Finite-Blocklength Performance of Sequential Transmission over BSC - - PowerPoint PPT Presentation

Finite-Blocklength Performance of Sequential Transmission

• ver BSC with Noiseless Feedback

Hengjie Yang and Richard D. Wesel UCLA 2020 IEEE International Symposium on Information Theory (ISIT) Los Angeles, CA, USA

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 1 / 17

Outline

1

Introduction

2

Our contributions New non-asymptotic upper bound on average blocklength Markovian analysis Comparison of results

3

Summary

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 2 / 17

Motivation Stop feedback vs. Full feedback:

Encoder DMC Decoder θ Xt Yt ACK/NACK ˆ θ

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 3 / 17

Motivation Stop feedback vs. Full feedback:

Encoder DMC Decoder θ Xt Yt Y t−1 ˆ θ

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 3 / 17

Motivation Stop feedback vs. Full feedback:

Encoder DMC Decoder θ Xt Yt Y t−1 ˆ θ

Two important literatures:

• Polyanskiy et al. showed that the stop-feedback code can obtain a much higher

achievable rate than fixed-length code without feedback.

• Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,” IEEE Trans. Inf. Theory,
• vol. 56, no. 5, pp. 2307-2359, May 2010.
• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 3 / 17

Motivation Stop feedback vs. Full feedback:

Encoder DMC Decoder θ Xt Yt Y t−1 ˆ θ

Two important literatures:

• Polyanskiy et al. showed that the stop-feedback code can obtain a much higher

achievable rate than fixed-length code without feedback.

• Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,” IEEE Trans. Inf. Theory,
• vol. 56, no. 5, pp. 2307-2359, May 2010.
• Naghshvar et al. proposed a novel scheme known as small-enough-difference (SED)

encoder to attain the capacity and optimal error exponent of the BSC.

• M. Naghshvar, T. Javidi, and M. Wigger, Extrinsic Jensen-Shannon divergence: Applications to variable-length coding,

IEEE Trans. Inf. Theory, vol. 61, no. 4, pp. 2148-2164, April 2015.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 3 / 17

Motivation Stop feedback vs. Full feedback:

Encoder DMC Decoder θ Xt Yt Y t−1 ˆ θ

Two important literatures:

• Polyanskiy et al. showed that the stop-feedback code can obtain a much higher

achievable rate than fixed-length code without feedback.

• Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,” IEEE Trans. Inf. Theory,
• vol. 56, no. 5, pp. 2307-2359, May 2010.
• Naghshvar et al. proposed a novel scheme known as small-enough-difference (SED)

encoder to attain the capacity and optimal error exponent of the BSC.

• M. Naghshvar, T. Javidi, and M. Wigger, Extrinsic Jensen-Shannon divergence: Applications to variable-length coding,

IEEE Trans. Inf. Theory, vol. 61, no. 4, pp. 2148-2164, April 2015.

Issue: The non-asymptotic upper bound on average blocklength of full-feedback codes by Naghshvar et al. is above that of stop-feedback codes by Polyanskiy.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 3 / 17

Variable-length coding over the BSC

Encoder BSC (p) Decoder θ Xt Yt Y t−1 ˆ θ

System parameters: target prob. of error ǫ

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 4 / 17

Variable-length coding over the BSC

Encoder BSC (p) Decoder θ Xt Yt Y t−1 ˆ θ

System parameters: target prob. of error ǫ An (M, ǫ) variable-length code:

1 Message set Ω = {1, 2, . . . , M},

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 4 / 17

Variable-length coding over the BSC

Encoder BSC (p) Decoder θ Xt Yt Y t−1 ˆ θ

System parameters: target prob. of error ǫ An (M, ǫ) variable-length code:

1 Message set Ω = {1, 2, . . . , M}, 2 Encoding function et : Ω × Yt−1 → X,

Xt = et(θ, Y t−1), t = 1, 2, . . .

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 4 / 17

Variable-length coding over the BSC

Encoder BSC (p) Decoder θ Xt Yt Y t−1 ˆ θ

System parameters: target prob. of error ǫ An (M, ǫ) variable-length code:

1 Message set Ω = {1, 2, . . . , M}, 2 Encoding function et : Ω × Yt−1 → X,

Xt = et(θ, Y t−1), t = 1, 2, . . .

3 Decoding function d : Yτ → Ω,

ˆ θ = d(Y τ) where τ is the random stopping time.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 4 / 17

Variable-length coding over the BSC

Encoder BSC (p) Decoder θ Xt Yt Y t−1 ˆ θ

System parameters: target prob. of error ǫ An (M, ǫ) variable-length code:

1 Message set Ω = {1, 2, . . . , M}, 2 Encoding function et : Ω × Yt−1 → X,

Xt = et(θ, Y t−1), t = 1, 2, . . .

3 Decoding function d : Yτ → Ω,

ˆ θ = d(Y τ) where τ is the random stopping time. Goal: minimizeτ,et(·),d(·) E[τ] subject to Pe Pr{ˆ θ = θ} ≤ ǫ.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 4 / 17

Naghshvar et al.’s scheme Naghshvar et al. proposed the following scheme

• The belief state:

ρ(t) [ρ1(t), ρ2(t), . . . , ρM(t)], t = 1, 2, . . . where ρi(t) Pr{θ = i|Y t}, i ∈ Ω and ρi(0) = 1/M by default.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 5 / 17

Naghshvar et al.’s scheme Naghshvar et al. proposed the following scheme

• The belief state:

ρ(t) [ρ1(t), ρ2(t), . . . , ρM(t)], t = 1, 2, . . . where ρi(t) Pr{θ = i|Y t}, i ∈ Ω and ρi(0) = 1/M by default.

• The SED encoding rule: at time t, upon receiving yt−1 thanks to the noiseless

feedback, update ρ(t − 1) by Bayes rule. Then partition Ω into two subsets S0(t − 1) and S1(t − 1) such that 0 ≤ π0(t − 1) − π1(t − 1) ≤ min

i∈S0(t−1) ρi(t − 1),

where π0(t − 1)

• i∈S0(t−1)

ρi(t − 1), π1(t − 1)

• i∈S1(t−1)

ρi(t − 1).

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 5 / 17

Naghshvar et al.’s scheme Naghshvar et al. proposed the following scheme

• The belief state:

ρ(t) [ρ1(t), ρ2(t), . . . , ρM(t)], t = 1, 2, . . . where ρi(t) Pr{θ = i|Y t}, i ∈ Ω and ρi(0) = 1/M by default.

• The SED encoding rule: at time t, upon receiving yt−1 thanks to the noiseless

feedback, update ρ(t − 1) by Bayes rule. Then partition Ω into two subsets S0(t − 1) and S1(t − 1) such that 0 ≤ π0(t − 1) − π1(t − 1) ≤ min

i∈S0(t−1) ρi(t − 1),

where π0(t − 1)

• i∈S0(t−1)

ρi(t − 1), π1(t − 1)

• i∈S1(t−1)

ρi(t − 1).

• Decoding (or stopping) rule:

τ = min{t : max

i∈Ω ρi(t) ≥ 1 − ǫ}

ˆ θ = arg max

i∈Ω

ρi(τ).

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 5 / 17

Two phases in the variable-length coding problem Communication Phase Confirmation Phase

∃j ∈ Ω, ρj(t) ≥ 1/2 ∀i ∈ Ω, ρi(t) < 1/2

Assume θ = i ∈ Ω henceforth. Communication phase: ∀j ∈ Ω, ρj(t) < 1/2. Confirmation phase: ∃j ∈ Ω, ρj(t) ≥ 1/2.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 6 / 17

Average step size provided by the SED encoder Consider the log-likelihood ratio of the true message θ = i ∈ Ω, Ui(t) log ρi(t) 1 − ρi(t).

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 7 / 17

Average step size provided by the SED encoder Consider the log-likelihood ratio of the true message θ = i ∈ Ω, Ui(t) log ρi(t) 1 − ρi(t).

Lemma (Naghshvar et al., 2012)

With the SED encoding rule over the BSC(p), {Ui(t)}∞

t=1 forms a submartingale w.r.t.

the filtration Ft = σ{Y t}, satisfying E[Ui(t + 1)|Ft] ≥Ui(t) + C, if Ui(t) < 0 E[Ui(t + 1)|Ft] =Ui(t) + C1, if Ui(t) ≥ 0 |Ui(t + 1) − Ui(t)| ≤C2 where C = max

PX I(X; Y ) = 1 − H(p),

C1 = max

x,x′∈X D

• PY |X=xPY |X=x′

= p log p q + q log q p, (q 1 − p) C2 = max

y∈Y log maxx∈X PY |X(y|x)

minx∈X PY |X(y|x) = log q p.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 7 / 17

Previous non-asymptotic results

Theorem (Naghshvar et al., 2015)

Naghshvar et al.’s scheme for symmetric binary-input channels satisfies E[τ] ≤ log M + log log M

ǫ

C + log 1

ǫ + 1

C1 + 96 · 22C2 CC1 .

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 8 / 17

Previous non-asymptotic results

Theorem (Naghshvar et al., 2015)

Naghshvar et al.’s scheme for symmetric binary-input channels satisfies E[τ] ≤ log M + log log M

ǫ

C + log 1

ǫ + 1

C1 + 96 · 22C2 CC1 .

Theorem (Polyanskiy’s VLF bound, Williamson et al., 2015)

For a given ǫ > 0 and a positive integer M, there exists a stop-feedback VLF code for BSC(p), satisfying E[τ] ≤ log(M − 1) C + log 1

ǫ

C + 2(1 − p) C .

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 8 / 17

An illustration of bounds: BSC(0.05) 0.5 1 1.5 2 ·104 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average Blocklength Rate Capacity of BSC(0.05)

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 9 / 17

An illustration of bounds: BSC(0.05) 0.5 1 1.5 2 ·104 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average Blocklength Rate Capacity of BSC(0.05) Naghshvar et al., 2015

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 9 / 17

An illustration of bounds: BSC(0.05) 0.5 1 1.5 2 ·104 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average Blocklength Rate Capacity of BSC(0.05) Naghshvar et al., 2015 VLF lower bound, 2015

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 9 / 17

Outline

1

Introduction

2

Our contributions New non-asymptotic upper bound on average blocklength Markovian analysis Comparison of results

3

Summary

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 9 / 17

Main results

Corollary (A consequence of Naghshvar’s submartingale result)

Appealing to the general submartingale result by Naghshvar et al., the SED encoding scheme for the BSC(p) also satisfies E[τ] ≤ log M C + log 1−ǫ

ǫ

C1 + 3C2

2

CC1 .

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 10 / 17

Main results

Corollary (A consequence of Naghshvar’s submartingale result)

Appealing to the general submartingale result by Naghshvar et al., the SED encoding scheme for the BSC(p) also satisfies E[τ] ≤ log M C + log 1−ǫ

ǫ

C1 + 3C2

2

CC1 .

Theorem

The average blocklength E[τ] of Naghshvar et al.’s scheme for the BSC(p) satisfies E[τ] ≤ log M C + log 1−ǫ

ǫ

C2

• C2

C1 + pC2 C1 C1 + C2 C − C2 C1

• + C1

C .

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 10 / 17

Outline

1

Introduction

2

Our contributions New non-asymptotic upper bound on average blocklength Markovian analysis Comparison of results

3

Summary

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 10 / 17

The genie-aided decoder Genie-aided decoder: τi(ǫ) min{t : ρi(t) ≥ 1 − ǫ}, i ∈ Ω.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 11 / 17

The genie-aided decoder Genie-aided decoder: τi(ǫ) min{t : ρi(t) ≥ 1 − ǫ}, i ∈ Ω. Implication: E[τ] ≤ E[τi(ǫ)|θ = i].

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 11 / 17

The genie-aided decoder Genie-aided decoder: τi(ǫ) min{t : ρi(t) ≥ 1 − ǫ}, i ∈ Ω. Implication: E[τ] ≤ E[τi(ǫ)|θ = i]. Decomposition of E[τi(ǫ)|θ = i]: E[τi(ǫ)|θ = i] = E[τi(1/2)|θ = i]

• communication phase

+ E[τi(ǫ) − τi(1/2)|θ = i]

• confirmation phase

.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 11 / 17

Analysis of E[τi(1/2)|θ = i]

Lemma

With the SED encoder over the BSC(p), the stopping time τi(1/2) of the transmitted message θ = i ∈ Ω satisfies E[τi(1/2)|θ = i] <log M C + C1 C .

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 12 / 17

Analysis of E[τi(1/2)|θ = i]

Lemma

With the SED encoder over the BSC(p), the stopping time τi(1/2) of the transmitted message θ = i ∈ Ω satisfies E[τi(1/2)|θ = i] <log M C + C1 C . Proof sketch:

• First, consider ηt = Ui(t)

C

− t, then {ηt}∞

t=0 is also a submartingale.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 12 / 17

Analysis of E[τi(1/2)|θ = i]

Lemma

With the SED encoder over the BSC(p), the stopping time τi(1/2) of the transmitted message θ = i ∈ Ω satisfies E[τi(1/2)|θ = i] <log M C + C1 C . Proof sketch:

• First, consider ηt = Ui(t)

C

− t, then {ηt}∞

t=0 is also a submartingale.

• Let T τi(1/2). By Doob’s optional stopping theorem,

− log(M − 1) C =E[η0] ≤ E[ηT ] =E[Ui(T) − Ui(T − 1)] + E[Ui(T − 1)] C − E[T] ≤C1 + 0 C − E[T]. Namely, E[T] < log M

C

+ C1

C .

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 12 / 17

Analysis of E[τi(ǫ) − τi(1/2)|θ = i] First, appealing to the tower property, E[τi(ǫ) − τi(1/2)|θ = i] = E

• E[τi(ǫ) − τi(1/2)|θ = i, Ui(τi(1/2)) = u]
• .
• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 13 / 17

Analysis of E[τi(ǫ) − τi(1/2)|θ = i] First, appealing to the tower property, E[τi(ǫ) − τi(1/2)|θ = i] = E

• E[τi(ǫ) − τi(1/2)|θ = i, Ui(τi(1/2)) = u]
• .

Lemma

With the SED encoder over the BSC(p), the true message θ = i ∈ Ω satisfies, for any 0 ≤ u < C2, E[τi(ǫ)−τi(1/2) | θ = i, Ui(τi(1/2)) = u] ≤ log 1−ǫ

ǫ

C2

• C2

C1 + pC2 C1 C1 + C2 C − C2 C1

• .
• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 13 / 17

Analysis of E[τi(ǫ) − τi(1/2)|θ = i] First, appealing to the tower property, E[τi(ǫ) − τi(1/2)|θ = i] = E

• E[τi(ǫ) − τi(1/2)|θ = i, Ui(τi(1/2)) = u]
• .

Lemma

With the SED encoder over the BSC(p), the true message θ = i ∈ Ω satisfies, for any 0 ≤ u < C2, E[τi(ǫ)−τi(1/2) | θ = i, Ui(τi(1/2)) = u] ≤ log 1−ǫ

ǫ

C2

• C2

C1 + pC2 C1 C1 + C2 C − C2 C1

• .

Fact: Ui(t) form a Markov chain in the confirmation phase, Pr{Ui(t + 1) = u + C2 | Ui(t) = u, u ≥ 0} =q, Pr{Ui(t + 1) = u − C2 | Ui(t) = u, u ≥ 0} =p.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 13 / 17

Proof: time of first-passage on the generalized Markov chain

u∗ S0 p u∗ + C2 S1 p q u∗ + 2C2 S2 q p · · · p q u∗+(n − 1)C2 Sn−1 p q u∗ + nC2 Sn q 1

Generalized Markov chain: Sj = [jC2, jC2 + C2), j = 0, 1, . . . , n; each time only one value u∗ + jC2 ∈ Sj remains active, where u∗ ∈ [0, C2), n log 1−ǫ

ǫ

C2

• .
• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 14 / 17

Proof: time of first-passage on the generalized Markov chain

u∗ S0 p u∗ + C2 S1 p q u∗ + 2C2 S2 q p · · · p q u∗+(n − 1)C2 Sn−1 p q u∗ + nC2 Sn q 1

Generalized Markov chain: Sj = [jC2, jC2 + C2), j = 0, 1, . . . , n; each time only one value u∗ + jC2 ∈ Sj remains active, where u∗ ∈ [0, C2), n log 1−ǫ

ǫ

C2

• .

Position-invariant stopping rule: τ ∗

i (ǫ) = min

• t :
• Ui(t)

C2

• ≥ n
• .
• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 14 / 17

Proof: time of first-passage on the generalized Markov chain

u∗ S0 p u∗ + C2 S1 p q u∗ + 2C2 S2 q p · · · p q u∗+(n − 1)C2 Sn−1 p q u∗ + nC2 Sn q 1

Generalized Markov chain: Sj = [jC2, jC2 + C2), j = 0, 1, . . . , n; each time only one value u∗ + jC2 ∈ Sj remains active, where u∗ ∈ [0, C2), n log 1−ǫ

ǫ

C2

• .

Position-invariant stopping rule: τ ∗

i (ǫ) = min

• t :
• Ui(t)

C2

• ≥ n
• .

Lemma 1: τ ≤ τi(ǫ) ≤ τ ∗

i (ǫ).

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 14 / 17

Proof: time of first-passage on the generalized Markov chain

u∗ S0 p u∗ + C2 S1 p q u∗ + 2C2 S2 q p · · · p q u∗+(n − 1)C2 Sn−1 p q u∗ + nC2 Sn q 1

Generalized Markov chain: Sj = [jC2, jC2 + C2), j = 0, 1, . . . , n; each time only one value u∗ + jC2 ∈ Sj remains active, where u∗ ∈ [0, C2), n log 1−ǫ

ǫ

C2

• .

Position-invariant stopping rule: τ ∗

i (ǫ) = min

• t :
• Ui(t)

C2

• ≥ n
• .

Lemma 1: τ ≤ τi(ǫ) ≤ τ ∗

i (ǫ).

Lemma 2 (Expected time of first-passage): E[τi(ǫ)−τi(1/2)|θ = i, Ui(τi(1/2)) = u∗] ≤ E[τ ∗

i (ǫ) − τi(1/2)|θ = i, Ui(τi(1/2)) = u∗]

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 14 / 17

Proof: time of first-passage on the generalized Markov chain

u∗ S0 p u∗ + C2 S1 p q u∗ + 2C2 S2 q p · · · p q u∗+(n − 1)C2 Sn−1 p q u∗ + nC2 Sn q 1

Generalized Markov chain: Sj = [jC2, jC2 + C2), j = 0, 1, . . . , n; each time only one value u∗ + jC2 ∈ Sj remains active, where u∗ ∈ [0, C2), n log 1−ǫ

ǫ

C2

• .

Position-invariant stopping rule: τ ∗

i (ǫ) = min

• t :
• Ui(t)

C2

• ≥ n
• .

Lemma 1: τ ≤ τi(ǫ) ≤ τ ∗

i (ǫ).

Lemma 2 (Expected time of first-passage): E[τi(ǫ)−τi(1/2)|θ = i, Ui(τi(1/2)) = u∗] ≤ E[τ ∗

i (ǫ) − τi(1/2)|θ = i, Ui(τi(1/2)) = u∗]

= n 1 − 2p + p 1 − 2p

• 1 −
• p

1 − p n (∆0 − ∆∗

0)

where ∆0, ∆∗

0 are the actual and ideal self-loop time from S0 to S0, respectively.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 14 / 17

Proof: time of first-passage on the generalized Markov chain

u∗ S0 p u∗ + C2 S1 p q u∗ + 2C2 S2 q p · · · p q u∗+(n − 1)C2 Sn−1 p q u∗ + nC2 Sn q 1

Generalized Markov chain: Sj = [jC2, jC2 + C2), j = 0, 1, . . . , n; each time only one value u∗ + jC2 ∈ Sj remains active, where u∗ ∈ [0, C2), n log 1−ǫ

ǫ

C2

• .

Position-invariant stopping rule: τ ∗

i (ǫ) = min

• t :
• Ui(t)

C2

• ≥ n
• .

Lemma 1: τ ≤ τi(ǫ) ≤ τ ∗

i (ǫ).

Lemma 2 (Expected time of first-passage): E[τi(ǫ)−τi(1/2)|θ = i, Ui(τi(1/2)) = u∗] ≤ E[τ ∗

i (ǫ) − τi(1/2)|θ = i, Ui(τi(1/2)) = u∗]

= n 1 − 2p + p 1 − 2p

• 1 −
• p

1 − p n (∆0 − ∆∗

0)

≤nC2 C1 + pC2 C1 C1 + C2 C − C2 C1

• ,

where ∆0, ∆∗

0 are the actual and ideal self-loop time from S0 to S0, respectively.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 14 / 17

Outline

1

Introduction

2

Our contributions New non-asymptotic upper bound on average blocklength Markovian analysis Comparison of results

3

Summary

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 14 / 17

Comparison of results: BSC(0.05) 100 200 300 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average Blocklength Rate Capacity of BSC(0.05)

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 15 / 17

Comparison of results: BSC(0.05) 100 200 300 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average Blocklength Rate Capacity of BSC(0.05) VLF lower bound, 2015

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 15 / 17

Comparison of results: BSC(0.05) 100 200 300 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average Blocklength Rate Capacity of BSC(0.05) VLF lower bound, 2015 Corollary

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 15 / 17

Comparison of results: BSC(0.05) 100 200 300 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average Blocklength Rate Capacity of BSC(0.05) VLF lower bound, 2015 Corollary New lower bound

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 15 / 17

Comparison of results: BSC(0.05) 100 200 300 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Average Blocklength Rate Capacity of BSC(0.05) VLF lower bound, 2015 Corollary New lower bound Simulation of SED

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 15 / 17

Summary

1 We proved a tightened non-asymptotic upper bound on average blocklength of

full-feedback code by Naghshvar et al.’s scheme.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 16 / 17

Summary

1 We proved a tightened non-asymptotic upper bound on average blocklength of

full-feedback code by Naghshvar et al.’s scheme.

2 Our bound, as desired, is better than that of Polyanskiy which characterizes the

achievability of a stop-feedback code for moderately large crossover prob. p.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 16 / 17

Summary

1 We proved a tightened non-asymptotic upper bound on average blocklength of

full-feedback code by Naghshvar et al.’s scheme.

2 Our bound, as desired, is better than that of Polyanskiy which characterizes the

achievability of a stop-feedback code for moderately large crossover prob. p.

3 Future work is to extend our analysis to general discrete memoryless channels.

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 16 / 17

Thank you!

• H. Yang (UCLA)

ISIT 2020, Los Angeles, CA, USA June 2020 17 / 17