Rate Adaptive Distributed Source-Channel Coding Using IRA Codes for Wireless Sensor Networks

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1 Rate Adaptive Distributed Source-Channel Coding Using IRA Codes for Wireless Sensor Networks Saikat Majumder and Shrish Verma Department of Electronics and Telecommunication, National Institute of Technology, Raipur, India {smajumder.etc, Abstract. In this paper we propose a scheme for rate adaptive lossless distributed source coding scheme for wireless sensor network. We investigate the distributed source-channel coding of correlated sources when correlation parameter is not fixed or may change during sensor network operation. For achieving rate adaptability we propose the puncturing and extension of IRA code depending on the value of correlation between two sources and the quality of channel. In our scheme we need to transmit only incremental redundancy for decreased correlation or fall in channel quality to meet energy constraints and reducing computation cost. Keywords: Distributed source-channel coding, IRA code, sensor networks. 1 Introduction Distributed lossless compression of information is required in many applications like wireless sensor networks. To achieve efficient transmission the sources may be compressed at the individual nodes independently and sent to the fusion center through wireless channel. Compression is required before transmission because wireless sensor nodes are characterized by low power constraint and limited computation and communication capabilities. If there is correlation between the data sent by sensors, then, though they do not communicate with each other, exploiting the correlation would lead to further reduction in number of transmitted bits. Slepian-Wolf theorem [1] shows that lossless compression of two separate correlated sources can be as efficient as if they were compressed together, as long as decoding is done jointly at the receiver. Such a scheme is known as distributed source coding (DSC). Practical schemes for exploiting the potential of Slepian-Wolf theorem were introduced based on channel codes [2], [3], [4], [5] and some of them used modern error correcting codes (e.g. LDPC, Turbo codes) to achieve performance very close to theoretical Slepian-Wolf limit. Schemes for distributed joint source-channel coding has been proposed in literature [6], [7], [8] and the references therein which use a good channel code for jointly performing the operation of compression (source coding) and adding error correction capability (channel coding). The basic idea of distributed source coding with side information at the decoder is shown in the Fig. 1. It is a communication system of two binary sources X and Y with V.V. Das, G. Thomas, and F. Lumban Gaol (Eds.): AIM 2011, CCIS 147, pp , Springer-Verlag Berlin Heidelberg 2011

2 208 S. Majumder and S. Verma conditional probability mass function P(X Y) = p. For many sensor network applica- tions, this statistical dependency between X and Y may not be known before deploy- ment. Instead of designing the code for lowest correlation probability p min, the rate of the code should adapt to the correlation parameter. Designing code for lowest correla- Punctured low density parity check (LDPC) codes were used [10] for achieving the tion requires larger generator and parity check matrices and it results in longer codes. required rate. Though puncturing reduces the number of transmitted bits, it does not in any way ease the computational and memory requirements of the encoder and decod- and er. It is because puncturing channel code removes bits after encoding process reinserts null bits before decoding. Other variations of LDPC code has been used for achieving rate-adaptive codes for distributed source coding [11]. 2 System Model Fig. 1. Distributed source-channel coding with side information at the decoder The primary contribution of this paper is the development of a scheme for rate- requirements than the methods using puncturing only for achieving rate-compatibility. Though puncturing has been the accepted method for achieving rate adaptation, code extension has not been applied in the context of distributed source-channel coding. We apply efficient puncturing [12] and extending [13] methods for obtaining the adaptive distributed source-channel coding that has lesser computational and memory desired code rate for given value of source correlation and channel quality. Our idea of applying code extension methods for distributed source-channel coding reduces the memory and computational requirements at both sensor nodes and fusion center, besides conserving battery power. The paper is outlined as follows; Section 2 describes the overall system and exand plains the relation and tradeoffs between correlation parameter, channel quality desired BER. Section 3 discusses the methods employed for puncturing and extending IRA codes. Finally in section 4 and 5 we present simulation results and conclusion. Consider the system of Fig. 1 with the following notations used for rest of the paper. There are two correlated source vectors X = [x 1, x 2,..., x k ] and Y = [y 1, y 2,..., y k ] of length k. The dependency between these two sources is given by conditional probabil- before deployment, optimal rate of transmission has to be decided during run time. Y ity mass function P x y. Since the correlation parameter p may not be known is available lossless and error free at the decoder. We try to encode X as efficiently as possible; encoding operation being joint source-channel coding. In this paper we apply systematic irregular repeat accumulate (IRA) codes as joint source-channel encoder using the method in [6].

3 Rate Adaptive Distributed Source-Channel Coding Using IRA Codes 209 Fig. 2. Rate adaptive source-channel IRA encoder. H u is the mother code matrix, whereas matrices E l are obtained by code extension. All the generated parity bits are concatenated to produce the final code word Z. Puncture is applied when increase in code rate is required. IRA codes, a special class of LDPC codes, are used here in the context of sensor network because they enjoy extremely simple encoding and low complexity decoding. Parity check matrix for systematic IRA code has the form H = [H u H p ], where H u is a sparse matrix and H p is an dual diagonal matrix. The output of the systematic encoder is Z = [X P], where P is the parity vector. We design the IRA encoder in Fig. 2 which provides incremental redundancy and is based on such an LDPC encoder in [14]. In the encoder matrix H T u followed by accumulator block provides the parity check bits P 0 of mother code. This part is same as any conventional IRA encoder. The matrices E 1, E 2... E lmax take into account extra parity bits (for decreasing code rate) provided by code extension. The puncturing block is for increasing the code rate more than the rate of mother code. For example, when rate index l is specified, the concatenated code is given by Z = [X P 0 P 1 P 2... P l ]. Parity bits are obtained as P XH H and P XE for i 1,, [13], [15]. When sources are highly correlated or if the channel is good, only Z = [X P 0 ] is produced and code is punctured to get the required rate. Code design for given rate index (or rate) using extension and puncturing of IRA code is discussed in section 3. The generated code Z and lossless source Y can be jointly decoded using a message passing decoder [15]. The decoding process is same as any IRA decoder, only difference being the initialization of the log-likelihood ratios (LLR) [6]. For unit code word energy and code rate r, LLR of the parity bit nodes is 2P 1 4r N and that of information bit nodes is 2Y 1 ln 1 2X 1 4r N where, P = [P 0 P 1 P 2... P l ] and function f(x) indicates effect of channel on signal x. Next we derive the relation for code rate under channel constraints and varying amount of correlation between the sources. Since Y is a vector of equiprobable binary random variable, its rate is its entropy H Y kh y k. From Slepian-Wolf theorem, the theoretical limit on rate after lossless compression of X is H Z kh x y kh. For nonideal channel we have to account for the limited channel capacity. For error free communication of a binary source of per bit entropy H(p), using a channel code of rate r, the bit energy E b required is related as E N 2 1 /2r where, N 0 /2 is the noise power spectral density. Appropriate value

4 210 S. Majumder and S. Verma of rate r for given channel condition and conditional entropy H(p) can be found by solving this equation. For finite length codes and continuous channel, the SNR (E b /N 0 ) entered has to be few db higher than actual value. This shift depends on code length and minimum BER required and can be obtained empirically. For example, in our case of IRA code of 1024 information bits and tolerable BER of 10-4, shift required would be of 1.45 db. 3 Rate Adaptive Code Design We now elaborate the code design method which uses extended and punctured IRA code for a obtaining a specific rate r. Rate-compatible IRA codes through deterministic extending based on congruential extension sequences [13] is used in our study. Compared to other code extension methods, this method uses low-complexity, algebraic operations without any post-construction girth conditioning. As already mentioned, puncturing of code though reduces the transmitted energy; it does not reduce the memory and computational requirements of encoding and decoding operation. So, starting out with a channel code of lowest anticipated rate and puncturing saves transmission power but not computational resource. Therefore, we consider a mother code of moderate rate r k/n and length n 0. Corresponding parity check matrix H = [H u H p ] is of size m n. Let r,r,,r with r r r to be set of desired target rates lower than r 0 for corresponding parity check matrices H,H,,H. For a particular rate index l, starting from H 0, we construct H H H 0 (1) E 0 I where, E is obtained by concatenation of sub-matrices E 1, E 2,..., E l as E T = [E T 1, E T 2,..., E T l ]. Each of the submatrices E l is of size ε k ε sε, 0 are all zero matrices, I is a ε ε identity matrix. s k/q, where q = ε if ε is odd, otherwise q = ε 1. ε is the number of rows in E l and is chosen to have the desired code rates. Two random sequences, : 0,, 1 and : 0,, 1, with elements from GF(q) is generated. Care must be taken so that no two element in same sequence are same. A matrix A = [a ij : 0 i l - 1, 0 j s - 1] of dimension s is formed with its elements q, where we have taken d = 1. Using the elements of A, a new s matrix L is formed with its each element I(a ij ) being ε ε identity matrices with rows cyclically shifted to the right by a ij positions. Finally, our extended matrix E is obtained from L by method of masking [9]. To obtain code rates 1,,r,,r,r with 1 r r r we use the puncturing method in [12]. In this method only parity bits are punctured and punctured node is chosen such that it is at equal distance (in terms of number of nodes) from neighboring unpunctured nodes. The punctured codes of different rates obtained from this type of deliberate puncturing are not rate compatible. To obtain rate compatibility, they have proposed this simple algorithm which performs puncturing in reverse way. In their algorithm, mother code is deliberately punctured to the highest code rate, then unpuncturing those punctured nodes to obtain lower rates.

5 Rate Adaptive Distributed Source-Channel Coding Using IRA Codes Simulation Resultss For simulation we have used mother code with node-wise degree-distribution of , rate r 0 = 0.5 and k = 1024 as in [13]. All the simulations are done for a binary phase shift keying (BPSK) modulated AWGN channel. The iterative sum product algorithm (SPA) was used for decoding, and the maximum number of decoding iterations is 100. We simulate the error performance of distributed joint source-channel coding of two correlated sources for various rates and correlation parameter. We then compare the memory requirements for obtaining various rates. In our case mother code rate is 0.5. Code of rate 0.8 and 1/3 is obtained by punctur- r = ing and extension, respectively. Fig. 3 gives the BER plot for two different rates {1/3, 0.8} and different values of correlation parameter. These plots clearly show that for given p, any suitable rate r l can be chosen in accordance to the channel condition. As can be seen from the plots the code performance is the about 2 db away from the Shannon s limit. This gap can be further reduced by increasing the code length, which may not be always possible for low cost sensor nodes. Fig. 3. Simulation results for joint source-channel decoding of source X for different value of correlation parameter p. The values in parenthesis beside correlation values are code rates. Table 1 compares the memory requirements with an algorithm based on [10] but using systematic IRA code instead of systematic LDPC code. The numbers are indica- tor values only and are obtained for simulation program running on Matlab. It can be seen that if we start out with a medium rate code, our memory requirements are sig- nificantly less than the case where one starts with lower rate code and uses puncturing only.

6 212 S. Majumder and S. Verma Table 1. Memory utilization (in kilobytes) for encoding and decoding processes at different code rates. Two different rate adaptation methods are compared, our method utilizing both puncturing and extension, and other one uses code puncturing only. Code rate Rate adaptation by puncturing and extension (our method) Rate adaptation by puncturing only (mother code rate is 1/3) Encoder Decoder Encoder Decoder 8/ / / Conclusion We proposed a scheme for lossless distributed source-channel coding of correlated sources with side information using extended and punctured IRA codes. Extension and puncturing is used for achieving the required rate, which in turn depends on correlation between sources and channel condition. The simulation results confirm that proposed method can achieve any required rate for given correlation and channel condition. We have shown that the use of code extension for rate adaptation for joint source-channel coding requires lesser memory than the methods using puncturing only. Thus, besides saving transmission power by adapting to most suitable rate, our method also minimizes the memory requirement for same code rate. References 1. Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley-India, New Delhi (2006) 2. Pradhan, S.S., Ramchandran, K.: Distributed Source Coding Using Syndromes (DISCUS): Design and Construction. In: Proceedings of IEEE DCC, pp (1999) 3. Stankovic, V., Liveris, A.D., Xiong, X., Georghiades, C.N.: On Code Design for Slepian- Wolf Problem and Lossless Multiterminal Network. IEEE transactions on Information Theory 52(4) (2006) 4. Garcia-Frias, J., Zhao, Y.: Compression of Correlated Binary Sources Using Turbo Codes. IEEE Communication Letters 5(10) (2001) 5. Fresia, M., Vandendorpe, L., Poor, H.V.: Distributed Source Coding Using Raptor Codes for Hidden Markov Cources. IEEE Transactions on Signal Processing 57(7) (2009) 6. Liveris, A.D., Xiong, Z., Georghiades, C.N.: Joint Source-Channel Coding of Binary Sources with Side information at the Decoder Using IRA Codes. In: Proceedings of IEEE Workshop on Multimedia Signal Processing, pp (2002) 7. Garcia-Frias, J.: Joint Source-Channel Decoding of Correlated Sources Over Noisy Channels. In: Proceedings of IEEE DCC (2001)

7 Rate Adaptive Distributed Source-Channel Coding Using IRA Codes Zhao, Y., Garcia-Frias, J.: Turbo Compression/Joint Source-Channel Coding of Correlated Sources With Hidden Markov Correlation. Elsevier Signal Processing 86, (2006) 9. Xu, J., Chen, L., Djurdjevic, I., Lin, S., Abdel-Ghaffar, K.: Construction of Regular and Irregular LDPC Codes: Geometry Decomposition and Masking. IEEE Transactions on Information Theory 53(1) (2007) 10. Sartipi, M., Fekri, F.: Distributed Source Coding Using LDPC Codes: Lossy and Lossless Cases with Unknown Correlation Parameter. In: Proceedings of Allerton Conference on Communication, Control and Computing (2005) 11. Varodayan, D., Aaron, A., Girod, B.: Rate-adaptive Codes for Distributed Source Coding. Elsevier Signal Processing 86, (2006) 12. Yue, G., Wang, X., Madihian, M.: Design of Rate-Compatible Irregular Repeat Accumulate Codes. IEEE Transactions on Communications 55(6) (2007) 13. Benmayor, D., Mathiopoulos, T., Constantinou, P.: Rate-Compatible IRA Codes Using Quadratic Congruential Extension Sequences and Puncturing. IEEE Communications Letters 14(5) (2010) 14. Li, J., Narayanan, K.: Rate-Compatible Low Density Parity Check Codes for Capacity- Approaching ARQ Schemes in Packet Data Communications. In: Proceedings of IASTED CIIT, pp (2002) 15. Ryan, W.E., Lin, S.: Channel Codes: Classical and Modern. Cambridge University Press, New York (2009) 16. Hou, J., Siegel, P.H., Milstein, L.B.: Performance Analysis and Code optimization of Low Density Parity-Check Codes on Rayleigh Fading Channels. IEEE Journal on Selected Areas in Communication 19(5) (2001)