Zipping Characterization of Chaotic Sequences Used in Spread Spectrum Communication Systems

Transcription

1 Zipping Characterization of Chaotic Sequences Used in Spread Spectrum Communication Systems L. De Micco, C. M. Arizmendi and H. A. Larrondo Facultad de Ingenieria, Universidad de Mar del Plata (UNMDP). Juan B. Justo 432 (76) Mar del Plata, Argentina. Abstract. In this paper finite sequences, known as Pseudo Noise codes (PN-codes) are studied. PN-codes are used in Spread Spectrum Digital Systems (SSDS), a widely employed wireless communication technique. The selection of the best "set of PN-codes" (PN-family) is an important issue in SSDS. This selection is based on a very involved statistical analysis. This paper presents three contributions in this field: the first contribution is the definition of new global quantifiers instead of those used in the engineering literature. The second contribution is to show that a quantifier based on the Zipping Complexity Measure may be a global optimization quantifier of the whole system. Finally a comparison between usual PN-families and chaotic generated PN-families is presented, showing that chaotic systems are competitive in this application field. Keywords: chaos, zipping, spread spectrum, PN-codes PACS: 5.45+b INTRODUCTION Since the introduction of chaos synchronization [] the application of chaos to communication systems has attracted attention and many different systems has been designed. The particular case considered here, SSDS is widely employed nowadays in wireless communications (cell phones, wireless hubs, etc.). As in any other communication system a signal (message) is sent from a transmitter to a receiver. In SSDS the transmitter encode the message (text, voice, image, etc.) as a binary signal. Then a finite length binary sequence (PN-code), that is independent of the encoded message, is employed as a modulation waveform to produce a modulated signal to be transmitted. Each user of the same communication system, has a different PN-code. The set of all the PN-codes employed in the same communication system is known as the PN-family. When a user (the transmitter) wants to send a message to another user (the receiver) it must employ the PN-code assigned to the receiver (pre-agreed code). Each receiver continuously evaluates the correlation between the received signals and the particular PN-code assigned to it. When the correlation exceeds a threshold an incoming call is supposed to exist (this is the detection step) and the receiver looks for the maximum of the correlation between the incoming signal and the receiver PN-code, making a circular shift of it (this is the synchronization step). Then the demodulation and decoding processes start (this is the decoding step). In this way many users can transmit simultaneously sharing the same frequency band but with different PN-codes according to which receiver they want to contact with, and that particular receiver picks up the signal. The name "spread spectrum" comes from the fact that the modulated signal has a spectrum that covers a whole frequency band larger that the one covered by the original encoded message. This is a requirement common to all digital systems in order to reduce the Electromagnetic Interference (EMI) they produce. The influence of the statistical properties of the PN-family on the overall performance of the system is of upmost importance in the design process [2, 3,4] and different PN-families were proposed as is explained in the next section. More recently PN-codes generated by chaotic maps were also considered [5, 6, 7, 8]. Such chaotic systems present a potential advantage over conventional SSDS communication in the opportunity they provide to simultaneously coding and spreading the information signal and they can provide families with a high number of PN-codes. CHAOTIC AND NON CHAOTIC FAMILIES A PN-code is a N-bit binary string and then there exist 2 N number of possible different strings. This number increases exponentially with the number of bits but most families have a considerable lower number of usable PN-codes (family CP93, Nonequilibrium Statistical Mechanics and Nonlinear Physics, XV Conference edited by O. Descalzi, O. A. Rosso, andh. A. Larrondo 27 American Institute of Physics /7/S23. 39

2 members) because of the restrictions imposed by the following optimization requirements:. All the members of a given PN-family must be generated using only one algorithm: this is a hardware restrictions and there are several strategies to produce PN-families reported in the engineering literature. Each strategy gives rise to a different family. Chaotic maps are good candidates because they are very simple to be implemented. 2. Availability of a large number of elements: The number of members depends on the family generation method but usually it is p «2 N. this number determines the maximum number of users sharing the spectrum. Again chaotic maps are good candidates because they can provide a high p. 3. The autocorrelation of each PN-code must be a delta function: this makes each PN-code easily distinguishable from a circular time-shifted version of itself to make the synchronization step faster. This property is also important for reducing the effect of multipath propagation. 4. The cross correlation between any pair of PN-codes must be a null function: each sequence in the family must be easily distinguishable from (a possibly circular time-shifted version of) every other signal in the same family. This requirement is important during the detection step for minimizing the multiuser interference. 5. Spread spectrum: the modulated signal must have a uniform spectrum over the assigned bandwidth. This is a requirement common to all digital systems in order to reduce the Electromagnetic Interference (EMI). The problem is that the global optimization is not easy. For example, the auto-correlation optimization (item 3) is usually attained at the expense of a poorer cross correlation performance (item 4) and vice versa [2]. A similar situation occurs with the spreading properties of the sequence (item 5). Furthermore any optimization reduces the number p of usable sequences in the family (item 2). That is in fact the reason why several families coexist, each family using a different generating algorithm that partially optimizes over some of the requirements mentioned above. Chaotic families studied in this paper have two important advantages: the big number of sequences generated by a chaotic map and the ease of their generation. In this paper they are compared with the commonly used families in the engineering literature. A brief description of each studied family follows and their main characteristics (see section for a detailed comparison between them):. Non chaotic PN-codes studied in this paper are (a) M-sequences: They are the most widely studied PN-codes and they are the basis for other PN-families. Each family is generated by a m-stage shift register with linear feedback according to a primitive polynomial. The length of each PN-code is N = 2 m -. The spreading and autocorrelation properties are almost ideal but the cross-correlation between any pair of codes is a periodic function with high peaks. The number of members p of each family is small. (b) Gold-codes: Gold PN-codes are obtained by means of an exclusive OR operation between two preferred M- sequences of the same family. Preferred M-sequences are pairs with minimal cross correlation. Including the preferred pair, a total of p = 2 m + Gold Codes can be produced from any m-stage feedback shift register. The cross-correlation between any pair of Gold PN-codes is uniform and bounded. Gold PN-families have a large number of members. The spread spectrum properties are not optimal. (c) Walsh-Hadamard codes: These PN-codes have zero cross-correlation for zero circular time-shift, but for a shift different from zero their cross-correlation is very much dependent on the particular pair of codes used. For synchronous systems (where shift is always zero) the correlation properties of these codes are optimal. The spread spectrum does not cover the whole bandwidth, but only a number of discrete frequency components. This family has p=nmembers. 2. Chaotic PN-codes studied in this paper are generated by iterating two chaotic maps: the Three-Way Bernoulli Map (TWBM) and the Four-Way Tailed Shift Map (FWTSM)[5]. The sequences generated by each map are transformed into a binary code by means of the usual symbolics dynamics rule ([,.5) >, (.5,] > ). The number p of family members is limited by the periodicity of the pseudo chaotic digital sequence. Autocorrelation, cross correlation and Spreading Spectrum properties are studied below. GLOBAL PERFORMANCE QUANTIFIERS In this section three new quantifiers are proposed so as to compare the performance of the considered PN-families. The first one C is measure of both the autocorrelation (3) and the cross correlation (4) quality. In view of the decoding procedure the linear correlation properties of each family of PN sequences play a major role in the system efficiency, 4

3 TABLE. ficorr as a function of threshold for different families with N= 27 Cth M-seq Gold.496. ficorr FWTSM TWBM Walsh since they determine the level of multiple access interference, the self-interference due to multipath propagation, and the time required for code synchronization. The second proposed quantifier S measures the spreading of the spectrum over the whole possible frequencies. The third proposed global quantifier Z is proposed as a global parameter of the whole system. All these quantifiers are global, to compare the different families as a whole, it means they include all the members of each family instead of the more traditional approaches that use randomly selected representative members. Comparisons are made between families with members of the same length N. Furthermore the definition of global quantifiers is a first step towards a global optimization process. Global Correlation Quantifier C: this quantifier includes the auto correlation and cross correlation properties of each family. For a perfect detection and synchronization the autocorrelation is required to be a delta function and the cross correlation between the different members of the family must be zero as pointed in section. This is an ideal case but actually a peaked autocorrelation and low cross correlations are the best one can obtain. For this reason it is considered that a good family is one that has a cross correlation lower than the threshold used by the receiver in the detection step. This threshold is a percentage of the autocorrelation value at zero shift. The proposed global quantifier C is calculated as follows: ) Let {JCW, (j = l,...p)} be the members of a given family, each member being a TV-bit binary vector. Calculate correlations between any pair of them, for all possible circular time shifts. { ^N \s\ i j ^ A ZjcJo X l X k+s S > () c^{-s) s< The number of values is N cor r = 2p 2 N. 2) Choose a threshold level c t h defined as a fraction of the maximum autocorrelation located at zero shift AQ. 3) Count the number of elements higher than the threshold level «corr and normalize it: Hcorr WcorrI Ncorr Table shows that n corr is a decreasing function of Q/,. 4) Finally C is the minimum threshold Q/, giving n corr =. In Fig. for the studied families. On the x-axis families are ordered from best (left) to worst (right). Global Spectrum Quantifier S: the PN-codes are periodic and they have always a discrete spectrum but if the modulated signal energy is concentrated over a small number of discrete frequency components the spreading is not very efficient. The ideal case would be a constant spectrum over the whole band. Therefore the proposed quantifier S is calculated as follows: ) Evaluate the magnitude of the Fast Fourier Transform (FFT) for each PN-code (see Fig. 2a ). 2) Normalize each FFT magnitude by dividing it by its mean value. 3) Calculate the variance cr, of each normalized FFT vector, magnitude (see the label on each panel of Fig. 2a). 4) The spectrum quantifier S is S=< cr, >, the mean value over all the members of the family. Each subplot of Fig. 2b shows Oi for a given family, as a function of the N/2 discrete frequencies. The order of the families is from top \^) 4

4 .2 -J.8 Co I P A PN-families o i Gold M-seq A FWTSM o TWBM * Walsh FIGURE. (a) Global Correlation Quantifier C as a function of the threshold c^ for several PN-families withiv:^ 27 to bottom: M-sequences, Gold, FWSTM, TWBM, and Walsh. S is shown as a label and also as an horizontal line over each subplot. The M-sequence family has the lowest S and then this family has the most uniform spectrum. Gold, TWTSM and FWBSM have quite similar values. Finally the Walsh family has the highest S showing that the spectrum is spread over a small number of frequency components. For each family it is possible to delete those codes with high Oi's in order to improve the performance of the whole family. Of course this decision diminishes p, the number of allowed simultaneous users. Zipping complexity quantifier Z. For a string of characters the algorithmic complexity is defined as the length in bits of the smallest program that produces the string as output [9]. The problem with this definition is that it is impossible, even in principle, to find such a program. Nevertheless, the zippers or file compressors are algorithms conceived to do that job at least approximately. The Lempel and Ziv algorithm is used for most zippers and it is one of the best known file compressors [, ]. For a given finite length string the zipping complexity may be defined as lz/l where /, is the string length and lz is the zipped string length. The problem is that the compressing ability of a zipper depends on the string length and also on the order of the PN-codes inside the string. Then for our specific application it is necessary to construct surrogate series to obtain a parameter independent of these two factors. Then in this paper the zipping complexity quantifier Z is then evaluated as follows: ) Construct a binary string consisting of all the members of a given family one after the other. The PN-family with highest p defines the size of the string / = N p. The string corresponding to a family with a lower p is completed by repeating the family PN-codes. 2) Generate p\ surrogates by changing the order of the PN-codes inside the original string in all the possible ways. 3) Zip each surrogate to give z t. The global quantifier Z is given by Z =< z t >, the mean value for all the surrogates. Table 2 shows the results obtained for non chaotic and chaotic families. This parameter is not completely independent on the two previously defined because: - The string of a PN-family with high cross correlations between its PN-codes is more compressible. - A PN-family with a colored spectrum has periodicities and then the string will be more compressible. - A family with a few members must be repeated many times to complete the string, and then the string will be more compressible. Then Z decreases if correlation increases, spreading is not flat and the number of members decreases as Table 3 shows. 42

5 (^3,482 <PS>:292,9499 q, s :3,324 ^ <P,S_>:276,974 o Q. 3 * -AJiw-l^ A 5 5 frequency ^7,7588 <PS>:299,4985 q, s :255 <PS>:32, ^^^Avw^A,^^ <c>=,62828 <G>=2,239 2 k^^v^a^^^y^a^a^a^a^^^^!^s^^^aj**ts\h~j^^ <c>=2, <a>=34,4673 sequence FIGURE 2. (a) Typical spectrum of a PN-code for the different families the mean value and the variance cr, are shown as labels on each subplot; b) Variances a, for the PN-codes of the different families. The value of S is shown as a label on each subplot CONCLUSIONS Three global quantifiers were defined. They are the correlation quantifier C, the spreading quantifier S and the zipping quantifier Z. These quantifiers are global in the sense there is one value for each whole PN-family. The zipping 43

6 TABLE 2. Global Quantifier Z for different families and different lengths family Gold FWTSM TWBM Walsh N P / 2,2 2,2 2,2 2,2 Z M-seq Gold ,32.74 FWTSM ,56. TWBM ,56.3 Walsh , M-seq ,68.22 Gold ,24.96 FWTSM ,24.75 TWBM ,24.75 Walsh ,24.32 TABLE 3. Proposed global quantifiers for different classical and chaotic PN-families family Gold FWTSM TWBM M-seq Walsh N P C S Z quantifier appears as a possible quality measure to compare different families, with the highest Z corresponding to the best family, because Z is higher for strings consisting of correlated PN-codes and it also decreases if the spectrum is not flat. Using the proposed quantifiers classical and chaotic PN-families were studied. The results presented show that chaotic families can have a higher number of members than conventional ones, with equivalent correlation and spreading properties. Furthermore they can be easily implemented and consequently they are good candidates to be used in real systems. ACKNOWLEDGMENTS This work was supported by Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET) (PIP 5569/4), Agencia Nacional de Promotion Cientifica y Tecnologica (PICT 249/4 and PICTO 4-495) and University of Mar del Plata, Argentina. HAL is a CONICET researcher. REFERENCES. L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64(8) (99). 2. L. P. Welch, L.R. IEEE. Trans. Inform. Theory 2(3), (974). 3. D. V. Sarwate and M. B. Pursley Proc. of the IEEE. 68(5), , (98). 4. M. B. Pursley IEEE. Trans. Commun. 25(6), (977). 5. G. Mazzini, G. Setti and R. Rovatti, IEEE Trans. Circuits Sys. 44(), (997). 6. R. Rovatti, G. Mazzini and G. Setti, IEEE Trans. Circuits Sys. 5(7), (24). 7. R. Rovatti, G. Mazzini and G. Setti, IEEE Trans. Circuits Sys. 5(7), (24). 8. M. Sushchtk, L. S. Tsimring and A. R. Volkovskii, IEEE Trans. Circuits Sys. 47(2), (2). 9. J. R. Sanchez, F. Family, C. M. Arizmendi, Phys. Lett. A 249, (998).. J. Ziv and A. Lempel, IEEE Trans. Inf. Theory 22, 75-8 (976).. J. Ziv and A. Lempel, IEEE Trans. Inf. Theory 23, (977). 44