Generation of Orthogonal Logistic Map Sequences for Application in Wireless Channel and Implementation using a Multiplierless Technique KATYAYANI KASHYAP 1, MANASH PRATIM SARMA 1, KANDARPA KUMAR SARMA 1, NIKOS MASTORAKIS 2 Electronics and Communication Engineering Gauhati University 1, Technical University of Sofia 2 Guwahati-781014, Assam 1 ; Bulgaria 2, INDIA 1, ITALY 2 kashyap.katyayani@gmail.com, manashpelsc@gmail.com, kandarpaks@gmail.com Abstract: - In wireless communication system spread spectrum modulation plays a vital role. Therefore generation of spreading sequence is one of the crucial issues in such type of modulation technique. For secure communication, chaotic spreading sequence is advantageous. This paper presents a design of orthogonal chaotic spreading sequence for application in Direct Sequence Spread Spectrum (DS SS) system for a faded wireless channel. Enhancing the security of data transmission is a prime issue which can better be addressed with a chaotic sequence. Generation and application of orthogonal chaotic sequence is done and a comparison with non orthogonal chaotic sequence is presented. The comparison factors are bit error rate (BER), mutual information, computational time and signal power, which finally dictates the efficiency of generated code. Again, a novel multiplier-less chaotic sequence generator is proposed for lower power requirement than the existing one. Key-Words: - Chaotic code, Orthogonal chaotic code, DS SS. 1 Introduction Spread spectrum is a most important component of code division multiple access (CDMA) system. The consideration behind spread spectrum is to use more bandwidth than the original message while maintaining the same signal power and is masked by noise. This makes the signal more difficult to distinguish from noise and therefore more difficult to jam or intercept [1]. There are two predominant techniques to spread the spectrum, one is the frequency hopping (FH) technique, which makes the narrow band signal jump in random narrow bands within a larger bandwidth. Another one is the direct sequence (DS) technique which introduces rapid phase transition to the data to make it larger in bandwidth. Some of the spreading codes are pseudo noise (PN) code / Gold code etc. A PN code is one that has a spectrum similar to a random sequence of bits but is deterministically generated [2]. A Gold code is a type of binary sequence, used in telecommunication (CDMA) and satellite navigation (GPS). Gold codes have bounded small crosscorrelations, which is useful when multiple devices are broadcasting in the equal frequency range. Gold code sequences are consists of 2 n -1 sequences each one with a period of 2 n -1 [2]. But PN code and Gold code are limited to sequence length. Again flexibility is also poor because for same sequence length we cannot generate multiple numbers of sequences. Therefore, an efficient spreading sequence is generated using chaotic logistic map in faded environment and also a orthogonal spreading sequence is also generated and a comparison between orthogonal and non orthogonal is presented with the help of bit error rate (BER), computational time and mutual information and signal power. Also a scheme for generating a multiplierless logistic map equation is done for lower power requirement than the existing one. This paper presents a design of orthogonal chaotic spreading sequence for application in Direct Sequence Spread Spectrum (DS SS) system for a faded wireless channel. Enhancing the security of data transmission is a prime issue which can better be addressed with a chaotic sequence. Generation and application of orthogonal chaotic sequence is done and a comparison with non orthogonal chaotic sequence is presented. The comparison factors are bit error rate (BER), mutual information, computational time and signal power, which finally dictates the efficiency of generated code. Again, a novel multiplier-less chaotic sequence generator is proposed for lower power requirement than the existing one. Therefore, the proposed sequences can be effectively used as ISBN: 978-960-474-374-2 292
spreading sequences in high data rate communication. Binary sequence is generated using logistic map as shown in the Fig. 1. The paper is organized as follows: Section 1 provides an introduction. Section 2 describes Logistic map. Section 3 states about the proposed method and simulation. Section 4 describes results. Finally, conclusion and discussions are summarized in section 5. 2 Logistic Map A logistic map is a polynomial mapping having a degree of 2. It gives the idea of how a very complex, chaotic behaviour can occur from very simple nonlinear dynamical equations. Prediction thus becomes impossible, and then the system behaves randomly [3], [4]. Mathematically, the logistic map is written as 1 (1) where denotes a number between zero and one, (at year 0). The logistic map behaviour is totally dependent on, which is clearly seen in the Table 1. Table 1. Behaviour dependent on Range of r Behaviour of Between 0 and 1 Independent of the initial Between 1 and 2 Independent of the initial Between 2 and 3 Fluctuate around the value 1 for some time Greater than 3 Dependent of the initial 3 Proposed Approach Here, we describe the proposed system. Initially we generate a binary sequence from the logistic map generator by using two different methods and then make it orthogonal, after that use it as part of a DS SS system. The performance of the system is compared by using non orthogonal one. 3.1 Binary spreading sequence generation using logistic map There are two proposed methods to generate the binary spreading sequence using logistic chaotic map. These are Binary sequence generation using thresholding method. Binary sequence generation using floating point to bit conversion method. Fig.1: Flow diagram of both thresholding method and integer to bit conversion method Let x be the floating point valued chaotic sequence. For transforming this floating point valued x sequence to binary sequence we define a threshold function as shown below. 1; 0.5 0; 0.5 Using this function we can generate binary sequence from logistic map. Again, the Fig. 1 shows the binary sequence generation using floating to binary converter. Here, floating point values are converted to integer and then integer to binary conversion is done. Here to convert integer to binary, frame based sequence is used. The generated sequences for three different values of r are shown in the Table 2. ISBN: 978-960-474-374-2 293
Table 2. Sequences generated by logistic map for r=3.61, 3.65 and 3.69 3.3 Generation of multiplierless binary sequence using logistic map Value of r Binary sequence 3.61 00111000011010010100 3.65 01001001010010010101 3.69 01101001001110000110 3.2 Generation of orthogonal binary spreading sequence using logistic map The flow diagram to generate orthogonal chaotic sequence is shown in the Fig. 2. Fig. 3: Flow diagram of multiplierless system Flow diagram of multiplier less system is shown in the Fig. 3. In the equation of logistic map there are two multiplication factors, in multiplierless chaotic sequence generator one of the multiplication is replaced by repetitive addition to reduce the amount of power required. The generated sequence have the same property as the existing one because the generated sequence is same as that of the existing one, just one of the multiplication is replaced by repetitive addition for lower power requirement. By using XPE (xilinx power estimator ) power is measured and it is found that multiplierless logistic map required 1.448 W and existing logistic map required 1.579 W. Fig. 2: Flow diagram of orthogonal chaotic sequence generation 3.4 System Model To generate orthogonal chaotic sequence the first step is to make equal numbers of 0 s and 1 s. After that alternate bits are complemented and then the alternate complemented bits and the bits having equal numbers of 0 s and 1 s are Exclusive-ORed. After Exclusive-ORing the generated bit sequence is orthogonal in nature. Fig. 4: System model ISBN: 978-960-474-374-2 294
The overall system model is shown in the Fig. 4. This is a DS SS system. At transmitter side, first the data is generated from a random source. It consists of a series of ones and zeros. Data input bits are converted into symbol vector using modulation. Modulation scheme used to map the bits to symbols used in this work is BPSK. The modulated data is spreaded by a chaotic code in the transmitter side. Orthogonal chaotic code is generated using logistic map as described above. Thus a new chaotic sample is generated. Next, the signal is passed through a channel, which has Rician fading characteristics. In Rician fading, there is only one direct component, and all other signals reaching the receiver are reflected. AWGN noise to added to the signal. Fig. 5 shows that BER curves using orthogonal chaotic sequence and non orthogonal chaotic sequence in DS SS system. The BER curve for orthogonal chaotic sequence is more close to theoretical vale than the non orthogonal one. That means orthogonal chaotic sequence gives better results than non orthogonal one in case of BER. Again it is clear from the Fig. 6 that orthogonal chaotic sequence outperforms non orthogonal one as regards mutual information is concerned. Also computational time of the system model is calculated for both orthogonal and non orthogonal chaotic sequence. Table 3 shows that system model using orthogonal chaotic sequence requires more computational time than non orthogonal chaotic sequence. The final step of the communication system is the received signal. The received signal is first despreaded using a replica of the chaotic signal used at the transmitter side of the system. The received signal is demodulated using BPSK demodulator. Finally, BER is calculated between the transmitted bit and received bit. 4 Results and Discussions Here, we briefly include the results. These are described in terms of BER, computational time, mutual and signal power. 4.1 BER, Mutual Information and Computational Time (CT) Fig. 6: Mutual information for orthogonal and non orthogonal chaotic code in DS SS system Table 3: CT of the system model using orthogonal and non orthogonal chaotic sequence CT using orthogonal CT using non orthogonal chaotic code chaotic code 1.98 seconds 1.66 seconds 4.2 Signal Power MATLAB FUNCTION RMS dbm Power in dbm Fig. 7: Block diagram of signal power calculation Fig. 5: BER curve for orthogonal and non orthogonal chaotic sequence Signal power is a important factor in communication system. Fig. 7 shows the block diagram of signal power calculation. At first ISBN: 978-960-474-374-2 295
MATLAB function is written for both orthogonal and non orthogonal DS SS system, then root mean square (RMS) value is calculated after that it is converted to dbm using dbw converter. In dbw converter a load resistance is required. Here, for both orthogonal and non orthogonal system the load resistance is assumed to be 50 ohm. For non orthogonal DS SS system signal power is found to be 20.56 dbm and for orthogonal DS SS system the signal power is 18.43 dbm. Therefore, the orthogonal DS SS system requires less signal power than the non orthogonal DS SS system. 5 Conclusion This study has presented a chaos based secure DS SS communication system which is based on a novel combination of the orthogonal chaotic DS SS and non orthogonal chaos sequences. BER performance of the proposed system are described and investigated by means of the computational analysis and numerical simulation. It can be seen from the obtained results, that the orthogonal chaotic system gives better results than the non orthogonal one, due to the interference rejection, anti jamming, fading reduction, and low probability of interception and also it requires less signal power than the non orthogonal one. But computational time requirement is more in orthogonal chaotic sequence than the non orthogonal one. But for security purpose the proposed system maintains an approximately good performance than the non orthogonal one, also achieves significant improvement in case of mutual information also. All these features make the proposed system feasible and robust for the security required and DS SS based digital communication system. The 3 rd World Congress on Intelligent Control and Automation, Vol.4, pp. 2464-2467, Jun. 28-Jul. 2, 2000. [5] A. J. Lawrance and R. C.Wolff. Binary Time Series Generated by Chaotic Logistic Maps. International Journal of Bifurcation and Chaos, Vol. 3, pp. 529-544. Jun, 2003. [6] V. Patidar, K. K. Sud and N. K. Pareek. A Pseudo Random Bit Generator Based on Chaotic Logistic Map and its Statistical Testing. International Journal of Modern Physics, Vol. 251, pp. 441-452, 2009. [7] B. T. Krishna, Binary Phase Coded Sequence Generation Using Fractional Order Logistic Equation. Circuits, Systems, and Signal Processing, Vol. 31, pp. 401-411, Apr. 7, 2011. [8] P. Chengji and W. Bo, New Optimal Design Method of Arbitrary Limited Period Spreading Sequences based on Logistic Mapping. 4 th International Conference on Signal Processing Systems, pp. 246-250, Vol. 58, 2012. [9] C. Fatima and D. Ali. New chaotic binary sequences with good correlation property using logistic maps. IOSR Journal of Electronics and Communication Engineering (IOSR-JECE), Vol. 5, pp. 59-64, Mar. - Apr. 2013. [10] H. Khanzadi, M. Eshghi and S. E. Borujeni, Design and FPGA Implementation of a Pseudo Random Bit Generator Using Chaotic Maps. IETE Journal of Research, Vol. 59, pp. 63-73, Sept. 26, 2013. References: [1] T. S. Rappaport, Wireless Communications - Principles and Practice, Pearson Education, 2nd Ed., 1997. [2] D. Tse and P. Viswanath, Fundamentals of Wireless Communications, Cambridge, 2nd. Ed., 2005. [3] G. V. Reddy. Performance Evaluation of Different DS-CDMA Receivers Using Chaotic Sequences. International Conference on RF and Signal Processing Systems, Vol. 32, pp. 49-52 Aug., 2007. [4] Z. Xueyi, J. Lu,W. Kejun and L. Dianpu. Logistic-Map Chaotic Spreading Spectrum Sequences Under Linear transformation. ISBN: 978-960-474-374-2 296