Families of PN sequences must fulfill the following main requirements:

Availability of a of codes per family to be able to havemany users.

: each code in the family must be easilydistinguishable from a time-shifted version of itself to make thesynchronization process faster.This property is also important for multipath robustness; it means toreduce the effect of Multipath propagation.

: each sequence in the family must be easilydistinguishable from (a possibly time-shifted version of) every othersignal in the same family. This is important for minimizing themultiuser interference and to increase the capacity of separation ofsignals of different users

: the modulated information must have a uniformspectrum over the system bandwidth.

!

!

!

!

large number

Autocorrelation

Cross correlation

Spread spectrum

Statistical characterization of pseudorandom sequences used in spread spectrum communication schemeL. De Micco, C. M. Arizmendi, H. A. Larrondo

Facultad de Ingeniería. Universidad Nacional de Mar del Plata Argentina

Since the introduction of chaos synchronization the application of chaos to practical communication systems has attracted attention. The performance of asynchronous direct-sequence spread-spectrum(DS/SS) communication systems, using chaotic sequences as spectral spreading sequences is analyzed. Among the advantages of chaotic sequences the big number of them, the ease of their generation,

as well as their inherent improvement in transmission security. Sequences obtained by replication of a truncated and quantized chaotic time series are compared with classical pseudo noise (PN) andorthogonal sequences, such as gold codes, m-sequences, kasami codes, walsh codes, etc. The tools used for the analysis are the auto-correlation function, the cross-correlation function and the sequences

spectrum. The sequences complexity measured with zippers is studied as a global performance index.

ABSTRACT

The communication sistem performance is characterized by:

! Average signal-to-noise power ratio (SNR)! Binary error rate (BER)! Code synchronization! Multipath signal rejection! Number of simultaneous users (system capacity)

Spreading PN codes statistical properties play an important role in theimplementation of DS/CDMA systems and the selection of the codedirectly influences the system performance.

Any CDMA communication system uses one or several families of PNsequences. Each sequence in the family is used by one user andthen the number of simultaneous users is limited by the number ofavailable codes.

In view of the decoding procedure the correlation properties of eachfamily PN sequences play a major role in the code efficiency forDS/CDMA systems, since they determine the level of multiple accessinterference, the self-interference due to multipath propagation, and thetime required for code synchronization.

System Performance

It is a transmission technique in which a PN code, independent of the information data, is employed as a modulation waveform to “spread” the signal energyover a bandwidth much wider than the signal information bandwidth.At the receiver end the signal is “despread” using a synchronized replica of the PN code.

In Code Division Multiple Access (CDMA) systems with direct sequence different users transmit simultaneously over the same spectral band and aredistinguished only by means of their PN code.

Correlation properties of PN codes are essential for the success of the decoding process. The receiver makes the correlation between the particularsequence it uses and the received signals. When this correlation grows over a threshold the receiver synchronizes and decodes the signal.

DEFINITION OF SPREAD SPECTRUM (SS):

The problem is that optimization of the auto-correlation properties of thesequences in a given family is attained in general at the expense of poorer

cross-correlation properties and vice versa. Thus sequences in sets which havevery good cross-correlations properties usually have poorer auto-correlation

properties. A similar situation occurs with the spreading properties of thesequence. That is the reason why several families exist and each family

optimizes some of the requirements mentioned above.

Chaotic Spreading CodesChaotic sequences studied in this paper are obtained by iterating two chaoticmaps:the Three-Way Bernoulli Map (TWBM) and Four-Way Tailed Shift Map(FWTSM)

and then quantizing and truncating the values in order to obtain binarysequences. The number of family members is not limited in chaoticspreading codes.

pFig. 5 TWBM Fig. 6 FWTSM

PERFORMANCE QUANTIFIERS PROPOSED IN THIS PAPER:

All the performance quantifiers we propose in this paper include all the members of each family instead of the more traditional approaches that use randomly selected representative members. The purpose is to compare the different families as a whole.Comparisons are made between families with members of the same length .n

! M-sequences! Gold-codes! Kasami-codes.! Walsh-Hadamard codes

Kasami sequences

Walsh-Hadamard codes

A more involved procedure is used to generate Kasamisequences by means of preferred M-sequences. The mainadvantage of Kasami sequences is their very low cross-correlation.

There are two different sets of kasami sequences.The small set of kasami, using shift registers = 2 binarysequences of period = 2 -1 can be generated.The large set of kasami sequences again consists of sequencesof period = 2 -1, for even, and contains both the Goldsequences and the small set of Kasami sequences as subsets.All the values of cross correlations and autocorrelations frommembers of this set are limited to five values.

For the same length the number of different codes per family islarger in Kasami sequences than in Gold codes.

These sequences have zero cross-correlation for zero shift, buttheir cross-correlation is very much dependent on the particularpair of codes used, for a shift different from zero. For synchronoussystems this codes are optimal.

The codes do not have a single, narrow autocorrelation peak.The spectrum spreading does not cover the whole bandwidth, butonly a number of discrete frequency components. For thesefamily there are = members.

m p

n

n n

p

p n

m

m

m

/2M-sequences

Gold-codes

The Maximum-length sequences or M-sequences are the most widely knownPN codes.They are generated by a -stage shift register with linear feedback accordingto a primitive polynomial.

They are sequences of length = 2 - 1. Each value defines a differentfamily. Their main properties are:The spectrum spreading is almost optimal.The autocorrelation is almost ideal. It has only a peak value when both signalsare correctly aligned and a minimum level when both signals are shifted.The cross-correlation between any pair of codes is a periodic function havinghigh peaks.The number of different M-sequences for any particular is small.

Gold sequences are useful because of the large number of codes they supply.They can be chosen so that over a set of codes available from a givengenerator the cross-correlation between the codes is uniform and bounded.Gold codes are constructed by means of an EXOR operation between twopreferred M-sequences of the same family. Preferred M-sequences are pairswith minimal cross correlation. Including the preferred pair, a total of = 2 + 1Gold Codes can be produced from any -stage feedback shift register.

m

n m

m

pm

m

m

SOME KNOWN FAMILIES OF SPREADING CODES:

Correlation performance quantifier C:

The receiver correlates the received signal with the locally generated PN sequence to produce a measure of similarity between them (detection).This measure is then compared to a threshold to decide if there is an incoming signal or not. Once an incoming signal is detected the receiversynchronizes with it (synchronization).For proper detection and synchronization it is required a peaked autocorrelation and a low cross correlation between family members to prevent falsedetections.For this reason a “good” family is the one that has a cross correlation with no peaks greater than a percent of the autocorrelation value at zero shift.

The method consists on:

1. Calculate all the cross correlations between any pair of sequences of a given family, for all possible shifts. (See Fig.)

2. Choose a threshold level cth defined as a fraction of the autocorrelation at zero shift A , and evaluate , the number of

pairs of sequences having at least one correlation value over the threshold level.

3. is obtained dividing by .

0 N

C N

Table 1 shows theobtained values forc l a s s i c a l a n dchaotic families withperiod = 127.n

The best family is Gold because we have = 0 for very lowthreshold levels. M-sequence and the chaotic FWTSM present asimilar performance. The chaotic TWBM sequence is a littleworst, and finally the Walsh sequence has values of crosscorrelation higher than the maximum threshold level.

C

Fig. shows a more detailed information. Each subfigurecorresponds to a family. The graph shows all the cross-correlationsbetween all different members at different shifts. The x axis is theshift. The red lines are different threshold levels ( = 0.8, 0.4 and0.2). We can see that the Gold and M-sequences families have thelowest . The FWTSM and TWBM behave quite similar, and theWalsh family is the worst one.

2

C

S p e c t r u m q u a n t i f i e r S:

All the PN codes are periodic and their spectra are discrete. If the signal energy is concentrated over a small number of discrete frequency components the spreading is notvery efficient.The ideal case would be a constant spectrum over the whole band.Therefore we have calculated the normalized spectrum and it variance for each member of each family.Asmall variance indicates a more uniform spectrum.

The steps are: 1. Evaluate the FFT magnitude for each sequence of the family.

2. Calculate the mean value of each FFT magnitude.

3. The normalized FFT magnitude will be the FFT magnitude divided by its mean value.

4. Calculate the variance of the normalized FFT magnitude.

5. The spectrum quantifier is the mean value of with over all the members of the family.

i

s

S s i

i

i

In Fig. 3 we have plotted the variance of eachcode of the family as a blue dot and the red line isthe family variance mean value .From the graph we can see that the M-sequencefamily has the lowest and then this family has themost uniform spectrum. Gold, TWTSM andFWBSM have quite similar values. Finally theWalsh family has the highest showing that thespectrum is spread over a small number offrequency components.

It is also possible to delete codes with high valuesof variance in order to improve the performance

of the whole family.Of course this decision diminishes the number ofallowed simultaneous users.

S

S

S

si

Zipping complexity quantifier Z:

1) Construct a text file with a string consisting of all the members of a given family one afterthe other separated by return characters.

2) Zip the file using a file compressor (we used winzip© 9.0).

3) Find the ratio Z between the compressed file and the original file.

This quantifier is defined in the following way:

Table 2 shows the results obtained for classical and chaotic families.In this table we can see in the zip/orig column the higer values indicating the bestsfamilies.

Aglobal comparison between families is shown in Table 3.

CONCLUSION:

We have analyzed sequences obtained by repeating a truncated and quantizedchaotic time series and compared with classical sequences by means of some

performance indexes. This analysis reveals that, unlike conventionalsequences, chaotic spreading codes can be generated for any number of users

and allocated bandwidth.Results confirm the fact that chaos-based DS/CDMA outperforms classical

DS/CDMA.It is known that the compressing ability of any zipper depends not only on the

complexity of the zipped string but also on the file size. The order of thesequences inside the file is also important. Then it is necessary to constructsurrogate series to normalize this parameter . Surrogates are obtained by

changing the order of the family members inside the text file in all the possibleways. Z is the mean value for all the surrogates.

Z

Fig. 3 Fig.4

Fig. 1

Table 1

Fig. 2

cth/127Table 2

Table 3

Figure 1 shows the values ofthreshold level at witch the crosscorrelations of the families becomehigher.

Fig. 7