Rateless Feedback Codes

1. Rateless Feedback Codes Jesper H. Sørensen, Toshiaki Koike-Akino, and Philip Orlik 2012 IEEE International Symposium on Information Theory Proceedings

2. Outline • Introduction • Background of rateless codes • Rateless feedback codes • LT feedback codes • Raptor feedback codes • Numerical results

3. Introduction • On the internet • When a strong feedback channel is available • Using automatic repeat-request (ARQ) schemes. • Difficult to employ this scheme in many practical systems. • With no frequent feedback • A rateless coding can achieve reliable communications. • LT codes • Raptor codes • Complexity : O(klogk)

4. Introduction • If the receiver has m feedback opportunities (0 < m < k) , • What scheme should we applied ? • Doped fountain coding [3] • The receiver feeds back information on undecoded symbols. • Making the transmitter able to transmit input symbols • Accelerate the decoding process. • Real-time oblivious erasure correcting [4] • Telling how many of the k input symbols have been decoded • The transmitter chooses a fixed degree • Maximizes the probability of decoding new symbols.

5. In this paper • We propose LT feedback codes and Raptor feedback codes. • Can use any amount of feedback opportunities . • The feedback tells the encoder which source symbols have been recovered. • Helping the encoder to modify the degree distribution • Our goal • To decrease the coding overhead especially for short k. • We will show that the proposed feedback scheme can decrease both the coding overhead and the complexity

6. Background of rateless codes • LT codes encoding process : • Randomly choose a degree d by sampling Ω(d). • Choose uniformly at random d of the k input symbols. • Perform XOR of the d chosen input symbols. • LT decoding process : • Find all degree-1 output symbols and put the neighboring input symbol into ripple. • Symbols in the ripple are processed one by one • We potentially reduce a buffered symbols to degree one , we call this a symbol release

7. Rateless feedback codes • The ripple size is an important parameter in the design of LT codes. • This design contains two steps: • Find a suitable ripple evolution to aim for • Find a degree distribution which achieves that ripple evolution. • We first focus on the case of a single feedback located when f1ksymbols have been decoded.

8. LT feedback codes • The feedback informs the transmitter which input symbols have been decoded. • The first f1k symbols has no influence on the release of the symbols received after the feedback. • The probability that a symbol of degree d is released when L input symbols remain unprocessed :

9. LT feedback codes • Symbols received after the feedback • Based on the reduced set of input symbols of size (1- f1)k. • Their releases are independent of the processing of the first f1k input symbols. • Their release probabilities is • We can give the ripple a boost at an intermediate point. k = 100 f1 = 0.5

10. LT feedback codes • The ripple size should be kept larger than , for some positive constant c . • Every time a symbol is processed, the ripple size is either increased or decreased by one with equal probabilities. • Viewing the ripple evolution as a simple random walk. • The expected distance from the origin after z steps is

11. LT feedback codes • Due to the feedback, the ripple evolution can be viewed as two random walk . • Ripple evolution :

12. LT feedback codes • The next step in the design is to find a degree distribution which achieves the proposed ripple evolution . • We let Q(L) denotes the expected number of releases in the (k − L) –th decoding step. • We want to let the vector R(L) map into Q(L) .

13. LT feedback codes • The achieved Q(L) can be expressed as a function of the applied degree distribution, , through (1). • This is done for the general case without feedback as • For the case with feedback, we have contributions from two different degree distributions ,

14. LT feedback codes • The solution tells the degrees amount before and after the feedback in order to achieve the desired ripple evolution. • n1 and n2 are the free parameters. • Normalizing the solution vectors with n1 and n2 provides the degree distributions. • We propose to use the least-squares nonnegative solution for (7) to achieve close to the desired ripple evolution.

15. LT feedback codes

16. LT feedback codes • Consider m feedback opportunities whose locations are at fi , for i =1, 2,...,m. • The decoding can be viewed as m+1 random walks . • The proposed ripple evolution :

17. LT feedback codes

18. LT feedback codes • An important performance metric is the average degree, • Showing that LT feedback codes decrease the average degree, and thereby computational complexity.

19. Raptor feedback codes • Raptor coding is a concatenation of an LT code and a highrate block code. • A weight w : • To make it more strict. • To avoid the case that the ripple lacks robustness around the feedback point

20. Raptor feedback codes k = 128, f1 = 0.75, c1 = c2 = 1 and d= 2.7 Fig. 5. The ripple evolution achieved in the Raptor feedback code compared to the one proposed for LT feedback codes. • A slight lack of robustness near the feedback point • andnear the end of decoding

21. Numerical results • 5000 runs for c1 = c2 = 1 • The best performance : for k = 32, 64 and 96 • for k = 32, 64,96 and 128.

22. Numerical results

23. Numerical results • Raptor feedback codes ,for zero, one and two feedback opportunities, we choose d of 5, 3 and 2.5 respectively.

24. Numerical results • The average overhead and complexity of LT feedback codes for k= 128 as a function of the number of feedback opportunities.

25. Reference