UEP LT Codes with Intermediate Feedback Jesper H. Sørensen, Petar Popovski, and Jan Østergaard Aalborg University, Denmark IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 8, AUGUST

Outlines Introduction UEP LT Codes with Feedback Simulation Results Conclusion 2

Introduction Conventional rateless codes treat all data as equally important. Unequal Error Protection (UEP), different data segments have different error probabilities – Ex. video streaming The purpose [8] is to quantify the redundancy of these codes, when recovery of all data segments is desired. Improving the decoding performance of the more important data can significantly degrade the decoding performance of the less important data. 3 [8] J. Sørensen, P. Popovski, and J. Østergaard, “Feedback in LT codes for prioritized and non-prioritized data,” in Proc IEEE Vehicular Technology Conference – Fall, pp. 1–5.

Introduction Successful decoding of a data segment is reported to the transmitter through a feedback channel. It is very beneficial to add such a single intermediate feedback. We consider point-to-point communication, where it is justified to use the feedback. Since the intermediate feedback is limited to only a single bit per k input symbols, the scheme may find applicability in broadcast scenarios, where many low rate feedback messages come from different users. 4

Encoder The encoding process of an LT code can be broken down into three steps: – 1) Randomly choose a degree i by sampling π(i). – 2) Choose uniformly at random i of the k input symbols. The uniform distribution used for selection of input symbols is replaced by one which favors more important symbols Non-uniform random selection of symbols is performed. – 3) Perform bitwise XOR of the i chosen input symbols. This solution to UEP has no impact on the decoder. 5 [6] N. Rahnavard, B. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Trans. Inf. Theory., vol. 53, pp. 1521–1532, Apr

Notation X i is the i’th element of X The sum of all elements is denoted and the zero vector is denoted 0 The probability mass function of X is denoted f X (x) The k input symbols are divided into N subsets s 1, s 2,..., s N, each of size α 1 k, α 2 k,..., α N k, where We refer to these subsets as layers 6

7 [6] N. Rahnavard, B. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Trans. Inf. Theory., vol. 53, pp. 1521–1532, Apr

Notation α = [α 1, α 2,..., α N ] The probability of selecting input symbols from s j is p j (k)α j k, such that Assume that p i (k) ≥ p j (k) if i < j. If then all data are treated equally, as in the standard single layer LT code. We define the vector, β, with 8

Notation The N-dimensional degree is denoted j j n : the number of neighbors belonging to the n’th layer We refer to i’ as the reduced degree, where i’ n denotes the reduced number of neighbors belonging to the n’th layer. L = [L 1, L 2,..., L N ], where L n is the number of unprocessed input symbols from the n’th layer R = [R 1,R 2,..., R N ], where R n is the number of symbols in the ripple belonging to the n’th layer 9

Notation We denote the set of j which satisfy j n ≥ i’ n, n = 1, 2,...,N, and ˆj = i The number of encoded symbols collected by the receiver is denoted Δ Initially, all degree-1 output symbols are identified, which makes it possible to recover their neighboring input symbols. These are moved to a storage called the ripple. Symbols in the ripple are processed one by one, which means they are XOR’ed with all their neighbors and removed from the ripple. 10

Definition Definition 1. (Decoder State) A decoder state, D, is defined by the number of unprocessed symbols, L = [L 1, L 2,..., L N ] and the ripple, R = [R 1,R 2,..., R N ]. Hence, D = [L,R]. Whenever a symbol is processed, ˆL will decrease by one. We call this a decoding step, which can only be performed if the ripple is not empty. If the ripple is empty, decoding stops, and we are left with what we call a terminal state. 11

Definition Definition 2. (Terminal State) A terminal state, D X = [L X,R X ], is defined as a state in which R = 0. The probability of ending in a particular terminal state, d X, after having collected Δ symbols, is denoted f D X (d X |Δ) The number of symbols collected while in terminal state d X is denoted Δ d X The maximum value of Δ is denoted Δ max 12

13 The receiver collects a number of encoded symbols, denoted Δ, prior to decoding. We define the vector Ω = [Ω 1,..., Ω i,..., Ω k ], where, Ω i denotes the number of symbols with original degree i among the Δ collected symbols. After having identified all Ω 1 degree-1 symbols and created the initial ripple, we have what we refer to as an initial state, whose distribution function is defined in Definition 2. [9] J. Sørensen, P. Popovski, and J. Østergaard, “Analysis of LT codes with unequal recovery time,” technical note, Apr. 2012, arXiv: [cs.IT].

Example Consider the case of k = 10, N = 2, α 1 k = 6 and α 2 k = 4. Decoding is attempted at Δ = 10 and the received output symbols have the following degrees respectively: 2, 3, 2, 4, 7, 1, 2, 1, 4, 1. Hence, Ω = [3, 3, 1, 2, 0, 0, 1, 0, 0, 0]. The three degree-1 symbols constitute the initial ripple and two of them belong to layer 1. In this case the initial state will be as follows: L I = [6 4], R I = [2 1], C I = [ ]. 14 [9] J. Sørensen, P. Popovski, and J. Østergaard, “Analysis of LT codes with unequal recovery time,” technical note, Apr. 2012, arXiv: [cs.IT].

Example 15 [9] J. Sørensen, P. Popovski, and J. Østergaard, “Analysis of LT codes with unequal recovery time,” technical note, Apr. 2012, arXiv: [cs.IT].

Example 16 [9] J. Sørensen, P. Popovski, and J. Østergaard, “Analysis of LT codes with unequal recovery time,” technical note, Apr. 2012, arXiv: [cs.IT].

Example 17 Fig. 2. An example of a decoder state evolution with two terminal states before successful decoding. [9] J. Sørensen, P. Popovski, and J. Østergaard, “Analysis of LT codes with unequal recovery time,” technical note, Apr. 2012, arXiv: [cs.IT].

Reduced Degree Distribution Theorem 1. (N-Layer Reduced Degree Distribution [8]) In an N- layer UEP LT code using any π(i) and with parameters, α and β, where l n symbols remain unprocessed from n’th layer, 18 [8] J. Sørensen, P. Popovski, and J. Østergaard, “Feedback in LT codes for prioritized and non-prioritized data,” in Proc IEEE Vehicular Technology Conference – Fall, pp. 1–5.

Reduced Degree Distribution N = 2 – Base layer (n = 1) – Refinement layer (n = 2) L X B : the number of undecoded symbols from the base layer L X R : the number of undecoded symbols from the refinement layer α B =0.5, α R = 1 - α B =1-0.5=0.5, β=P B /P R =9 19

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UEP LT Codes without Feedback The Δ’th symbol is received in state d X, if the decoding of the first Δ − 1 symbols resulted in the terminal state d X. The expected number of symbols, E[Δ d x (Δ max )], received while being in state d X, equals the expected number of times decoding fails in that state. It is easy to obtain E[kε 0 (Δ max )] by multiplying by π’ β (0, l X ) and summing over all d X, for which l X ≠ 0. 21

UEP LT Codes without Feedback 22

UEP LT Codes with Feedback We can divide the transmission into two phases; – Before feedback (phase 1) – After feedback (phase 2) The number of symbols collected in phase 1 is denoted Δ 1 The total number of symbols collected in both phases is denoted Δ 2 In phase 2 the encoder only considers refinement layer symbols, which is the equivalent of β = 0, thus entailing the reduced degree distribution π 0 ’(0,l x ) Phase 1 continues as long as l X B = 0 and phase 2 continues as long as l X R ≠ 0. 23

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Simulation Results Settings – k = 100 – α = 0.5 – RSD with c = 0.1 and δ = 1 – E[Δ d x (Δ max )] through Monte Carlo simulations with 1000 iterations 25

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No Feedback 27

With Feedback

Converged redundancy 29 Note that the redundancy converges for increasing β in the scheme applying intermediate feedback.

Conclusion We analyze LT codes with unequal error protection (UEP). We propose a modification with a single intermediate feedback message. 30

References [4] S. Karande, K. Misra, S. Soltani, and H. Radha, “Design and analysis of generalized lt-codes using colored ripples,” in Proc IEEE International Symposium on Information Theory, pp. 2071–2075. [5] D. Sejdinovi´c, D. Vukobratovi´c, A. Doufexi, V. ˇSenk, and R. Piechocki, “Expanding window fountain codes for unequal error protection,” IEEE Trans. Commun., vol. 57, pp. 2510–2516, Sept [6] N. Rahnavard, B. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Trans. Inf. Theory., vol. 53, pp. 1521–1532, Apr [7] P. Cataldi, M. Grangetto, T. Tillo, E. Ma gli, and G. Olmo, “Sliding window raptor codes for efficient scalable wireless video broadcasting with unequal loss protection,” IEEE Trans. Image Process., vol. 19, pp. 1491–1503, June [8] J. Sørensen, P. Popovski, and J. Østergaard, “Feedback in LT codes for prioritized and non-prioritized data,” in Proc IEEE Vehicular Technology Conference – Fall, pp. 1–5. [9] J. Sørensen, P. Popovski, and J. Østergaard, “Analysis of LT codes with unequal recovery time,” technical note, Apr. 2012, arXiv: [cs.IT]. 31

32 Fig. 1. The relative number of undecoded input symbols as a function of the amount of received symbols for the simulated schemes.

33 Fig. 3. The relative number of undecoded input symbols from the individual layers as a function of the amount of received symbols.

34 Fig. 4. Average distortion as a function of the symbol error rate.