Low-Delay Codes Minimizing the Average Delay Among Lost Packets

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Yuval Cassuto joint work with:

Nitzan Adler

Technion, EE

BIRS 15w5150

[ISIT ’15: Low-Delay Erasure-Correcting Codes

with Optimal Average Delay]

Coding performance classification

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Deterministic Stochastic

worst-case , adversarial expected

average

Do not confuse!

Burst-erasure channel

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packets of

data & parity

symbols

Burst-erasure

B sequential

packets are erased

Low delay codes

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Trading-off information rate, error tolerance and decoding delay.

Decoding delay

Information

rate

Error

tolerance

Classical

Error-correcting

codes

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Decoding delay

The decoding delay is the number of packets from a packet erasure to its reconstruction.

i

B erased packets The packets that participate

in the reconstruction

T

The ‘i’-th packet is

reconstructed

Review: Low-delay codes for burst-erasure channels

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Any code with T (constant), B and R satisfies the lower bound:

B [packets]– the burst length

T [packets]– the constant delay

R – the code rate

• Martinian & Sundberg “Burst erasure correction codes with low decoding delay” (2004)

𝑇 ≥ 𝐵 ∙

𝑅

1 − 𝑅 , 𝑅 ≥

1

2

𝐵 , 𝑅 <1

2

Low delay codes – some previous work

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• Martinian & Trott, “Delay-optimal burst erasure code construction” (2007)

• Li, Khisti, Girod, “Correcting erasure bursts with minimum decoding delay” (2011)

• Badr, Khisti, Martinian, “Diversity embedded streaming erasure codes (de-sco): Constructions and optimality”(2011)

• …

Streaming in the old world

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6 5 4 3 2 1 6 5 4 3 2 1

Streaming in the new world

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6 5 4 3 2 1 6 5 4 3 2 1

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6 5 4 3 2 1

𝑇 → 𝑇

Low average delay is translated to more information available for processing

6 5 4 3 2 1

Heterogeneous delay codes - Parameter definitions

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packet : th-’iThe delay of the ‘

Average delay:

𝑇 = 𝑇𝑖𝐵𝑖=1

𝐵

B erased

packets

The packets that

participate in the

reconstruction

T1

Ti

TB

Reconstruction delay

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:reconstruction delayThe

𝜅𝑖= 𝑇𝑖 − (𝐵 − 𝑖)

i

B erased

packets

The packets that

participate in the

reconstruction

Ti

𝜅i

𝑇 = 𝜅 +𝐵−1

2

Example: Reconstruction delay for constant-delay codes

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B erased packets The packets that participate

in the reconstruction

𝜅 B

𝜅 B-1

𝜅 2

𝜅 1

𝜅1 + 𝐵 − 1 = 𝜅2 + (𝐵 − 2) = ⋯ = 𝜅𝐵−1 + 1 = 𝜅𝐵= 𝑇

Lower bound on average delay Theorem:

The average delay 𝑇 obtained for a decoding instance following an erasure burst of length B, must satisfy:

𝑇 ≥

𝑅

1 − 𝑅⋅𝐵 + 1

2+𝐵 − 1

2 , 𝑅 ≥

1

2

𝐵

2

𝑅

1 − 𝑅+ 1 ,

1

2> 𝑅 ≥

1

1 + 𝐵

𝐵 + 1

2 , 𝑅 <

1

1 + 𝐵

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Lower bounds comparison

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Ex.: 𝐵 = 4

Per packet: S data symbols, P parity symbols

* A similar proof can be done for non-systematic codes.

Lower bound on average delay- Proof

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1st reconstructed packet 𝜅𝑓𝑖𝑟𝑠𝑡≥

𝑆

𝑃

2nd reconstructed packet

𝜅𝑠𝑒𝑐𝑜𝑛𝑑≥

2𝑆

𝑃

last reconstructed packet

𝜅𝑙𝑎𝑠𝑡≥

𝐵𝑆

𝑃

𝜅 ≥

𝑆𝑃+

2𝑆𝑃

+⋯+𝐵𝑆𝑃

𝐵

Lower bound on average delay- Proof

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𝑹 ≥𝟏

𝟐:

𝜅 ≥

𝑆𝑃+

2𝑆𝑃

+⋯+𝐵𝑆𝑃

𝐵≥

𝑆𝑃+2𝑆𝑃+⋯+

𝐵𝑆𝑃

𝐵=

𝑅

1 − 𝑅⋅𝐵 + 1

2

𝑇 ≥

R

1 − R⋅𝐵 + 1

2+𝐵 − 1

2

The bound may only be met for 𝑅 =𝑚

1+𝑚 , 𝑚 ∈ ℤ+

Lower bound on average delay

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𝟏

𝟏+𝑩≤ 𝑹 <

𝟏

𝟐:

𝜅 ≥

𝑆𝑃+

2𝑆𝑃

+⋯+𝐵𝑆𝑃

𝐵≥ … ≥

𝑅

1 − 𝑅⋅𝐵

2+1

2

𝑇 ≥

𝐵

2

𝑅

1 − 𝑅+ 1

The bound may only be met for 𝑅 =1

1+𝑚 and

𝐵

𝑚∈ ℤ+ , m∈ ℤ+

𝑹 <𝟏

𝟏+𝑩:

κ ≥ 1 (trivial)

𝑇 ≥

𝐵 + 1

2

Desired reconstruction schedule

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𝐦𝐁 Burst

erasure

𝐦

time

Construction with constant delay [Martinian et. al 2004]

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𝒅𝒂𝒕𝒂 𝒑𝒂𝒓𝒊𝒕𝒚

𝐦 = 𝟐 , 𝐁 = 𝟑

R =𝟐

𝟑

Burst

erasure

time

𝑅 =𝑚

1 +𝑚

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

𝑖

P i = 𝑠0 𝑖 − 𝐵 + 𝑠1 𝑖 − 2𝐵

𝑠0 𝑠1

Martinian’s reconstruction schedule

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𝐦𝐁 Burst

erasure

𝐁

Nothing decoded!

time

Desired reconstruction schedule

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𝐦𝐁 Burst

erasure

𝐦

Encoder?

P{ }

P{ }

P{ }

P{ }

Optimal average-delay construction (R>1/2)

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𝒅𝒂𝒕𝒂 𝒑𝒂𝒓𝒊𝒕𝒚

Burst

erasure

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

𝑅 =𝑚

1 +𝑚

weren’t

erased

P i = 𝑠 𝑖 2 𝑖 − 𝑖 2 − 1 + 𝑠 𝑖 2 𝑖 − 𝑖 2 − 1 − 𝐵 +𝑠 𝑖 2 𝑖 − 𝑖 2 − 1 − 2𝐵

𝐦 = 𝟐 , 𝐁 = 𝟑

R =𝟐

𝟑

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Optimal average-delay

𝐦𝐁

κ =mB +m

2 T =

mB +m+ B − 1

2

𝐨𝐩𝐭𝐢𝐦𝐚𝐥!

for any m,B co-prime

Burst

erasure

P{ }

P{ }

P{ }

P{ }

Decoding a different burst

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𝒅𝒂𝒕𝒂 𝒑𝒂𝒓𝒊𝒕𝒚

Burst

erasure

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

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Burst phase shift

Phase shift (𝝓): 𝜙 = 𝑖 mod m

𝑖

B erased

packets

properly received

packets

properly received

packets

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Average decoding delay by phase shift

𝐦𝐁

𝐨𝐩𝐭𝐢𝐦𝐚𝐥

Burst

erasure

𝝓 = 𝟎

𝝓 = 𝟏

𝝓 = 𝟐

𝝓 = 𝟑

𝐰𝐨𝐫𝐬𝐭

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Comparison to known codes

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Optimal average delay for R<1/2

𝐅𝐨𝐫 𝐞𝐯𝐞𝐫𝐲 𝛟 T =B

2

1

𝑚+ 1

𝐨𝐩𝐭𝐢𝐦𝐚𝐥!

Decoding order

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MSB LSB

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P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

P{ }

BAD decoding order

GOOD news: we know how to fix that!

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Conclusion

• We know how to get to optimal average delay

• Method: specify decoding schedule, find realizing encoder

• Open problem: can we have optimal delay for all phase shifts when R>0.5?

• Future work (a nice metric):

Codes with average delays growing gracefully with B

Add random erasures

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Thank you!!!