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54 ADVANCES IN ELECTRONICS AND TELECOMMUNICATIONS, VOL. 1, NO. 2, NOVEMBER 2010 TABLE II THE RESULTSOF NIST TESTSFOR CPRNG WITH λ = 7, α = 4, K=3 Type of the test R(> 0.981) PT (> 0.0001) Final result Block Frequency 0.9900 0.86288 pass Serial* 0.9870 0.13728 pass Approximate Entropy 0.9920 0.13112 pass Linear Complexity 0.9920 0.68902 pass Universal 0.9850 0.00737 pass Overlapping Templates 0.9900 0.16170 pass Non-overlapping Templates* 0.9820 0.02979 pass Cumulative Sums* 0.9840 0.67661 pass Runs 0.9860 0.04198 pass Longest Runs of Ones 0.9930 0.89348 pass Rank 0.9950 0.96019 pass Spectral DFT 0.9880 0.26757 pass Random Excursions* 0.9865 0.31094 pass Random Excursions Variant** 0.9828 0.09676 pass Frequency 0.9870 0.93900 pass *This test consists of several subtests: the worst result is shown. **The minimum pass rate for this test for a standard set of parameters is approximately 0.978. pliers λ enables us to construct a high-speed pseudorandom generatorwithlongperiodsofgeneratedstreams. Thesimplest method uses a field programmable gate array (FPGA). In this circuit, we implement r CPRNGs with different values of λ that work in parallel. In each step of generation, we obtain r pseudorandomnumbers.Consequently,the speedof producing pseudorandom numbers increases r times. This property can be used in cryptography and in multi-core processors for fast generation of high-quality pseudorandom numbers with long periods. REFERENCES [1] P. Bratley, B. L. Fox, and L. E. Schrage, A Guide to Simulation. New York: Springer-Verlag, 1987, ch. 6. [2] J. E. Gentle, Random Number Generation and Monte Carlo Methods. New York: Springer, 2003, ch. 1. [3] D. E. Knuth, The Art of Computer Programming, 2nd ed. Addison Wesley, 1981, vol. 2, ch. 3. [4] [online], http://csrc.nist.gov/rng/. [5] F. Gebhard, “Generating pseudo-random numbers by shuffling a Fibonacci sequence,” Mathematics of Computation, vol. 21, pp. 708–709, 1967. [6] C. Bays and S. D. Durham, “Improving a poor random number generator,” ACM Trans. on Mathematical Software, vol. 2, pp. 59–64, 1976. [7] M. P. Kennedy, R. Rovatti, and G. Setti, Chaotic Electronics in Telecommunications. Boca Raton: CRC Press, 2000, ch. 3. [8] L. Kocarev, G. Jakimoski, and Z.Tasev, Chaos and Pseudo-Randomness in Chaos Control, 2003, pp. 247–263. [9] T.Kohda and A. Tsuneda, “Statistics of chaotic binary sequences,” IEEE Trans. Inf. Theory, vol. 43, pp. 104–112, Jan. 1997. [10] T. Stojanovski and L.Kocarev, “Chaos-based random number generators – Part I: Analysis,” IEEE Trans. Circuits Syst. I, vol. 48, pp. 281–288, Mar. 2001. [11] M. Jessa, “Designing security for number sequences generated by means of the sawtooth chaotic map,” IEEE Trans. Circuits Syst. I, vol. 53, pp. 1140–1150, May 2006. Mieczyslaw Jessa was born in Poland in 1961. He received the M.Sc. degree with honors from Poznan University of Technology in 1985 and the Ph.D. degree in 1992 from the same University. Since 1985 he has been employed at the Institute of Electronics and Telecommunications in Poznan. Now, he works with the Chair of Telecommunication Systems and Optoelectronics of the same University. Initially, his research interest included phase-locked loops and PDH/SDH network synchronization. In the years 1995-1997 he was an expert of Polish Ministry of Communications in the field of digital network synchronization. His current research concerns randomness and pseudo-randomness, the applications of the chaos phenomenon, and mathematical models of systems evolution. He is the author or co-author of over one hundred journal and conference papers and fifteen patents.
ADVANCES IN ELECTRONICS AND TELECOMMUNICATIONS, VOL. 1, NO. 2, NOVEMBER 2010 55 New Tailbiting Convolutional Codes over Rings Abstract—In this paper a method of using convolutional codes over rings for packet data transmission over additive white gaussian noise (AWGN) channel is proposed. The tailbiting method is generalized and applied to convolutional codes based on ring of integers modulo-M. The codes were named tailbiting codes over ring (TBR). This paper presents a method to desing TBR codes obtained by the concatenation of feedback convolutional encoder over ring and M-QAM modulator. The paper describes how a systematic ring convolutional encoder with feedback can obtain the same starting and ending state. The best TBR codes with different number of encoder states for 16-QAM modulated symbol sequences of varying lengths are tabulated. Index Terms—Convolutional codes over rings, tailbiting codes Piotr Remlein and Dawid Szłapka I. INTRODUCTION P ACKETdatatransmissionschemesareoftenusedinwireless telecommunication systems. The convolutionalcodes are used in such systems as an efficient and powerful class of error correcting codes [1]. To be able to use convolutional codes (of rate R = k/n and m memory elements) in the packet transmission, we must convert these codes to block codes. There are some methods for this conversion. One of such methods is called tailbiting. In this method, no additional bits are appended to the codeword to drive the encoder to a known state [2]. The encoder starts and finishes the encoding process in the same state but this state is not known by the decoder. In the paper [3] it is shown that the turbo-codes generally provide the best error rate performance for long blocks(over150bits), but forshort blocks(under150bits) the tailbiting convolutional codes provide the best performance. The motivation for investigating the ring convolutional codes was to explore a natural relation between M-ary modulation and codes over the rings of integers modulo-M [4]. Up to now, the best tailbiting codes with the greatest minimum Hamming distance were published in the literature [2], [5], [6]. In case of search for the best convolutional codes for the signals transmitted over the AWGN channel, the quality the criterion is Euclidean distance [1]. In this article we assumed the Euclidean distance as a parameter to estimate the quality of TBR codes. To find the best TBR codes, one can search the full space of the codes. Such method gives the certainty that the found codes are the best. The fault of this method is the exponentially growing complexity with the growing number of memory cells of the encoder and the number of its inputs. In this paper we analysed the tailbiting codes over rings encoded by systematic ring convolutional encoders with feedback. We present the results of the search for the best convolutional encoders over ring modulo-4 with code rate R = 1/2 Piotr Remlein and Dawid Szłapka are with Poznan University of Technology, Faculty of Electronics and Telecommunications, Piotrowo 3a; 60-965 Poznan (Poland). used with 16-QAM modulation, with labeling as in [7]. We found new TBR codes for 16-QAM modulation with the best Euclidean distance. This paper is organised as follows. Section 2 describes the procedure of encoding TBR by using the systematic convolutionalencoderswith feedback.In Section 3 we present the results of computersearch for the best TBR codes. Finally, Section 4 gives the concluding remarks. II. TAILBITING CODES OVER RINGS – ENCODING METHOD In this article we generalize the tailbiting method onto convolutional codes over rings of integers modulo-m [2], [3] and we name the resultingcodestailbiting convolutionalcodes over rings (TBR). In the proposed method we encode and decode a block of N (M-ary) symbols without a known tail, thus keeping the effective rate of transmission equal to the code rate. This is done by letting the encoder start and end in the same state, unknown for the decoder. The encoding procedure to achieve this is not difficult if the structure of the encoder is feedforward. In this case, the starting state depends on the m last information symbols in the transmited packet, wherem is the numberof memorycells in the encoder.Incase of convolutional encoder with feedback Fig. 1, the starting state depends on all the information symbols in the packet. Finding the initial state wherein the encoder should start encoding and – after N symbol intervals – end encoding in the same state is complex. One of the methods for finding this initial state was proposed in [8] and extended for multilevel codes in [9]. InFig.1,weshowtherealizationofthesystematicfeedback convolutional encoder over ring of integers modulo-M [4], [6] with the code rate R = k/n, n = k + 1. At time t, the information vector Ut with M-ary elements u (i) t belonging to the ring ZM = 0, 1, 2, ..., M − 1, (ℜ = ZM) inputs the encoder. Ut = (u (1) t , u (2) t , ..., u (k) t ) (1) The convolutional encoder produces a coded sequence of symbols which belong to the same ring ZM Vt = (v (1) t , v (2) t , ..., v (n) t ) (2) where n = k + 1. The coefficients in the encoder Fig. 1 are taken from the set 0, ..., M − 1. The memory cells are capable of storing ring elements. Multipliers and adders perform multiplication and addition, respectively, in the ring of integers modulo-M. The encoding process can be described as mapping of the information vector (1) into the encoded vector (2)
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