1

LT-AF Codes: LT Codes with Alternating Feedback

Ali Talari and Nazanin Rahnavard

Oklahoma State University

IEEE ISIT (International Symposium on Information Theory) 2013

2

Outlines

• Introduction• LT-AF codes• Simulation results• Conclusions

3

Introduction• Rateless codes require only one feedback that is issued by the

decoder to inform the encoder of a successful LT decoding. • LT codes, the available feedback channel remains under utilized

during the transmission.• As the data-block length decreases the performance of LT

codes significantly deteriorates. • Existing work have proposed to employ feedbacks to inform

the encoder – the number of successfully decoded input symbols, – a suitable input symbol to be sent for enhancing the decoding, – the index of some recovered input symbols.

4

Introduction• We propose LT Codes with Alternating Feedback (LT-AF

Codes) that considerably improve the performance of LT codes for short-block lengths when Belief Propagation decoder is in use.

• The decoder alternatively issues two types of feedbacks based on the dependencies of the still undecoded received output symbols and the number of decoded input symbols.

• In contrast to other existing work, we design LTAF codes with a realistic feedback channel assumption, where the feedback channel can have unknown or varying erasure rate εfb [0, 1).

5

LT-AF codes

• γsucc : the required coding overhead to have a successful decoding with high probability

• γsucc × k coded symbols are enough to decode k input symbols with high probability

• γ : the received coding overhead (meanwhile the transmission is in progress), 0 < γ < γsucc

• γ × k : the number of received output symbols at receiver• We exploit the feedback channel to obtain a much

smaller γsucc for a finite k in LT-AF coding.

6

LT-AF codes

• Ω k,n(.) : the degree distribution of LT-AF codes for a data-block of length k when n input symbols are already recovered at decoder.

• We adopt the idea of SLT codes [6], and propose to shift Ω k,n(.) based on n.

• We allow the decoder to issue the first type of feedback referred to as fb1, which is used to keep the encoder updated with the current value of n.

• The encoder generates an output symbol of degree-one (containing a randomly selected input symbol) as acknowledgment.

[6] A. Hagedorn, S. Agarwal, D. Starobinski, and A. Trachtenberg, “Rateless coding with feedback,” IEEE INFOCOM 2009, pp. 1791 –1799, 2009. http://www.powercam.cc/slide/24647

7

Inefficiency of LT Codes for our Problem

k+ Encoded Symbols

Decoding host

Decode

Input Symbols

n out of k input symbols are known a priori at the decoding host

Many redundant encoded symbols

http://www.powercam.cc/slide/24647

Inefficiency of LT Codes for our Problem

•The number of these redundant encoded symbols grows with the ratio of input symbols known at the decoder (n) to the total input symbols (k)

•If n input symbols are known a priori, then an additional LT-encoded symbol will provide no new information to the decoding host with probability

…which quickly approaches 1 as n → k

d

i

k

dk ik

ind01

)(

8http://www.powercam.cc/slide/24647

9

Intuitive Fix • n known input symbols serve the function of degree 1

encoded symbols, disproportionately skewing the degree distribution for LT encoding

• We thus propose to shift the Robust Soliton distribution to the right in order to compensate for the additional functionally degree 1 symbols

• Questions – 1) How?– 2) By how much?

0 200 400 600 800 1000-6

-5

-4

-3

-2

-1

0

Degree

log 10

(Pro

babi

lity)

LT Code (Robust Soliton)

http://www.powercam.cc/slide/24647

10

Shifted Code Construction• Definition

The shifted robust soliton distribution is given by

– k : the number of input symbols in the system– n : the number of input symbols already known at the decoder– round(・ ) rounds to the nearest integer

• Intuitionn known input symbols at the decoding host reduce the degree of each encoding symbols by an expected fraction

j

kn

iij nknk

1roundfor )()(,

kn1

1

http://www.powercam.cc/slide/24647

11

Shifted Code Distribution

0 200 400 600 800 1000-6

-5

-4

-3

-2

-1

0

Degree

log 10

(Pro

babi

lity)

LT Code (Robust Soliton)Shifted Code

LT code distribution and proposed Shifted code distribution, with parameters k = 1000, c = 0.01, = 0.5. The number of known input symbols at the decoding host is set to n = 900 for the Shifted code distribution. The probabilities of the occurrence of encoded symbols of some degrees is 0 with the shifted code distribution. http://www.powercam.cc/slide/24647

12

LT-AF codes

• We propose to employ Ideal Soliton distribution in LT-AF coding and exploit the feedback channel and request a suitable input symbol (which is an output symbol of degree 1)

• We allow the decoder to request desired input symbols employing the second type of feedback referred to as fb2

• The encoder generates a degree-one output symbol if and only if it has received a fb1 or fb2

13

LT-AF codes• The lack of the arrival of an output symbol at the decoder with

degree one after issuing a fb1 or fb2 clearly indicates a feedback loss.

• All feedback packet losses are identified by the decoder and a feedback retransmission is performed.

• The decoding recovery rate of LT-AF codes does not considerably degrade at high feedback channel loss rates εfb [0, 1)

• A degree-one output symbol generated at the encoder after a fb1 contains a randomly selected input symbol. However, such an output symbol would contain the requested input symbol (selected by decoder) after a fb2

14

LT-AF codes

• Let ,where is the probability of selecting degree d to generate an LT-AF output symbol

• We do not allow the encoder to generate any degree-one symbol, we set

15

LT-AF codes

• The average degree of a check node (output symbol) generated employing Ω k,n(.) distribution is

16

Generating fb1

• The decoder is not always aware of n unless its knowledge about n is updated by a fb1

• Initially, the encoder assumes n = 0 and employs the degree distribution Ω k,0(.) to generate output symbols.

• nr : the most recent reported value of n employing a fb1 • We propose the encoder to generate a fb1 when

, i.e., average degree of Ω k,n(.) increases by at least

• ni : the threshold that for n ≧ ni the ith fb1 is generated

17

Generating fb1

18

Generating fb1

• At k = 102 the first and the second fb1’s are issued at n 39 and ≧ n 58≧• For k = 104 the first and the second fb1’s are issued at n 2740 and ≧ n 4346≧• ni / k decreases as k increases.

39 4346

2740

58

19

Generating fb2

• Transmission of fb2’s before γ = 1 does not considerably contribute to decoding progress.

• We propose to generate fb2’s only when γ surpasses 1.

• To have uniformly distributed fb2’s and to avoid feedback channel congestion, an LT-AF decoder issues a fb2 on the reception of every (ln k)th output symbol.

20

Generating fb2

• 1) Generating fb2 Based on Variable Node with Maximum Degree (VMD)– Decoder requests the variable node with the highest degree

in its current decoding graph to issue a fb2

– VMD greedily removes the largest possible number of edges from G and decreases the degree of many check nodes.

– Vun : the set of remaining undecoded variable nodes

– Cbuff : the set of buffered check nodes with a degree higher than one

– the ripple : the check nodes with degree 1

21

Generating fb2

• VMD would request v5

• On the arrival of c8 containing only v5

, the value of v5 will become known.• This removes all the edges

emanating from v5 to all other check nodes and reduces some to degree 1 (they are added to the ripple).

• C7 is added to the ripple, which recovers v7 in the next decoding iteration.

Fig. 2. The bipartite graph representing the input and the output symbols of an LT-AF code at the buffer of a decoder. Graph G at a decoder at γ = 1 for k = 7.

22

v1 v2 v3 v4 v5 v6 v7

c1 c2 c3 c4 c5 c6 c7

v1 v2 v3 v4 v5 v6 v7

c1 c2 c3 c4 c5 c6 c7 c8

23

v1 v2 v3 v4 v6 v7

c1 c2 c3 c4 c5 c6 c7 c8

• the ripple : c7

• Cbuff = {c1, c2, . . . , c6} • Vun = {v1, v2, v3, v4, v6}

24

Generating fb2

• 2) Generating fb2 Based on Full Variable Node Decoding(FVD)– A more complex method to generate fb2 is to run a

dummy decoding for all unrecovered input symbols. – The single input symbol whose delivery results in

the highest number of decoded input symbols is requested by a fb2

– Full Variable Node Decoding (FVD) has a much higher complexity than VMD.

25

Simulation Results

• Our results are obtained employing Monte-Carlo method by averaging over the results of at least 107 numerical simulations.

• The decoding bit-error-rate (BER) (average ratio of unrecovered input symbols to total number of input symbols 1−E[n/k])

• k = 1000• We set c = 0.9 and δ = 0.1 for SLT and LT codes as

proposed in [6].[6] A. Hagedorn, S. Agarwal, D. Starobinski, and A. Trachtenberg, “Rateless coding with feedback,” IEEE INFOCOM 2009, pp. 1791 –1799, 2009.

26

LT-AF Decoding Error Rate and Runtime

γsucc =1.09 γsucc =1.14 γsucc =1.31

27

LT-AF Decoding Error Rate and Runtime

• In LT-AF coding the average degree of output symbols is much higher than that of regular LT codes, which causes a higher encoding/decoding complexity.

28

Number of Feedbacks

• Not only LT-AF codes decrease the required coding overhead for a successful decoding γsucc , but also they need slightly smaller number of feedbacks compared to SLT codes. • The total number of feedbacks is much smaller than γsucc k.

29

Robustness to Erasure in Feedback Channel

• Assume that the loss rate of the feedback channel is εfb = 0.9 (which is not known to encoder and decoder), hence 90% of the feedbacks are lost in transmission.• To the best of our knowledge robustness against feedback loss had not been considered in any existing work and this significantly distinguishes LT-AF codes.

Fig. 5. Effect of 90% feedback loss on the performance of SLT and LTAF codes employing VMD. Note that the curves representing LT-AF codes’ performance for εfb = 0.9 and εfb = 0.9 fully overlap for all γ ".

30

Conclusions

• We proposed LT-AF codes that are LT codes with two types of feedback, which alleviate the low performance of LT codes for short data-block lengths.

• We showed that LT-AF codes require lower coding overhead for successful decoding and lower number of feedback.

31

References• [6] A. Hagedorn, S. Agarwal, D. Starobinski, and A. Trachtenberg, “Rateless

coding with feedback,” IEEE INFOCOM 2009, pp. 1791 –1799, 2009.• [7] A. Kamra, V. Misra, J. Feldman, and D. Rubenstein, “Growth codes:

Maximizing sensor network data persistence,” SIGCOMM Computer Communication Rev., vol. 36, no. 4, pp. 255–266, 2006.

• [8] A. Beimel, S. Dolev, and N. Singer, “RT oblivious erasure correcting,” IEEE/ACM Transactions on Networking, vol. 15, pp. 1321 –1332, dec. 2007.

• [9] S. Kokalj-Filipovic, P. Spasojevic, E. Soljanin, and R. Yates, “ARQ with doped fountain decoding,” ISSSTA, pp. 780 –784, aug. 2008.

• [10] J. Sørensen, P. Popovski, and J. Østergaard, “On the Role of Feedback in LT Codes,” Arxiv preprint arXiv:1012.2673, 2010.

• [11] J. H. Sorensen, T. Koike-Akino, and P. Orlik, “Rateless feedback codes,” in IEEE ISIT, pp. 1767–1771, 2012.