SCIENCE CHINA Information Sciences
. RESEARCH PAPER . Special Issue
February 2013, Vol. 56 022302:1–022302:10 doi: 10.1007/s11432-012-4764-7
Packet combining based on cross-packet coding LIN DengSheng1 ∗ , XIAO Ming2 & LI ShaoQian1 1National
Key Lab. of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu 611731, China; 2ACCESS Linnaeus Center, School of Electrical Engineering, Royal Institute of Technology (KTH), Stockholm 100044, Sweden Received November 27, 2012; accepted December 5, 2012
Abstract We propose a packet combining scheme of using cross-packet coding. With the coding scheme, one redundant packet can be used to ensure the error-correction of multiple source packets. Thus, the proposed scheme can increase the code rate. Moreover, the proposed coding scheme has also advantages of decoding complexity, reducing undetectable errors (by the proposed low-complexity decoder) and flexibility (applicable to channels with and without feedback). Theoretical analysis under the proposed low-complexity decoding algorithm is given to maximize the code rate by optimizing the number of source packets. Finally, we give numerical results to demonstrate the advantages of the proposed scheme in terms of code rates compared to the traditional packet combining without coding or ARQ (automatic repeat-request) techniques. Keywords
packet combining, coding, ARQ, feedback, error correction
Citation Lin D S, Xiao M, Li S Q. Packet combining based on cross-packet coding. Sci China Inf Sci, 2013, 56: 022302(10), doi: 10.1007/s11432-012-4764-7
1
Introduction
Packet-level error-control coding, e.g., rateless codes [1,2] or network codes [3–6], has recently attracted substantial research interest. However, for these packet-level coding or ARQ (automatic repeat-request), a packet is normally dropped, if any bit error is detected, e.g., by CRC (cyclic redundancy check). However, if the channel is in a good condition, the packets with lots of bits may only have a few erroneous bits. Thus, dropping the whole packet is inefficient. Packet combining (PC) including soft combining and hard combining, which belongs to a kind of the hybrid-ARQ technique, helps the erroneous packets further improve the transmission efficiency. These packet combining techniques usually retransmit more packets if the source packet is in error. If the retransmitted packets are in error, too, these erroneous packets are jointly used rather than dropped to recover the source packet. In recent researches, the authors study the joint design of turbo codes and packet combining in MIMO (multiple-input multiple-output) channel in [7] and relay channel in [8], respectively. Similarly, in [9], the authors propose a joint iterative decoding algorithm for turbo codes with packet combining. In [10,11], the authors focus on the theoretical evaluation of performance in ∗ Corresponding
author (email:
[email protected])
c Science China Press and Springer-Verlag Berlin Heidelberg 2013
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applying packet combining ARQ technique in TDMA (time division multiple access) system and singlecarrier frequency domain equalization (SC-FDE) system, respectively. However, these packet combining techniques involve joint decoding of channel codes and packet combining, which will greatly increase the complexity of decoding. Joint decoding of channel codes and packet codes also limits the flexibility of their application. Another packet combining technique is proposed in [12]. In the scheme, two or more erroneous packets including source packet and the retransmission packets are compared to find the locations of erroneous bits. With the locations, the receiver uses a brute-forced approach to search for correcting source packet. In [13], the modified PC is proposed to allow multiple retransmitted packets to do the packet combining. In [14], a packet reversed packet combining scheme is proposed to reduce the correlation between the source packet and the retransmitted packet that is generated by bit-level reversing the source packet. A throughput analysis on reversed packet combining scheme is given in [15]. The advantages of these PC schemes include avoiding channel codes involved and flexibility for any channel types, etc. Moreover, similar to the above PC schemes, the existing PC techniques are the extension or modification of the ARQ scheme. Clearly, PC is invoked only when consecutive retransmitted packets have errors, and also the retransmitted packets only can be used for recovering one source packet. This limits the efficiency and application of the PC schemes. Based on the observation, we have proposed an error correction scheme for combining cross-packet coding and PC in [16]. Different from previous PC techniques, a retransmitted packet is constructed for multiple erroneous source packets through coding. As shown in [16], this substantially increases the transmission efficiency. In this paper, we will further extend and develop the proposed coding scheme for PC. Firstly, both the transmission scheme on the channels with feedback and without feedback are investigated. Secondly, a reduced-complexity decoding algorithm is developed to reduce the complexity and the undetectable errors. Thirdly, theoretical analysis based on the reduced-complexity decoder is given to maximize the coded rate by optimizing the number of source packets.
2
System description
The proposed encoding scheme works in a systematic form. That is, k uncoded source (systematic) packets Sj , j = 1, 2, . . . , k are first transmitted. Then the received systematic packets can be denoted by ˆj = Sj ⊕ ej , S
j = 1, 2, . . . , k,
(1)
where ej denotes an error pattern (positions of erroneous bits) caused by e.g., noise or interference of physical channels. ⊕ means XOR calculation. Note that if channel codes are used, (1) are results after channel decoding. Thus, in this sense, our scheme is on top of channel coding. After the systematic blocks, the redundant packets are encoded by XORing some or all of the systematic packets after interleaving according to certain interleaving patterns which keep them independent. It can be denoted by πi,j (Sj ) mod 2, i = 1, 2, . . . , m, (2) Ri = j∈Ci
where πi,j (·) is the interleaver whose pattern depends on the indices of i and j, and m is the total number of redundant packets when transmission stops for all k packets. Ci is a set of the indices of the systematic packets used to encode the ith redundant packet. Note that coding operations among the systematic packets include their CRC bits. But redundant packets do not use CRC. Hence the length of redundant packets is equal to that of systematic packets with CRC. The proposed codes can work on the channel with or without feedback depending on whether the set of Ci is determined by the receiver or the transmitter. With feedback, the transmitter knows the status of previous transmission, which is helpful for optimizing the next transmission (designing Ci+1 ). In the following, we first discuss the scheme with feedback. The scheme without feedback will be discussed in Section 4.
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The received redundant packets are evaluated by ˆ i = Ri ⊕ e , R i
(3)
i = 1, 2, . . . , m,
where ei denotes another error pattern independent of ej , The receiver will decode immediately after receiving each Ri . Then, an error pattern is calculated as ˆi ⊕ Ei R
ˆj ) mod 2 = ei ⊕ πi,j (S
j∈Ci
πi,j (ej ) mod 2.
(4)
j∈Ci
ˆj s in Ci and R ˆ i . By (4), “1”s of Ei indicate the locations Ei is the binary sum of all error patterns of S ˆ i . Then a natural decoding approach is to find a correct S ˆj of all possible erroneous bits in Sˆj s and R ˆj (S ˆj ∈ Ci ). The testing of one S ˆj stops by exhaustive testing (inverting) each error bit for every S ˆj requires at most 2dp testings. when CRC indicates error-free. Assuming dp “1”s in Ei , decoding S The complexity can be quite high and there may be undetectable errors too. Actually, the number of erroneous bits in a systematic packets is usually much smaller than that of “1”s in Ei , especially when k is large. Thus, it is not necessary to test every possible error pattern based on Ei to correct a systematic packet. Consequently, we propose a reduced-complexity decoding algorithm, with which we can tradeoff the decoding complexity and performance by pre-defining a complexity parameter Nc,max . It denotes the maximum times of CRC check for each systematic packet and gives a tradeoff between complexity and code rate. We also use a performance parameter dd,max , which means the maximum distance of CRC check and gives a tradeoff between code rate and performance. In the proposed decoding algorithm, the decoder tests the error pattern with possible erroneous bits (denoted by d) from low to high till d can not longer be increased due to the limitation of decoding complexity related to Nc,max or the ability of error correction related to dd,max . Clearly, the maximum d should satisfy dmax = min(dd,max , dc,max ), where dc,max is given as
dp,dc,max dc,max
Nc,max
2,
2,
ke,d 2.
(22)
Because the decoder cannot separate the two types of errors if the criteria given above is not satisfied, it regards all the packets in Γd as the first type of errors. Following the results described in Section 3, each redundant packets can help roughly recover one half of packets in Γd . Yet at most log2 (ke,d ) redundant packets are needed to recover the first type of errors. Following the proposed decoding algorithm, the first type of errors can be decoded till d cannot increase due to the limit of decoding complexity related to parameter Nc,max or the ability of error correction related to dd,max . Thus the maximum d is given in (5). The sum of error bits in Γd is lkd Pb,d and there is ke,d “1”s overlapping, which cannot be shown in error patterns. One redundant packet only contains one half of packets in Γd except the first redundant packets. Therefore, dp,d is given by dp,d =
1 (lkd Pb,d − ke,d ) . 2
(23)
Finally, we have m1 =
d max d=1
log2 (ke,d ).
(24)
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Table 1
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Rate comparisons between ARQ and proposed scheme
Pb
Pb
Sim k
Anly k
ARQ rate
Sim rate
Anly rate
0.020
0.0093
114
116
0.75
0.83
0.87
0.018
0.0071
186
210
0.80
0.94
0.93
0.016
0.0052
330
350
0.85
0.97
0.96
0.014
0.0036
700
667
0.90
0.98
0.98
The rest packets with errors more than dmax bits belong to the second type of errors. They only can be recovered by ARQ. Thus we have m2 = kdmax .
(25)
Finally, by replacing (24) and (25) into (16) and (15), we obtain the relation between rate R and the number of systematic packets k given bit error probability of BSC channel Pb . Then based on the equation, we can maximize the rate by optimizing the value of k.
6
Numerical results
In what follows, we shall give the numerical results by simulations. In the simulation, we select a BCH code (127, 106, 3) [17] as the inner code. The code is capable of correcting any 3 bit errors. The packeterror-rate after decoding in BSC (binary symmetric channel) channel is given by Pf,3 by (17) by letting d be 3. Each packet has a separate CRC sequence of length 16 bits. In the proposed scheme, the CRC sequence also is used to help efficiently recover the erroneous packets. For comparison, we present two retransmission schemes: The first is ARQ, in which the erroneous BCH-based packets are recovered by simple retransmission. The rate equals 1 − Pf,3 ; the second is the proposed coding scheme as described above. In the proposed scheme, k systematic packets first are interleaved on bit-by-bit to let the erroneous bits independently distribute in all the k packets. Thus the bit error probability after BCH-based decoding Pb will be Pb,d based on (20) by letting d be 3 and ignoring the term of Pf,d . In all the simulation, we set the parameter Nc,max and dd,max to 1024 and 4, respectively. We also keep the packet error rate to be less than 10−2 after cross-packet decoding in the proposed scheme. Table 1 gives the comparisons between the two transmission schemes. In the table, the third column means the optimal k by simulations, and the related maximum rate it produces is shown in the sixth column. The fourth and seventh columns mean the optimal k and rate R given by analysis according to Section 5, respectively. From the table, firstly, by comparing the data in columns 3, 4 and 6, 7, we can see that the results by analysis match quite well with the simulations for all k and R. Secondly, the table also shows significant advantage of the proposed scheme in terms of rate comparing with the traditional ARQ scheme. For example, when Pb = 0.018, the rate increases from 0.80 for the ARQ scheme to 0.93 for the proposed scheme. Figure 4 shows the rates of the proposed code and ARQ scheme in Table 1. The figure also gives the rates of the proposed code for the channel without feedback and traditional packet combining technique. The parameters k, Nc,max and dd,max for the scheme without feedback are set to be the same as those of the scheme with feedback. The figure shows that the scheme without feedback performs almost as well as the one with feedback. The figure also shows the proposed schemes both significantly outperform the packet combining proposed in [12] and ARQ techniques. Especially, with decreasing Pb , the rate of PC rapidly trends to that of ARQ, but a significant gain still remains between the proposed schemes and ARQ. Note that several modified PC schemes, such as modified PC (MPC) and packet reversed PC (PRPC), are developed in [13,14]. But we do not give the comparisons with them in Figure 4 because they do not have obvious gain with the scheme proposed in [12] under our simulation condition.
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1.0
0.9
R
0.8
0.7 ARQ Traditional PC Proposed wo feedback Proposed wi feedback
0.6
0.5 0.020
0.018
0.016
0.014
Pb Figure 4
7
Rate comparisons among ARQ, traditional PC and the proposed schemes with and without feedback.
Conclusions
We propose a cross-packet coding scheme by combining multiple source packets to form each of redundant packets. Consequently, the coding scheme is able to decode more than one source packet by a redundant packet. Thus the code rate is substantially increased. Moreover, the proposed scheme can work on channels with or without feedback. Moreover, an efficient reduced-complexity decoder is presented to substantially reduce the decoding complexity. An optimization method is also provided to effectively design the coding scheme and theoretically evaluate the optimal code rate. Numerical results demonstrate the benefits of the proposed schemes. The results also closely match to our theoretical analysis and show the gain in terms of rate compared to existing schemes, such as packet combining and ARQ techniques.
Acknowledgements The work was supported by Key Laboratory of Universal Wireless Communications (Beijing University of Posts and Telecommunications), Ministry of Education, China. This work was also supported in part by National Basic Research Program of China (973 Program) (Grant No. 2013CB329001), National Science and Technology Major Projects of China (Grant No. 2010ZX03003-004-01), and Sino-Sweden Cooperative Program by MOST in China and VINNOVA in Sweden.
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