10 - Properties and applications of digital filters with nonlinearities

Abstract

This chapter elaborates the properties and applications of digital filters with nonlinearities. A possibility of applying digital filters associated with two's complement arithmetic in the security communication is presented. It is found that when the pole of first-order digital filters associated with two's complement arithmetic is between 1 and 2, the system matrix is unstable and so the digital filters exhibit chaotic behavior. One would expect that the state variables might reach any value between the maximum and minimum numbers in the possible region, and uniform probability distribution of the state variable is obtained. It is found that although the first-order digital filter associated with two's complement arithmetic exhibits random-like chaotic behavior, the possibility of occurrence of the state variable in the region is close to zero. It is observed that to explore the statistical property of state variables and symbolic sequences, Shannon entropies are employed. It is found that since the state variables will converge to zero, and no overflow will occur if both the eigenvalues of the second-order digital filters associated with two's complement arithmetic are real and inside the unit circle, the Shannon entropies of the symbolic sequences are therefore zero. The computer cryptography through digital filters associated with nonlinearities is also elaborated in the chapter.