Discrete time domain analysis of chaos-based wireless communication systems with imperfect sequence synchronization

Problem of sequence synchronization in chaos-based direct sequence spread-spectrum (DS-SS) systems and code division multiple access systems (CDMA) has been widely investigated. However, no mathematical expressions have been derived in closed form for the bit error probability in these systems when they operate with the imperfect time synchronization (represented by a delay between the received and reference spreading sequence generated in the receiver). Precise derivatives for this bit error probability are necessary to quantify the effect of imperfect synchronization on the overall properties of the system. To implement a random delay between the received and the receiver reference sequence, all signals in this paper are represented in the discrete time domain. To represent finite and random discrete delays between the sequences (which occur in a limited interval) the Gaussian and uniform probability density functions are expressed in discrete form. Furthermore, due to the finite value of possible random discrete delays, the expressions of related truncated density functions are expressed in closed form. Following this approach, the expressions for the bit error probability in closed form for chaotic and random spreading sequences have been derived.