for the DMC and the AWGN channel are well-understood M (n, ε, S): maximum codebook size for n uses of the AWGN channel with SNR S and average error probability ε Hayashi (2009) and Polyanskiy-Poor-Verdú (2010) showed that log M (n, ε, S) = nc(s) + nv(s)φ 1 (ε) + o( n) nats, where the Gaussian capacity and Gaussian dispersion functions are defined as C(S) := 1 2 log(1 + S), V(S) := S(S + 2) 2(S + 1) 2 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

5 Second-Order Asymptotics for Point-to-Point Channels Second-order asymptotics for the DMC and the AWGN channel are well-understood M (n, ε, S): maximum codebook size for n uses of the AWGN channel with SNR S and average error probability ε Hayashi (2009) and Polyanskiy-Poor-Verdú (2010) showed that log M (n, ε, S) = nc(s) + nv(s)φ 1 (ε) + o( n) nats, where the Gaussian capacity and Gaussian dispersion functions are defined as C(S) := 1 2 log(1 + S), V(S) := S(S + 2) 2(S + 1) 2 Second-order coding rate = Largest coefficient of n term = V(S)Φ 1 (ε). Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

6 Second-Order Asymptotics in Networks Second-order asymptotics in networks has been more modest Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

7 Second-Order Asymptotics in Networks Second-order asymptotics in networks has been more modest Tan-Kosut (2014) and Nomura-Han (2012) Slepian-Wolf problem Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

8 Second-Order Asymptotics in Networks Second-order asymptotics in networks has been more modest Tan-Kosut (2014) and Nomura-Han (2012) Slepian-Wolf problem R 2 Let L(ε; R 1, R 2 ) be the set of all (L 1, L 2 ) R 2 such that there exists length-n codes of sizes (M 1,n, M 2,n ) and errors ε n such that lim sup n 1 (log M j,n nr n j ) L j, j = 1, 2, lim sup ε n ε. n θ = tan(l 2/L 1) H 2 H 2 1 θ (R 1, R 2 ) H 1 2 H 1 R 1 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

9 Second-Order Asymptotics in Networks Second-order asymptotics in networks has been more modest Tan-Kosut (2014) and Nomura-Han (2012) Slepian-Wolf problem R 2 Let L(ε; R 1, R 2 ) be the set of all (L 1, L 2 ) R 2 such that there exists length-n codes of sizes (M 1,n, M 2,n ) and errors ε n such that lim sup n 1 (log M j,n nr n j ) L j, j = 1, 2, lim sup ε n ε. n θ = tan(l 2/L 1) (L 1, L 2 ) L(ε; R 1, R 2) H 2 H 2 1 θ (R 1, R 2 ) H 1 2 H 1 R 1 implies exists ε-reliable codes with log M j,n nr j + n L j + o( n) Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

10 Second-Order Asymptotics in Networks R 2 θ = tan(l 2/L 1) H 2 H 2 1 θ (R 1, R 2 ) H 1 2 H 1 R 1 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

11 Second-Order Asymptotics in Networks R 2 θ = tan(l 2/L 1) H 2 H 2 1 θ (R 1, R 2 ) H 1 2 H 1 R 1 where L(ε; H 1, H 2 1 ) = {(L 1, L 2 ) : Ψ(L 2, L 1 + L 2 ; V 2,12 ) 1 ε}. Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

12 Second-Order Asymptotics in Networks R 2 θ = tan(l 2/L 1) H 2 H 2 1 θ (R 1, R 2 ) H 1 2 H 1 R 1 where L(ε; H 1, H 2 1 ) = {(L 1, L 2 ) : Ψ(L 2, L 1 + L 2 ; V 2,12 ) 1 ε}. z2 z3 Ψ(z 2, z 3 ; V) := N (0, V) du, and ( [ ] ) V 2,12 := Cov log px2 X 1 log p X1 X 2 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

13 Second-Order Asymptotics in the MAC The next-easiest network problem is the MAC Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

14 Second-Order Asymptotics in the MAC The next-easiest network problem is the MAC Numerous attempts to characterize second-order asymptotics for the MAC including Tan-Kosut (2014), Huang-Moulin (2012), MolavianJazi-Laneman (2012), Haim-Erez-Kochman (2012), Scarlett-Martinez-Guillén i Fàbregas (2013) etc. Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

15 Second-Order Asymptotics in the MAC The next-easiest network problem is the MAC Numerous attempts to characterize second-order asymptotics for the MAC including Tan-Kosut (2014), Huang-Moulin (2012), MolavianJazi-Laneman (2012), Haim-Erez-Kochman (2012), Scarlett-Martinez-Guillén i Fàbregas (2013) etc. No tight local-dispersion characterization yet for the DM-MAC or Gaussian MAC Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

16 Second-Order Asymptotics in the MAC The next-easiest network problem is the MAC Numerous attempts to characterize second-order asymptotics for the MAC including Tan-Kosut (2014), Huang-Moulin (2012), MolavianJazi-Laneman (2012), Haim-Erez-Kochman (2012), Scarlett-Martinez-Guillén i Fàbregas (2013) etc. No tight local-dispersion characterization yet for the DM-MAC or Gaussian MAC Some problems (primarily the converse): 1 Difficulty in characterizing the boundary of the CR 2 Independent input distributions Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

17 Second-Order Asymptotics in the MAC The next-easiest network problem is the MAC Numerous attempts to characterize second-order asymptotics for the MAC including Tan-Kosut (2014), Huang-Moulin (2012), MolavianJazi-Laneman (2012), Haim-Erez-Kochman (2012), Scarlett-Martinez-Guillén i Fàbregas (2013) etc. No tight local-dispersion characterization yet for the DM-MAC or Gaussian MAC Some problems (primarily the converse): 1 Difficulty in characterizing the boundary of the CR 2 Independent input distributions We consider a MAC-like model that retains the main characteristics of MAC but skirts the problems above Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

18 Gaussian MAC with Degraded Message Sets Z N (0, I n n ) M 1 f 1,n X 1 Y + ϕ ( ˆM 1, ˆM 2 ) n X 2 M 2 f 2,n x j 2 2 ns j, j = 1, 2 Encoder 1 has access to both messages Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

19 Gaussian MAC with Degraded Message Sets Z N (0, I n n ) M 1 f 1,n X 1 Y + ϕ ( ˆM 1, ˆM 2 ) n X 2 M 2 f 2,n x j 2 2 ns j, j = 1, 2 Encoder 1 has access to both messages Capacity region is well known [Exercise 5.18(b), El Gamal and Kim (2012)]; achieved using superposition coding Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

20 Gaussian MAC with Degraded Message Sets R2 (nats/use) The Capacity Region CR Boundary ρ =0 ρ =1/3 ρ =2/3 R 1 C ( (1 ρ 2 )S 1 ) R 1 +R 2 C ( S 1 +S 2 +2ρ S 1 S 2 ) C(x) = 1 log(1 + x) R1 (nats/use) ρ [0, 1] parametrizes curved boundary and indicates the amount of correlation between users codewords. Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

21 Second-Order Asymptotics for Gaussian MAC w/ DMS (R 1, R 2 ): arbitrary point on the boundary of the capacity region Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

22 Second-Order Asymptotics for Gaussian MAC w/ DMS (R 1, R 2 ): arbitrary point on the boundary of the capacity region L(ε; R 1, R 2 ): set of all (L 1, L 2 ) R 2 such that there exists length-n codes of sizes (M 1,n, M 2,n ) and average errors ε n such that lim inf n 1 (log M j,n nr n j ) L j, j = 1, 2, lim sup ε n ε n and x j 2 2 ns j, for j = 1, 2 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

23 Second-Order Asymptotics for Gaussian MAC w/ DMS (R 1, R 2 ): arbitrary point on the boundary of the capacity region L(ε; R 1, R 2 ): set of all (L 1, L 2 ) R 2 such that there exists length-n codes of sizes (M 1,n, M 2,n ) and average errors ε n such that lim inf n 1 (log M j,n nr n j ) L j, j = 1, 2, lim sup ε n ε n and x j 2 2 ns j, for j = 1, 2 If (L 1, L 2 ) L(ε; R 1, R 2 ), there exists ε-reliable codes s.t. log M j,n nr j + n L j + o( n), j = 1, 2. Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

24 Second-Order Asymptotics for Gaussian MAC w/ DMS (R 1, R 2 ): arbitrary point on the boundary of the capacity region L(ε; R 1, R 2 ): set of all (L 1, L 2 ) R 2 such that there exists length-n codes of sizes (M 1,n, M 2,n ) and average errors ε n such that lim inf n 1 (log M j,n nr n j ) L j, j = 1, 2, lim sup ε n ε n and x j 2 2 ns j, for j = 1, 2 If (L 1, L 2 ) L(ε; R 1, R 2 ), there exists ε-reliable codes s.t. log M j,n nr j + n L j + o( n), j = 1, 2. Main Contribution: A complete characterization of L(ε; R 1, R 2 ) First complete characterization of second-order asymptotics for a channel-type network information theory problem Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

25 Some Basic Definitions Mutual informations [ ] [ I1 (ρ) C ( (1 ρ I(ρ) := = 2 ) ] )S 1 I 12 (ρ) C ( S 1 + S 2 + 2ρ ) S 1 S 2 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

26 Some Basic Definitions Mutual informations [ ] [ I1 (ρ) C ( (1 ρ I(ρ) := = 2 ) ] )S 1 I 12 (ρ) C ( S 1 + S 2 + 2ρ ) S 1 S 2 Derivative of mutual informations D(ρ) := ρ I(ρ) = [ S 1 ρ 1+S 1 (1 ρ 2 ) S1 S 2 ] 1+S 1 +S 2 +2ρ S 1 S 2 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

27 Some Basic Definitions Mutual informations [ ] [ I1 (ρ) C ( (1 ρ I(ρ) := = 2 ) ] )S 1 I 12 (ρ) C ( S 1 + S 2 + 2ρ ) S 1 S 2 Derivative of mutual informations D(ρ) := ρ I(ρ) = [ S 1 ρ 1+S 1 (1 ρ 2 ) S1 S 2 ] x(y+2) 2(x+1)(y+1) 1+S 1 +S 2 +2ρ S 1 S 2 Dispersions V(x, y) := and V(x) := V(x, x) [ ] V1 (ρ) V V(ρ) := 1,12 (ρ) V 1,12 (ρ) V 12,12 (ρ) where V 1 (ρ) := V ( (1 ρ 2 ) )S 1, V12,12 (ρ) := V ( S 1 + S 2 + 2ρ ) S 1 S 2 V 1,12 (ρ) := V ( (1 ρ 2 )S 1, S 1 + S 2 + 2ρ ) S 1 S 2 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

28 Generalization of Inverse CDF of a Gaussian For a positive semi-definite matrix V, Ψ(z 1, z 2, V) = z1 z2 N (0, V) du Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

29 Generalization of Inverse CDF of a Gaussian For a positive semi-definite matrix V, Ψ(z 1, z 2, V) = Given ε (0, 1), z1 z2 N (0, V) du Ψ 1 (V, ε) = {(z 1, z 2 ) : Ψ( z 1, z 2, V) 1 ε} ρ = ρ = z2 z ε = 0.01 ε = z ε = 0.01 ε = z1 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

30 The Main Result: Vertical Boundary Points on vertical boundary reduce to scalar dispersion as sum rate constraint is in error exponents regime [Haim-Erez-Kochman (2012)] L(ε; R 1, R 2) = {(L 1, L 2 ) : L 1 V 1 (0)Φ 1 (ε)} The Capacity Region CR Boundary ρ =0 ρ =1/3 ρ =2/ R2 (nats/use) R1 (nats/use) (R 1, R 2 ) Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

31 The Main Result: Vertical Boundary Points on vertical boundary reduce to scalar dispersion as sum rate constraint is in error exponents regime [Haim-Erez-Kochman (2012)] L(ε; R 1, R 2) = {(L 1, L 2 ) : L 1 V 1 (0)Φ 1 (ε)} The Capacity Region CR Boundary ρ =0 ρ =1/3 ρ =2/3 Following expansion holds for cardinality of first codebook R2 (nats/use) R1 (nats/use) (R 1, R 2 ) log M 1,n ni 1 (0) + nv 1 (0)Φ 1 (ε) Far from sum rate constraint 1 lim n n log(m 1,nM 2,n ) < I 12 (0) Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

32 The Main Result: Curved Boundary Radically different behavior in the curved region { [ ] L(ε; R 1, R L 2) = (L 1, L 2 ) : 1 } βd(ρ) + Ψ 1 (V(ρ), ε) L 1 + L 2 β R The Capacity Region CR Boundary ρ =0 ρ =1/3 ρ =2/3 R2 (nats/use) (R 1, R 2 ) R1 (nats/use) Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

33 The Main Result: Curved Boundary Radically different behavior in the curved region { [ ] L(ε; R 1, R L 2) = (L 1, L 2 ) : 1 } βd(ρ) + Ψ 1 (V(ρ), ε) L 1 + L 2 β R The Capacity Region CR Boundary ρ =0 ρ =1/3 ρ =2/3 D(ρ) doesn t appear for SW R2 (nats/use) (R 1, R 2 ) R1 (nats/use) Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

34 The Main Result: Curved Boundary Radically different behavior in the curved region { [ ] L(ε; R 1, R L 2) = (L 1, L 2 ) : 1 } βd(ρ) + Ψ 1 (V(ρ), ε) L 1 + L 2 β R R2 (nats/use) The Capacity Region CR Boundary ρ =0 ρ =1/3 ρ =2/3 (R 1, R 2 ) D(ρ) doesn t appear for SW Ψ 1 (V(ρ), ε): corresponds to only using N (0, Σ(ρ)) where [ S Σ(ρ) = 1 ρ ] S 1 S 2 ρ S 1 S 2 S R1 (nats/use) Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

35 The Main Result: Curved Boundary Radically different behavior in the curved region { [ ] L(ε; R 1, R L 2) = (L 1, L 2 ) : 1 } βd(ρ) + Ψ 1 (V(ρ), ε) L 1 + L 2 β R R2 (nats/use) The Capacity Region CR Boundary ρ =0 ρ =1/3 ρ =2/3 (R 1, R 2 ) D(ρ) doesn t appear for SW Ψ 1 (V(ρ), ε): corresponds to only using N (0, Σ(ρ)) where [ S Σ(ρ) = 1 ρ ] S 1 S 2 ρ S 1 S 2 S 2 Non-empty regions in CR not in trapezium achievable by N (0, Σ(ρ)) R1 (nats/use) Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

36 The Main Result: Curved Boundary Radically different behavior in the curved region { [ ] L(ε; R 1, R L 2) = (L 1, L 2 ) : 1 } βd(ρ) + Ψ 1 (V(ρ), ε) L 1 + L 2 β R R2 (nats/use) The Capacity Region CR Boundary ρ =0 ρ =1/3 ρ =2/3 (R 1, R 2 ) R1 (nats/use) D(ρ) doesn t appear for SW Ψ 1 (V(ρ), ε): corresponds to only using N (0, Σ(ρ)) where [ S Σ(ρ) = 1 ρ ] S 1 S 2 ρ S 1 S 2 S 2 Non-empty regions in CR not in trapezium achievable by N (0, Σ(ρ)) Use N (0, Σ(ρ n )) and ρ n = ρ + β/ n Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

37 Illustration of Second-Order Coding Rates Ψ 1 (V,ε) L(ε;R 1,R 2) L2(nats/ use) S 1 = S 2 = 1 and ρ = L 1 (nats/ use) Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

38 Illustration of Second-Order Coding Rates Ψ 1 (V,ε) L(ε;R 1,R 2) L2(nats/ use) S 1 = S 2 = 1 and ρ = L 1 (nats/ use) Second-order rates achieved using a single input distribution N (0, Σ(ρ)) is not optimal Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

39 Illustration of Second-Order Coding Rates Ψ 1 (V,ε) L(ε;R 1,R 2) L2(nats/ use) S 1 = S 2 = 1 and ρ = L 1 (nats/ use) Second-order rates achieved using a single input distribution N (0, Σ(ρ)) is not optimal The optimal second-order coding rate region is a half-space Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

40 Main Ideas in Converse Proof: Part I By a standard n n + 1 argument [Shannon (1959)], we may consider codes with equal power constraints x j 2 2 = ns j, j = 1, 2. Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

41 Main Ideas in Converse Proof: Part I By a standard n n + 1 argument [Shannon (1959)], we may consider codes with equal power constraints x j 2 2 = ns j, j = 1, 2. Reduction from average to maximal error probability via expurgation of polynomially many codeword pairs Not possible for standard MAC [Dueck (1978)] Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

42 Main Ideas in Converse Proof: Part I By a standard n n + 1 argument [Shannon (1959)], we may consider codes with equal power constraints x j 2 2 = ns j, j = 1, 2. Reduction from average to maximal error probability via expurgation of polynomially many codeword pairs Not possible for standard MAC [Dueck (1978)] Reduction to constant correlation type classes { x 1, x 2 ( k 1 T n (k) = (x 1, x 2 ) : x 1 2 x 2 2 n, k ] }, k = 1, 2,..., n. n Without too much loss in rate Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

43 Main Ideas in Converse Proof: Part II Verdú-Han-type converse: For any γ > 0 and any (Q Y X2, Q Y ), have the following non-asymptotic converse bound ( ε n Pr j 1 (X 1, X 2, Y) 1 n log M 1,n γ or j 12 (X 1, X 2, Y) 1 ) n log(m 1,nM 2,n ) γ 2e nγ where j 1 (x 1, x 2, y)= 1 n log Wn (y x 1,x 2 ) Q Y X2 (y x 2 ) and j 12(x 1, x 2, y)= 1 n log Wn (y x 1,x 2 ) Q Y (y). Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

44 Main Ideas in Converse Proof: Part II Verdú-Han-type converse: For any γ > 0 and any (Q Y X2, Q Y ), have the following non-asymptotic converse bound ( ε n Pr j 1 (X 1, X 2, Y) 1 n log M 1,n γ or j 12 (X 1, X 2, Y) 1 ) n log(m 1,nM 2,n ) γ 2e nγ where j 1 (x 1, x 2, y)= 1 n log Wn (y x 1,x 2 ) Q Y X2 (y x 2 ) and j 12(x 1, x 2, y)= 1 n log Wn (y x 1,x 2 ) Q Y (y). Let j = [j 1, j 12 ] T. Choose (Q Y X2, Q Y ) and for (x 1, x 2 ) T n (k), E [ j(x 1, x 2, Y) ] I(ρ) and Cov [ n j(x 1, x 2, Y) ] V(ρ) where k 1 n ρ k n Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

45 Main Ideas in Converse Proof: Part III By evaluating Verdú-Han using multivariate Berry-Esseen, there exists a sequence {ρ n } n 1 [0, 1] satisfying ( ) 1 ρ n = ρ + O n such that [ 1 log M 1,n ] n log(m 1,n M 2,n ) I(ρ n ) + Ψ 1 (V(ρ n ), ε) n ( ) 1 + o n 1 Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

46 Main Ideas in Converse Proof: Part III By evaluating Verdú-Han using multivariate Berry-Esseen, there exists a sequence {ρ n } n 1 [0, 1] satisfying ( ) 1 ρ n = ρ + O n such that [ 1 log M 1,n ] n log(m 1,n M 2,n ) I(ρ n ) + Ψ 1 (V(ρ n ), ε) n ( ) 1 + o n 1 By a Taylor expansion of I(ρ n ) around I(ρ) I(ρ n ) I(ρ) + (ρ n ρ)d(ρ) and the Bolzano-Weierstrass theorem, we establish the converse for a point on the curved boundary Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

47 Conclusion and Perspectives Why is this possible vis-à-vis second-order analysis for DM-MAC? Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

48 Conclusion and Perspectives Why is this possible vis-à-vis second-order analysis for DM-MAC? Gaussianity: Allows us to identify the boundary of the CR and time-sharing not necessary Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

49 Conclusion and Perspectives Why is this possible vis-à-vis second-order analysis for DM-MAC? Gaussianity: Allows us to identify the boundary of the CR and time-sharing not necessary Degraded Message Sets: Codeword pairs do not have to be independent for the strong converse Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

50 Conclusion and Perspectives Why is this possible vis-à-vis second-order analysis for DM-MAC? Gaussianity: Allows us to identify the boundary of the CR and time-sharing not necessary Degraded Message Sets: Codeword pairs do not have to be independent for the strong converse Key distinction relative to existing second-order works: D(ρ) = ρ I(ρ) Gaussian broadcast channel is similar to this problem Techniques may carry over Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

51 Conclusion and Perspectives Why is this possible vis-à-vis second-order analysis for DM-MAC? Gaussianity: Allows us to identify the boundary of the CR and time-sharing not necessary Degraded Message Sets: Codeword pairs do not have to be independent for the strong converse Key distinction relative to existing second-order works: D(ρ) = ρ I(ρ) Gaussian broadcast channel is similar to this problem Techniques may carry over Full version: Vincent Tan (ECE-NUS) Gaussian MAC with DMS ITA / 17

Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities

Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities Vincent Y. F. Tan Dept. of ECE and Dept. of Mathematics National University of Singapore (NUS) September 2014 Vincent Tan