Design of Logistic Map-Based Spreading Sequence Generation for Use in Wireless Communication

Introduction

Over the last decades there has been an exponential growth in wireless communication systems. Among many techniques develop for communication through the wireless medium, spread spectrum modulation (SSM) finds an important place in wireless communication due to many striking features like robustness to noise and interference, low probability of intercept, application to Code Division Multiple Access (CDMA) and so on. The idea behind SSM is to use more bandwidth than the original message while maintaining the same signal power. A spread spectrum signal does not have a clearly distinguishable peak in the spectrum. This makes the signal more difficult to distinguish from noise and therefore more difficult to jam or intercept. 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 (Rappaport, 1997). Pseudo noise (PN) sequence, Gold code etc are the spreading codes which play a prominent role in SSM techniques. There are used as spreading code in SSM. A PN code is one that has a spectrum similar to a random sequence of bits but is determinately generated (Tse & Viswanath, 2005). A Gold code is a type of binary sequence, used in telecommunication primarily in CDMA and satellite navigation system like GPS (Tse & Viswanath, 2005). Gold codes have bounded small cross-correlations within a set, which is useful when multiple devices are broadcasting in the same frequency range. A set of Gold code sequences consists of 2ⁿ-1 sequences each one with a period of 2ⁿ-1. But PN and Gold codes are limited to fixed sequence lengths with a system configuration. Again flexibility is also poor because for same sequence length we cannot generate multiple numbers of sequences. Traditionally, to generate PN sequence, linear feedback shift register (LFSR) and certain sum-store blocks are used. Since Gold code is generated by doing exclusive-OR of two PN sequences, here also LFSR is required. Therefore, to generate PN or Gold code, a definite physical structure is required which consume significant power. The fixed length of LFSRs improve further constraints. The PN sequence length become confined within the LFSR size. In fading situations or in conditions where there are variations in the propagation medium, a varying length PN sequence shall be more suitable than a fixed length one primarily to use the advantages of SSM to counter detrimental effects observed in wireless channels. Continuous researches are going on to design devices that save power and demonstrate dynamic behaviour with respect to channel conditions required. Therefore, in this chapter an efficient spreading sequence generation method is presented using chaotic logistic map. The sequence thus generated is used as part of a SSM system designed for application method in faded environment. A logistic map is a polynomial mapping having a degree of 2. It gives the idea of how complex the generated sequence is. The chaotic behaviour arises from simple non-linear dynamical equations. Chaos happens when a small difference initially in the system leads to very big difference in the final state. A small error initially could lead to a big one in the final state. Prediction thus becomes impossible, and then the system behaves randomly (Reddy, 2007). Therefore, logistic map sequences have several advantages over Gold code. First, flexibility is more because the period of logistic map sequence is no longer limited to 2ⁿ-1 like Gold code For same spreading code length we can generate extended number of spreading sequences, which is not possible in case of Gold code. Next, bit error rate (BER) performance in Direct Sequence Spread Spectrum System (DS SS) for different spreading code lengths and for different modulation schemes (BPSK, QPSK, DPSK) using logistic map code is better than Gold code. Further, computational times using different spreading code lengths and also using BPSK and QPSK modulation schemes are lesser than Gold code as observed in implementations. Also, mutual information (for different spreading code lengths and for different modulation schemes) are better than Gold code. Performance of the proposed sequence with SSM is compared with Gold code. The comparison parameters are bit BER, mutual information and computational time. The generated dynamic and recursive logistic map sequences have moderate correlation property. Therefore, the proposed sequences can be effectively used as spreading sequences in high data rate modulation schemes. Again the general form of logistic map equation contains two multiplication terms which may yield more area and power. We further modify the logistic map sequence generator to a multiplierless system which lead to less area and less power consumption than the existing logistic map systems. Thus, the proposed system is an efficient sequence generator suitable for wideband systems demonstrating lower BER levels, computational time and power requirements compared to traditional LFSR based approaches.