2015 IEEE 14th Canadian Workshop on Information Theory (CWIT)
Overlapped Fountain Coding for Delay-Constrained Priority-Based Broadcast Applications Khaled F. Hayajneh, Student Member, IEEE
Shahram Yousefi, Senior Member, IEEE
Dept. of Electrical & Computer Eng. Queen’s University Kingston, Ontario, Canada Email:
[email protected]
Dept. of Electrical & Computer Eng. Queen’s University Kingston, Ontario, Canada Email:
[email protected]
Abstract—In this paper, we propose a new rateless code based on overlapped generations. This proposed scheme works very well especially in a broadcast scenario, where channels with different characteristics are involved in the system. In the proposed scheme, we allow the adjacent generations to overlap. Our main focus is on the design of rateless codes to make any receiver able to recover all the source file with reduction in the overhead and the delay time. Indeed, we show that for several channels characteristics, our proposed scheme outperforms the regular rateless codes such as Luby transform (LT) code. The good advantage of our proposed scheme that it can be used with any kind of degree distributions in the literature and the same improvements are guaranteed.
fill the bucket, that is, receive enough information or symbols until the data is properly decoded. For each generation, the required number of symbols is slightly larger than k when the channel is the binary erasure channel (BEC). The receiver recovers the k source symbols and sends an acknowledgment to the transmitter. Once the transmitter receives acknowledgments from all the intended receivers, it moves forward to the next generation. The advantage of using this scenario is that each receiver’s latency, quality of service and realized rate is dependent on the quality of its own link to the transmitter.
I. I NTRODUCTION
LT codes were originally designed for the BEC which is a simple model and yet quite realistic for may scenarios like that of the generic Internet communications. They were subsequently applied with much success to noisy channels [4]–[7].
Fountain codes which are a subclass of rateless codes have received much attention over the recent years. They are used as erasure codes to improve the user experience in a variety of applications in terms of both latency and quality while also improving the energy and bandwidth usage [1]. Fountain codes are a shift in paradigm: the required system adaptivity is moved from the transmitter end to the receiver end. This allows for better bandwidth utilization and lower outage in general: much more attractive performance-complexity-rate tradeoffs.
Fountain codes belong to the class of low-density generator matrix (LDGM) codes. They possess two important features that make them an ideal choice for a wide range of systems. First, the encoding and decoding complexities are small. Second, they are able to operate at operating points very close to the capacity without suffering from outage or failure. In terms of operation, they are not very different from network codes. Random combinations of the k source symbols are used to generate the transmitted symbol. These are mostly linear. In the LT code case, the number of source symbols to be selected for generating each encoded symbol is chosen according to a practical probability distribution. These distributions are the distinguishing feature of fountain codes controlling the performance as well as the complexity incurred. They lead to sparse graphical representations to be utilized for efficient belief propagation (BP) decoding at the receiver side.
The digital fountain idea is that any receiver can successfully decode the data from any subset of the transmitted symbols as long as the subset is slightly larger than the size of the source data [2]. Luby transform (LT) codes are the first practical realization of the digital fountain method [3]. In LT codes, the source file is partitioned into a number of generations, say g, each of size k packets. These packets are further composed of bits or symbols: typically octals. The transmitter is reminiscent of a water fountain that generates a limitless number of water drops, that is, encoded symbols. They are transmitted through the available channel and any receiver who wishes to receive the file, holds the analogous of a bucket under the fountain and collects enough drops to
With error probability at most δ, an LT code needs on average O(ln(k/δ)) symbol operations for encoding and the same number of operations for decoding over BEC [3]. To reduce these complexities, Shokrollahi introduced Raptor codes by precoding the LT code with a good high-rate error correcting code such as a low-density parity-check (LDPC)
This work was supported by Yarmouk University, Jordan and NSERC, Canada.
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2) d source symbols are chosen uniformly at random from the k source symbols, and, 3) the value of ci is the bitwise XOR of the d source symbols. To make sure that the decoder is aware of the composition of each encoded symbol, either each encoded symbol carries this information in its header (packet overhead in practice) or two synchronous pseudorandom number generators are used in the encoding and decoding sides. At the receiver, a BP decoder is used to recover the source symbols. Once the receiver obtains a sufficient number of encoded symbols, i.e., slightly larger than k, it starts the decoding process. The standard BEC BP process as described by Luby in the original work is used to recover the data [3].
code [8], [9]. In Raptor codes, the encoding and decoding complexities are reduced to O(log(1/ε)) where ε = n/k − 1. During the past decade, the attention has been focused on designing good degree distributions for a variety of channels and applications. In the asymptotic regime (practically when k > 10000), the optimal degree distribution for the BEC is given by Luby in his original work: the robust soliton distribution (RSD) is optimal and universal in the sense of achieving the BEC capacity for any erasure rate [3]. For the finite length regime, there are many improvements to the LT code. In LT codes, the sampling of the k information symbols to create the encoded symbol is a uniform sampling. Since 1998, many improvements to the LT code have been reported. Most of these works design improved degree distributions to outperform the RSD in terms of error rates and/or realized rates [10]–[13]. More recently, Hayajneh et al. provided a new method to design better rateless systems via modifying the sampling method [6], [14]. This technique is applicable to any degree distribution and when tuned, outperforms LT and all other degree distributions known to date [15]. The concept of sliding window has been studied for Raptor codes by optimizing the peak signal-to-noise ratio [16], we address it in different contexts.
RSD, Ω(d), is asymptotically optimal and is given by: Ω(d) =
ρ(d) + τ (d) β
(1)
P where β = d (ρ(d) + τ (d)) is a normalization constant, ρ(d) is ideal soliton distribution given by: ( 1 , if d = 1 (2) ρ(d) = k 1 , if d = 2, 3, ..., k, d(d−1) and τ (d) is a positive function given by: S if d = 1, ..., Sk − 1 dk , S S (3) τ (d) = k ln( δ ), if d = Sk k 0, if d = S , ..., k, √ in which S = c k ln( kδ ), c and δ are the expected number of degree-one symbols, a suitable constant of order one, and the allowable failure probability, respectively. c and δ constitute the tuneable parameters of the distribution [3].
In this paper, we introduce the concept of overlapped generations for fountain codes. We focus on broadcast scenarios and particularly target the improvement of latency and complexity for cases where users with a wide range of erasure rates listen to a single fountain source through BEC links. We show that our method provides better delays and complexities at essentially the same error rates or symbol redundancies. The new scheme is equally applicable to any degree distribution in the literature (including the works mentioned above). To show its potential, we study its application to the RSD.
III. OVERLAPPED FOUNTAIN CODES In this section, we propose a new encoding technique for fountain codes in which the code generations overlap. The proposed scheme divides the source file, K symbols, into g overlapped generations each of size k symbols. Any two adjacent generations overlap by α symbols. Clearly:
The rest of this paper is organized as follows. In section II, we briefly review the basics of LT codes required to make this work self-contained. Then, we propose the concept of overlapped LT codes in section III. Simulation results and conclusions are provided in sections IV and V, respectively.
K (4) k−α where 0 ≤ α ≤ k − 1. If α = 0, there is no overlap between the generations as with regular fountain coding. g=
II. LT C ODES Consider the transmission of a K-bit information vector. LT coding is applied to each of the g = d K k e generations separately. Each session handles the encoding and decoding processes of a single generation. The encoding process of LT codes is as follows. Let u = (u1 , u2 , ..., uk ) be the source symbols in a given generation. The encoder creates a limitless number of encoded symbols c = (c1 , c2 , ...). Each encoded symbol, ci ; i = 1, 2, ..., is generated as follows [3]: 1) randomly choose a degree d from a specified degree distribution, Ω(d),
We propose a solution suitable for data composed of portions with different priority such as those motivated in priority encoding transmission schemes [17] and multi-level coding. In general, we consider that each generation is subdivided into a number of partitions, each partition with a different priority. The first partition has the source symbols u1 , u2 , ..., um with the highest priority for encoding. This is the partition with the highest level of priority; we refer to
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2015 IEEE 14th Canadian Workshop on Information Theory (CWIT)
it as partition π1 . π2 includes symbols um+1 , um+2 , ..., u2m k and so on. There are a total of m = ν levels in general: πν is the least important partition.
B. Decoding overlapped LT codes Each receiver is equipped with a standard BP decoder for the BEC. A receiver collects coded symbols on the fly and decodes them on the decoding factor graph progressively. Because of the higher priority of the first source partition, the decoder decodes π1 faster on the average. When a decoder recovers π1 fully, it sends an acknowledgment to the encoder. When all the receivers on the other ends of BEC links with different erasure rates communicate their acknowledgments to the source successfully, the encoder shifts up to the next generation.
Allowing for the overlap of generations intelligently provides more flexibility in the system design. This as we will show later can lead to better a wider range of performance-complexity trade-offs. In particular, we do not need the entire generation to decode, before we can move on to the next. We implement this by allowing the last α symbols of each generation to coincide with the first α symbols of the following generation. We allow for α to encompass the first γ partitions of the following generation, that is, α = γ.m. This is referred to as a (K, k, α, m, γ) overlapped fountain code. Our proposal is equally applicable to any degree distribution; we focus on the LT code in what follows. Similar returns are expected for any other left degree distribution exploited.
IV. S IMULATION RESULTS The performance of the proposed scheme is compared to that of regular LT codes for K = 1024 via Monte-Carlo simulations. The RSD parameters are c = 0.02 and δ = 0.1. To show the performance over a wide range of conditions, we consider a broadcast to five users with erasure probabilities of e = [0.3 0.2 0.1 0.01 0.001]. We have picked a 4 to 1 priority for the π1 and π2 partitions, that is p = 0.8.
To clearly show how these codes function, let us consider the case of k = 2m. Figure 1 shows the (1024, 256, 128, 128, 1) overlapped fountain code schematically.
Figure 2 shows the resulting coding rate (R) versus erasure probability (e) for three schemes. The first scheme is the regular LT code with a generation size of k = 256: in this scheme we need four generations to send the whole data set. The second scheme is the overlapped LT codes (1024, 256, 128, 128, 1): we need g = 8 overlapped generations to cover all the source symbols. The third scheme is the regular LT code with a generation size of k = 1024: we need only one generation for the entire sequence. Our novel scheme performs at similar realized rates as the LT code with k = 1024 while it affords significant reductions in complexity as seen in Figure 3.
K=1024 k=256 g1
u1 ,u2 ,...,u128
u129 ,...,u256 k=256
g2
u129 ,...,u256
u257,,...,u384 k=256
g3
u257,,...,u384
Fig. 1: Overlapped fountain (1024, 256, 128, 128, 1).
code
u385,,...,u512
with
parameters
0.9
0.85
A. Encoding overlapped LT codes
0.8
The encoding process for overlapped LT codes is similar to that of regular LT codes. The difference lies in how the symbols are sampled. In similarity with fountain codes for unequal error protection (UEP), we sample the information nodes with different probabilities [18], [19]. The encoder chooses symbols from the two partitions π1 and π2 with probabilities p and (1 − p), respectively. When p > 1 − p, we prime the decoder to recover the symbols in π1 faster. The main benefit of this simple system is that the decoding delay for each receiver in a broadcast session is reduced on the average. That is, the realized code rate is improved. These improvements are achieved at a slight increase in the overall encoding and decoding complexity as the average number of edges in the corresponding factor graphs increases.
0.75 Rate, R
Regular LT, k=256 Overlapped LT, k=1024 Regular LT, k=1024 0.7
0.65
0.6
0.55 0 10
−1
−2
10
10
−3
10
Erasure Probability, e
Fig. 2: Coding rate versus erasure probability for different schemes.
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2015 IEEE 14th Canadian Workshop on Information Theory (CWIT)
extra edge in the factor graph. If we compare this result with the performance of LT codes at k = 1024, our proposed scheme has almost the same coding rate, however, LT codes at k = 1024 has an extra one symbol over our scheme (two symbols over LT codes at k = 256) on average in the encoding complexity. In addition, our proposed scheme reduces the waiting time between the generations in the decoding process.
Figure 3 shows the encoding and decoding complexity for the three schemes discussed above. The complexity is measured in terms of the average number of edges per symbol. As we can see, the regular LT code at k = 1024 has the highest encoding complexity no matter what the channel characteristic is. Also, the regular LT code at k = 256 has the lowest complexity. However, our proposed scheme lies between the two schemes. In addition, for the channel with high erasure probability, our proposed scheme is approaching the performance of regular LT code at k = 256.
R EFERENCES
20
19
Average number of edges
18
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16
Regular LT, k=256 Overlapped LT, k=1024 Regular LT, k=1024
15
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11 0 10
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−2
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−3
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Erasure Probability, e
Fig. 3: Average number of edges versus erasure probability for different schemes. From the previous two figures, we can observe that our proposed scheme reaches the performance of regular LT code at k = 1024 in terms of coding rate while reducing the complexity of regular LT codes at k = 1024 to get closer to the complexity of regular LT codes at k = 256. In addition, our proposed scheme reduces the delay time for each receiver. Similar returns can be achieved for other sets of parameters. V. C ONCLUSIONS We have proposed a new rateless coding scheme suitable for broadcast scenarios with a wide range of channel conditions. Our proposed scheme uses the concept of overlapped generations for LT codes; it is equally applicable to any other coding distributions. We show, through MonteCarlo simulations, that our proposed scheme improves the performance of regular LT codes in terms of realized coding rate and as well as delay. The cost of this scheme is a slight increase in the computational complexity. For instance, for a channel with erasure probability of e = 0.01 and packet length k = 256, the improvement in terms of the coding rate is 6.1% at an average cost of an
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