Balancing forward and feedback error correctionsahai/Presentations/Nov6UCB.pdf · In extending the...

Balancing forward and feedback error correction

Anant Sahaibased in part on joint work with:

Stark Draper, Tunc Simsek, and Paul Liu

[email protected] Foundations

Department of Electrical Engineering and Computer SciencesUniversity of California at Berkeley

Major Support from NSF ITR

November 6th 2006

Anant Sahai (UC Berkeley) Limited Feedback Nov 6, 2006 1 / 44

Information theory studies:

Goals: high rate, low delay, and low probability of error.



Feedback

Goals: high rate, low delay, and low probability of error.What if there is feedback?



Feedback


◮ If noise is memoryless, capacity is unchanged.◮ What about delay and probability of error?



Feedback


◮ If noise is memoryless, capacity is unchanged.◮ What about delay and probability of error?◮ How should we use it?



Feedback


◮ If noise is memoryless, capacity is unchanged.◮ What about delay and probability of error?◮ How should we use it?◮ How much feedback do we need?


Flashback and Warning: 1973 (Bob Lucky)

“Feedback communications was an area of intense activity in1968. . . A number of authors had shown constructive, even simple,schemes using noiseless feedback to achieve Shannon-like behav-ior. . . The situation in 1973 is dramatically different. . .The subjectitself seems to be a burned out case. . .

In extending the simple noiseless feedback model to allow formore realistic situations, such as noisy feedback channels, bandlim-ited channels, and peak power constraints, theorists discovered acertain“brittleness” or sensitivity in their previous results. . . ”


Flashback and Warning: 1973 (Bob Lucky)

“Feedback communications was an area of intense activity in1968. . . A number of authors had shown constructive, even simple,schemes using noiseless feedback to achieve Shannon-like behav-ior. . . The situation in 1973 is dramatically different. . .The subjectitself seems to be a burned out case. . .

In extending the simple noiseless feedback model to allow formore realistic situations, such as noisy feedback channels, bandlim-ited channels, and peak power constraints, theorists discovered acertain“brittleness” or sensitivity in their previous results. . . ”

Our goal: show“robustness” of gains due to feedback.


Outline

1 Problem and Background2 Erasure channels: packet-wise hard deadlines with limited feedback3 BSC: bitwise soft deadlines with truly noisy feedback


Communication with a latency requirement

-AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA

-

? ? ? ? ? ? ? ? ?

? ? ?? ? ? ? ? ? ? ? ? ?

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 Y13 Y14 Y15 Y16 Y17 Y18 Y19 Y20 Y21 Y22 Y23 Y24 Y25 Y26

Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z13 Z14 Z15 Z16 Z17 Z18 Z19 Z20 Z21 Z22 Z23 Z24 Z25 Z26

fixed delayd = 7

bB6 bB8bB1 bB2 bB3 bB4 bB5 bB7 bB9

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13

Bits/packets arrive steadily at rateR per channel use.End-to-end latency requirement ofd:




-

? ? ? ? ? ? ? ? ?

? ? ?? ? ? ? ? ? ? ? ? ?



fixed delayd = 7


B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13


◮ Hard : Declared or undetected errors are equally bad.




-

? ? ? ? ? ? ? ? ?

? ? ?? ? ? ? ? ? ? ? ? ?



fixed delayd = 7


B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13


◮ Hard : Declared or undetected errors are equally bad.◮ Soft: Undetected errors are very bad, but declared errors shouldbe

infrequent.


Review: Fixed block-codes for hard deadlinesBuffer-upnRbits and use a block-codeMust decode after a furthern channel uses


Review: Fixed block-codes for hard deadlinesBuffer-upnRbits and use a block-codeMust decode after a furthern channel usesStudy the probability of block error in the limit of largenBlock error exponents:Pe ∝ exp(−nE(R))



◮ Generic channels: (Haroutunian-77)

E+(R) = infG:C(G)<R

max~q

D (G||P|~q)

◮ Holds with or without feedback



◮ “Sphere-packing” bound:

Esp(R) = max~q

minG:I(~q,G)≤R

D (G||P|~q)

= supρ≥0

E0(ρ) − ρR

E0(ρ) = max~q

− ln∑

z

[

∑

y

qyp1

1+ρ

z|y

](1+ρ)

◮ Holds without feedback

◮ Same asE+(R) for symmetric channels!(Feedback useless)Anant Sahai (UC Berkeley) Limited Feedback Nov 6, 2006 7 / 44

Review: Fixed blocks

x

x

x

x x

x

x x

x

Hard decision regionscover space

Feedback is pointless


Review: Fixed blocks, Soft deadlines

x

x

x

x x

x

x x

x��

��

��

��

��

��

��

��

��

Hard decisions regions butcheck hash signatures



x

x

x

x x

x

x x

x��

��

��

��

��

��

��

��

��


1 bit feedback can requestretransmissions

Can interpret as expectedblock-length



x

x

x

x x

x

x x

x��

��

��

��

��

��

��

��

��




Run close to capacity

Use≈ n(C− R) bits forsignatures



x

x

x

x x

x

x x

x��

��

��

��

��

��

��

��

��




Run close to capacity

Use≈ n(C− R) bits forsignatures

Linear slope for error exponent


Review: Fixed blocks, Soft deadlines: Forney-68

x

x

x

x x

x

x x

x

Refuse to decide whenambiguous





x

x

x

x x

x

x x

x




Decision regions catch the typicalsets only



x

x

x

x x

x

x x

x




Decision regions catch the typicalsets only

Better error exponents at lowerrates


Review: Fixed blocks, Soft deadlines: Burnashev-76

Is more feedback helpful?Burnashev said yes:




◮ Considered expected stopping time and used Martingale arguments.




◮ Considered expected stopping time and used Martingale arguments.◮ ShowedC1(1− R

C) was a bound whereC1 = maxi,j D(pi ||pj)




◮ Considered expected stopping time and used Martingale arguments.◮ ShowedC1(1− R

C) was a bound whereC1 = maxi,j D(pi ||pj)

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt

average communication rate (bits/channel use)

Burnashev exponentForney exponentSphere packing exponentSphere−packing bound

Random signature performance

Forney performance

Burnashev bound


Review: Yamamoto-Itoh-79 strategy attains theBurnashev bound

Ack/Nak

( 1 − λ )

Enc Dec

λ n

Decision feedback = m

.....retransmissions

possible .....

Block length n

Data Transmission

n

Data Transmission:λn channel uses for block code atR≃ C



Ack/Nak

( 1 − λ )

Enc Dec

λ n



possible .....

Block length n

Data Transmission

n


Decision Feedback:m sent back



Ack/Nak

( 1 − λ )

Enc Dec

λ n



possible .....

Block length n

Data Transmission

n



Confirm/Deny: (1− λ)n channel uses toACK or NAK



Ack/Nak

( 1 − λ )

Enc Dec

λ n



possible .....

Block length n

Data Transmission

n




If confirmed, decode tomotherwise erase.



Ack/Nak

( 1 − λ )

Enc Dec

λ n



possible .....

Block length n

Data Transmission

n





Pr[err]=Pr[NAK→ACK]=2−(1−λ)nC1 ≃2−nC1(1−RC )



Ack/Nak

( 1 − λ )

Enc Dec

λ n



possible .....

Block length n

Data Transmission

n





Pr[err]=Pr[NAK→ACK]=2−(1−λ)nC1 ≃2−nC1(1−RC )

Moral: Collective reward/punishment is good for reliability


Outline

1 Problem and Background2 Erasure channels: packet-wise hard deadlines with limited feedback

◮ Mildly delayed, but high-rate and reliable, feedback◮ Low-rate, but reliable, feedback◮ Unreliable feedback

3 BSC: bitwise soft deadlines with truly noisy feedback


Nonblock coding for BEC with perfect feedback

f

f

f

f

f

PPPPPPP

��

1 − δ

1 − δ

δ

δ

1

0

1

0

e

Simple capacity 1− δ bitsper channel use

With perfect non-delayedfeedback, simple toachieve: retransmit until itgets through


Nonblock coding for BEC with perfect feedback

f

f

f

f

f

PPPPPPP

��

1 − δ

1 − δ

δ

δ

1

0

1

0

e

Simple capacity 1− δ bitsper channel use

With perfect non-delayedfeedback, simple toachieve: retransmit until itgets through

Hard deadlines bounds:

0

0.2

0.4

0.6

0.8

1

1.2

0.1 0.2 0.3 0.4 0.5 0.6Rate (in bits)

Err

or

Exp

on

en

t (b

ase

2)

Without Feedback:Sphere-packing boundD(1− R||δ)

With Feedback:Focusing bound(E0(ρ)

ρ, E0(ρ))


k-delayed feedback packet erasure case

First approach: treat as k parallel unit-delay channels


k-delayed feedback packet erasure case

First approach: treat as k parallel unit-delay channels

0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4 0.5 0.6Rate (in bits)

Err

or E

xpon

ent (

base

e)

Serious penalty to waitingk steps between retransmissions


The traditional approach: block-coding

Group bits into packets with lengthnR′

◮ Transmit using a rateR′ > R random block-code of lengthn ≫ k




◮ Transmit using a rateR′ > R random block-code of lengthn ≫ k◮ Use feedback to ACK/NAK only (low rate)◮ Retransmit block if unsuccessful





Effective block-lengthn + k ≈ n





Effective block-lengthn + k ≈ nPerformance

◮ Er(R′) < E0(ρ) governs probability of block retransmission





Effective block-lengthn + k ≈ nPerformance

◮ Er(R′) < E0(ρ) governs probability of block retransmission◮ Bad, even without accounting for queuing delay


Low-rate (“bandlimited”) feedback picture

�

-Forward BEC uses

S1 S2 S3 S4 S5 S6

Rate1c noiseless feedback channel uses

c is part of the problem, not tied to the block-length.


Low-rate (“bandlimited”) feedback picture

�

-Forward BEC uses

S1 S2 S3 S4 S5 S6

Rate1c noiseless feedback channel uses

c is part of the problem, not tied to the block-length.

Assume target latencyd ≫ c

Assume round-trip timek ≪ c


Incremental Redundancy Hybrid ARQ

Gentler retransmission:



Gentler retransmission:1 Group bits into blocks of sizenR. (c ≪ n ≪ d)



Gentler retransmission:1 Group bits into blocks of sizenR. (c ≪ n ≪ d)2 Hold blocks in a FIFO queue



Gentler retransmission:1 Group bits into blocks of sizenR. (c ≪ n ≪ d)2 Hold blocks in a FIFO queue3 Service using an∞-length random codebook. (rateless code)



Gentler retransmission:1 Group bits into blocks of sizenR. (c ≪ n ≪ d)2 Hold blocks in a FIFO queue3 Service using an∞-length random codebook. (rateless code)4 Feedback “ACK” when we can decode.




Three delays:




Three delays:◮ Assembly:n insignificant relative tod




Three delays:◮ Assembly:n insignificant relative tod◮ Queuing: Wait before servicing starts




Three delays:◮ Assembly:n insignificant relative tod◮ Queuing: Wait before servicing starts◮ Transmission: Service-time distribution


Transmission-delay: operational interpretation of E0(ρ)

Block transmission time T can be bounded by a constant plus a geometricrandom variable.

0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4Rate (in nats)

Err

or E

xpon

ent (

base

e)

Rρ

ρ

ρ=3

E ( )0

Need to do list-decoding at low rates.Anant Sahai (UC Berkeley) Limited Feedback Nov 6, 2006 19 / 44

How to pick ρ for bounding

PickR < Rρ < C and aim forE+a (Rρ) = E0(ρ) exponent.

0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4

Rate (in nats)

Err

or E

xpon

ent (

base

e)

Target

R Rρ

ρE( )0

ρ−


Queuing delay: the point message view

Consider the queue at the level of blocks:◮ Arrive everyn channel uses



Consider the queue at the level of blocks:◮ Arrive everyn channel uses◮ When serviced, immediately consume a constantn R

Rρchannel uses




Rρchannel uses

◮ After that, geometric service-time withδρ = exp(−E0(ρ))




Rρchannel uses


Delay> d implies dn messages waiting in the queue




Rρchannel uses



Suppose the queue last renewedr messages ago◮ rn channel uses since then




Rρchannel uses



Suppose the queue last renewedr messages ago◮ rn channel uses since then◮ At mostq = r − d

n blocks serviced




Rρchannel uses




n blocks serviced◮ So removeqn R

Rρchannel uses




Rρchannel uses





Rρchannel uses

◮ Leavingd + qn(1− RRρ

) “slack” uses of which onlyq were successful.




Rρchannel uses





Rρchannel uses

◮ Leavingd + qn(1− RRρ

) “slack” uses of which onlyq were successful.

This is exactly like a bit-rate 1n(1− R

Rρ)

code on a perfect-feedback BEC

with erasure probabilityδρ.


Queuing delay: reduction to the low-rate erasure casePickR < Rρ < C and aim forE+

a (Rρ) = E0(ρ) exponent.

0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4Rate (in nats)

Err

or E

xpon

ent (

base

e)

Target

R Rρ

ρE( )0

ρ−

If n large, effective point-message rate(n(1− RRρ

))−1 is small.


Queuing delay: reduction to the low-rate erasure casePickR < Rρ < C and aim forE+

a (Rρ) = E0(ρ) exponent.

0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4Rate (in nats)

Err

or E

xpon

ent (

base

e)

Target

R Rρ

ρE( )0

ρ−

If n large, effective point-message rate(n(1− RRρ

))−1 is small.Erasure focusing bound is approximately flat at low rates so queuing delayexponent≈ − ln δρ = E0(ρ).


Performance

0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4 0.5 0.6Rate (in bits)

Err

or E

xpon

ent (

base

e)

List size 1 limit

List size 4 limit

List size 2 limit


Performance

0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4 0.5 0.6Rate (in bits)

Err

or E

xpon

ent (

base

e)

List size 1 limit

List size 4 limit

List size 2 limit

What is the price to get list-decoding gains?


How to use list-decoding

At low rates, forward error correction suffers from ambiguity.



At low rates, forward error correction suffers from ambiguity.1 Group bits into blocks of sizenR. (k ≪ n ≪ d)2 Hold blocks in a FIFO queue3 Transmit using an∞-length random codebook. (rateless code)



At low rates, forward error correction suffers from ambiguity.1 Group bits into blocks of sizenR. (k ≪ n ≪ d)2 Hold blocks in a FIFO queue3 Transmit using an∞-length random codebook. (rateless code)4 Feedback “ACK” when we canalmostdecode.



At low rates, forward error correction suffers from ambiguity.1 Group bits into blocks of sizenR. (k ≪ n ≪ d)2 Hold blocks in a FIFO queue3 Transmit using an∞-length random codebook. (rateless code)4 Feedback “ACK” when we canalmostdecode.5 Send back anO((L − 1) log(nR)) mini-message requesting clarification of

specific bits within the block




specific bits within the block6 Use repetition codes to clarify




specific bits within the block6 Use repetition codes to clarify7 ACK the clarifications





A block corresponds toL point messages — low rate relative toO(n)slack.





A block corresponds toL point messages — low rate relative toO(n)slack.

Approaches the focusing bound.


Unreliable feedback picture

�

-Forward packet-erasure channelsδ

S1 S2 S3 S4 S5 S6

Rate1c packet-erasure feedback channelsδf

Bothchannels drop packets randomly



�


S1 S2 S3 S4 S5 S6



Assume packets of moderate size that can have “header bits”



�


S1 S2 S3 S4 S5 S6



Assume packets of moderate size that can have “header bits”

Assume target latencyd ≫ c, k


How to deal with unreliable feedback

Basic 4-Part Strategy:◮ Incremental transmission of message block◮ Feedbacka request for disambiguation message◮ Repetition-code the disambiguation◮ Feedbacka final ACK




Add one header bit to forward packets◮ 0: main message packet◮ 1: first disambiguation message packet◮ 0: second disambiguation message packet

...





...

Repetition-code thefeedback messages





...


◮ Geometric service time as(δ1cf )d





...


◮ Geometric service time as(δ1cf )d

Each message now acts likeO(2 + (L − 1) log(nR)) point messages.◮ Low rate relative toO(n) slack◮ As long as−1

c logδf > E0(ρ) target exponent, no loss in achievedend-to-end delay exponent!


Performance with unreliable feedback channel

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Del

ay E

rror

Exp

onen

t (ba

se e

)

Rate (in packets per slot)

Balanced forward+feedback strategy

Forward only strategy

Rat

e lim

it w

ith ro

und−

trip

qual

ity

Cap

acity

Lim

it du

e to

forw

ard

qual

ity

Naive feedback strategywith free feedback


Performance with unreliable feedback channel

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Del

ay E

rror

Exp

onen

t (ba

se e

)


Balanced forward+feedback strategy


Limited with free 1/3 rate feedback


Shared channel picture

�

-c−1

c forward packet-erasure channelsδ

S1 S2 S3 S4 S5 S6

Rate1c packet-erasure feedback channelsδ

Bothusers share a single physical channel

Half-duplex constraint: only one can use at a time

Assume we must schedule them in advance

Same erasure probability in both directions


Shared unreliable channel used for feedback

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Del

ay E

rror

Exp

onen

t (ba

se e

)


If feedback were free

Pure feedbackstrategy


1:1 split



0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Del

ay E

rror

Exp

onen

t (ba

se e

)



feedback and forward


2:1


Dividing time slots between



0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Del

ay E

rror

Exp

onen

t (ba

se e

)



Dividing time slots betweenfeedback and forward


2:1

4:1

8:1



Final comments on packet erasure channels

Perfect feedback performance as long as−1c logδf is high enough.




It is worth allocating slots for feedbackeven if this means taking themaway from the forward channel.





The code is “anytime” in that it is delay universal — application can pickwhat latency is needed.

The code is universal over erasure probabilities.







The code is mildly “broadcast-friendly.”







The code is mildly “broadcast-friendly.”

Computation depends only onn, not on delay.


Outline

1 Problem and Background2 Erasure channels: packet-wise hard deadlines with limited feedback3 BSC: bitwise soft deadlines with truly noisy feedback

◮ The opportunity◮ Noiseless feedback◮ Optimality since matches “Hallucination bound”◮ Noisy feedback


A streaming data perspective on soft deadlines

end−to−end delay

TxPacketizer Queue Rx

App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream





App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream

Queue is optional. Needed if retransmissions are required





App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream

Queue is optional. Needed if retransmissions are required

If retransmissions are rare, thenexpectedend-to-end delay is dominatedby the Tx to Rx delay.


An opportunity presents itself

Data Blocks Nak

Acks

decodingerror

Erasures are rare so most messages are confirmed.



Data Blocks Nak

Acks

decodingerror


We are wasting channel uses.



Data Blocks Nak

Acks

decodingerror



What if we only sentNAKs when needed?



Data Blocks Nak

Acks

decodingerror



What if we only sentNAKs when needed?

Have a special message for this purpose.


Sliding blocks with collective punishment only(Kudryashov-79)

decision delaydecision delayFinal decision on blue msg

(unless detect Nak)Final decision on purple msg

(unless detect Nak)

Emergency msg,fixed length = decision delayData Blocks

Make packet sizen much smaller than soft deadlined.





(unless detect Nak)



A NAK collectively denies the pastdn − 1 packets





(unless detect Nak)



A NAK collectively denies the pastdn − 1 packets

Error only if dn − 1 NAKs are all missed


Unequal error protection required in codebook


Unequal error protection required in codebook

Typical noise sphere

Hypothesistestingthreshold:

or is it data?Is it a Nak

Unequal Error Protection


Specialize to BSC case

N

N0

Data Blocks

N N N

∆Nak for blocks = ∆

y

composition

q

outputcomposition

codewordBSC (p)

p

q

Gap

q (1−p) + (1−q) p0.5

1

0

Use all zero forNAK



N

N0

Data Blocks

N N N


y

composition

q

outputcomposition

codewordBSC (p)

p

q

Gap

q (1−p) + (1−q) p0.5

1

0

Use all zero forNAK

Use compositionq codefor data:R < H(q) − H(p)



N

N0

Data Blocks

N N N


y

composition

q

outputcomposition

codewordBSC (p)

p

q

Gap

q (1−p) + (1−q) p0.5

1

0

Use all zero forNAK


Probability of missedNAKis 2−ND(qy||p)



N

N0

Data Blocks

N N N


y

composition

q

outputcomposition

codewordBSC (p)

p

q

Gap

q (1−p) + (1−q) p0.5

1

0

Use all zero forNAK


Probability of missedNAKis 2−ND(qy||p)

Get∆ chances:2−∆ND(qy||p)


Resulting exponents

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt


Delay exponentBurnashev exponentForney exponentSphere packing exponent


Resulting exponents

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt



Positive error exponentD(12||p) even near capacity!


Resulting exponents

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt



Positive error exponentD(12||p) even near capacity!

Also far better than the focusing bound.


Matches the “Hallucination Bound”

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt



No converse for Forney,unlike Burnashev andSphere-packing.



0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt




What about here?



0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt




What about here?

Decoding correctlyrequires a certain “typical”volume of outputsequences



0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt




What about here?


If the channel forces you tohallucinate fordtime-steps, you aredoomed.



0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt




What about here?



With feedback, you canchoose the channel input tominimize this probability.



0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt




What about here?



With feedback, you canchoose the channel input tominimize this probability.

Matches achievability.


Noisy feedback: two distinct issues



App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream





App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream

Retransmission control: maintaining synchronization





App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream

Retransmission control: maintaining synchronization◮ Can model the state of the receiver as a random walk with drift.





App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream

Retransmission control: maintaining synchronization◮ Can model the state of the receiver as a random walk with drift.◮ Unstable process, but it can be tracked using ananytimecode.





App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream

Retransmission control: maintaining synchronization◮ Can model the state of the receiver as a random walk with drift.◮ Unstable process, but it can be tracked using ananytimecode.◮ Since retransmissions are rare, can afford extra latency toallow state

estimates to converge.





App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream


estimates to converge.◮ Synchronization rate is less than 1 bit per packet.





App

licat

ion

feedbackchannel

channel

App

licat

ion

data

stream


estimates to converge.◮ Synchronization rate is less than 1 bit per packet.

How toNAK?


What is needed to NAK?

N

misseddetection

detectederror

N0

Forward

Reverse

Sliding window binning latency constraint (playback deadline)

Data Blocks ∆∆Nak for blocks =

∆

��

��

��

��

��

��

Must know if any errors in the sliding window.



N

misseddetection

detectederror

N0

Forward

Reverse



∆

��

��

��

��

��

��


Identification problem since encoder knows what it wants to hear.



N

misseddetection

detectederror

N0

Forward

Reverse



∆

��

��

��

��

��

��



Use a random hash of entire sliding window.



N

misseddetection

detectederror

N0

Forward

Reverse



∆




Compare Pr(hash collision) with Pr(missed NAK)



N

misseddetection

detectederror

N0

Forward

Reverse



∆




Compare Pr(hash collision) with Pr(missed NAK)

If Cfb > D(qy||p), no loss in net exponent!


Feedback capacity acts as a ceiling to reliability

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt


Delay exp w/ noisy FBBurnashev exponentForney exponentSphere packing exp

Hash collision dominates

Missed Nak dominates

Feedback reliability gains are robust to noisy feedback.


Feedback capacity acts as a ceiling to reliability

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

erro

r ex

pone

nt


Delay exp w/ noisy FBBurnashev exponentForney exponentSphere packing exp

Hash collision dominates

Missed Nak dominates

Feedback reliability gains are robust to noisy feedback.Open problem: does this also hold in the general hard deadline case?


Summary of reliability for symmetric channels

Hard Deadlines Soft Deadlines(undetected errors only)Bound Achievable Robust Bound Achievable Robust

Block Sphere-packing Yes trivial Burnashev Yes Mild*No FB Sphere-packing Yes trivial Telatar Forney Yes*

Delay Focusing* Partial* Partial* Hallucination* Yes Yes*No FB Sphere-packing Yes trivial Unknown

* entriesare our contributions withbold for those in this talk.

Trivial robustness is when feedback is not used at all.Partial achievability of the focusing bound is:

◮ Tight for erasure channels and all DMCs withC0,f > 0◮ Robust for packet erasure channels with erasure feedback◮ Better than sphere-packing bound for essentially all channels









Forasymmetricchannels, Haroutunian vs sphere-packing style gaps existbetween bounds and achievable codes everywhere exceptBlock, No-FB.









Forasymmetricchannels, Haroutunian vs sphere-packing style gaps existbetween bounds and achievable codes everywhere exceptBlock, No-FB.

Ultimately, there should be a continuum between perfect feedback andno feedback, as well as between hard and soft deadlines.


Balancing forward and feedback error correctionsahai/Presentations/Nov6UCB.pdf · In extending the...

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