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Bingo Wing-kuen Ling - One of the best experts on this subject based on the ideXlab platform.

Peter Kwong-shun Tam - One of the best experts on this subject based on the ideXlab platform.

  • nonlinear behaviors of first and second order complex digital filters with two s Complement Arithmetic
    IEEE Transactions on Signal Processing, 2006
    Co-Authors: Bingo Wing-kuen Ling, Peter Kwong-shun Tam
    Abstract:

    For first-order complex digital filters with two's Complement Arithmetic, it is proved in this correspondence that overflow does not occur at the steady state if the eigenvalues of the system matrix are inside or on the unit circle. However, if the eigenvalues of the system matrix are outside the unit circle, chaotic behaviors would occur. For both cases, a limit cycle behavior does not occur. For second-order complex digital filters with two's Complement Arithmetic, if all eigenvalues are on the unit circle, then there are two ellipses centered at the origin of the phase portraits when overflow does not occur. When limit cycle occurs, the number of ellipses exhibited on the phase portraits is no more than two times the periodicity of the symbolic sequences. If the symbolic sequences are aperiodic, some state variables may exhibit fractal behaviors; at the same time, irregular chaotic behaviors may occur in other phase variables

  • Nonlinear Behaviors of First and Second-Order Complex Digital Filters With Two's Complement Arithmetic
    IEEE Transactions on Signal Processing, 2006
    Co-Authors: Bingo Wing-kuen Ling, Peter Kwong-shun Tam
    Abstract:

    For first order complex digital filters with two’s Complement Arithmetic, it is proved in this paper that overflow does not occur at the steady state if the eigenvalues of the system matrix are inside or on the unit circle. However, if the eigenvalues of the system matrix are outside the unit circle, chaotic behaviors would occur. For both cases, a limit cycle behavior does not occur. For second order complex digital filters with two’s Complement Arithmetic, if all eigenvalues are on the unit circle, then there are two ellipses centered at the origin of the phase portraits when overflow does not occur. When limit cycle occurs, the number of ellipses exhibited on the phase portraits is no more than two times the periodicity of the symbolic sequences. If the symbolic sequences are aperiodic, some state variables may exhibit fractal behaviors, at the same time, irregular chaotic behaviors may occur in other phase variables

  • Normalized Histogram of the State Variable of First-order Digital Filters with Two's Complement Arithmetic
    International Journal of Bifurcation and Chaos, 2005
    Co-Authors: Bingo Wing-kuen Ling, Peter Kwong-shun Tam
    Abstract:

    In this Letter, the normalized histogram of the state variable of first-order digital filters with two's Complement Arithmetic is investigated. When the pole of the digital filter is between 1 and 2, it is found that the possibility of occurrence of state variable in certain regions is close to zero no matter what is the initial condition. Some analytic results are given to account for this phenomenon.

  • DETECTION OF CHAOS IN SOME LOCAL REGIONS OF PHASE PORTRAITS USING SHANNON ENTROPIES
    International Journal of Bifurcation and Chaos, 2004
    Co-Authors: Bingo Wing-kuen Ling, Peter Kwong-shun Tam
    Abstract:

    This letter demonstrates the use of Shannon entropies to detect chaos exhibited in some local regions on the phase portraits. When both eigenvalues of the second-order digital filters with two's Complement Arithmetic are outside the unit circle, the Shannon entropies of the state variables are independent of the initial conditions and the filter parameters, except for some special values of the filter parameters. At these special values, the Shannon entropies of the state variables are relatively small. The state trajectories corresponding to these filter parameters either exhibit random-like chaotic behaviors in some local regions or converge to some fixed points on the phase portraits. Hence, by measuring the Shannon entropies of the state variables, these special state trajectory patterns can be detected. For completeness, we extend the investigation to the case when the eigenvalues of the second-order digital filters with two's Complement Arithmetic are complex and are inside or on the unit circle. It...

  • Autonomous response of a third-order digital filter with two's Complement Arithmetic realized in cascade form: Research Articles
    International Journal of Circuit Theory and Applications, 2004
    Co-Authors: Bingo Wing-kuen Ling, Wai-fung Hung, Peter Kwong-shun Tam
    Abstract:

    In this letter, results on the autonomous response of a third-order digital filter with two's Complement Arithmetic realized as a first-order subsystem cascaded by a second-order subsystem are reported. The behaviour of the second-order subsystem depends on the pole location and the initial condition of the first-order subsystem, because the transient behaviour is affected by the first-order subsystem and this transient response can be viewed as an excitation of the original initial state to another state. New results on the set of necessary and sufficient conditions relating the trajectory equations, the behaviours of the symbolic sequences and the sets of the initial conditions are derived. The effects of the pole location and the initial condition of first-order subsystem on the overall system are discussed. Some interesting differences between the autonomous response of the second-order subsystem and the response due to the exponentially decaying input are reported. Some simulation results are given to illustrate the analytical results. Copyright © 2004 John Wiley & Sons, Ltd.

Zhugang Yuan - One of the best experts on this subject based on the ideXlab platform.

Tao Shen - One of the best experts on this subject based on the ideXlab platform.

Wing-kuen Ling - One of the best experts on this subject based on the ideXlab platform.

  • Nonlinear Digital Filters - 10 – PROPERTIES AND APPLICATIONS OF DIGITAL FILTERS WITH NONLINEARITIES
    Nonlinear Digital Filters, 2007
    Co-Authors: Wing-kuen Ling
    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.

  • Nonlinear Digital Filters - 5 – AUTONOMOUS RESPONSE OF DIGITAL FILTERS WITH TWO'S Complement Arithmetic
    Nonlinear Digital Filters, 2007
    Co-Authors: Wing-kuen Ling
    Abstract:

    This chapter discusses autonomous response of digital filters with two's Complement Arithmetic. The two's Complement Arithmetic involves a periodic discontinuous nonlinear function, and the dynamics of digital filters could be very complicated even when no input signal is applied. Limit cycle and chaotic behaviors could occur even for second-order digital filters. The second-order real digital filters are represented by a state space model. The necessary and sufficient conditions for the nonlinear system to behave as a linear system after a number of iterations are given. It is found that once the initial conditions—the filter parameters and the input signal—are given, the state variables and the symbolic sequences can be uniquely defined by equations. It is found that the system would behave as a linear system after a number of iterations if and only if the state vector toggles between two points on a particular straight line of the phase plane. It is observed that the state vector toggles between two states in a steady state on a particular straight line. Several conditions for exhibiting eventually periodic state vector are also presented in the chapter.

  • 8 two s Complement Arithmetic in complex digital filters
    Nonlinear Digital Filters#R##N#Analysis and Applications, 2007
    Co-Authors: Wing-kuen Ling
    Abstract:

    This chapter discusses the two's Complement Arithmetic in complex digital filters. Nonlinear behavior of digital filters realized in the normal form is presented. It is found that when the accumulators are implemented through the two's Complement Arithmetic, then the complex digital filter associated with two's Complement Arithmetic can be implemented by the state space equation which has been discussed later in the chapter. It is found that the occurrence of nonlinear behaviors of digital filters associated with two's Complement Arithmetic depends on the realization of the system. It is suggested that chaotic behaviors will occur and the trajectory will neither converge to some fixed point nor exhibit limit cycle behavior. It is observed that chaotic behavior can be avoided if the system matrix is strictly or marginally stable and occurs if the system matrix is unstable. These behaviors are independent of the initial condition. It is found that as the orientations of the ellipses depend on the matrices, the orientations of two ellipses may be different.

  • 9 quantization and two s Complement Arithmetic in digital filters
    Nonlinear Digital Filters#R##N#Analysis and Applications, 2007
    Co-Authors: Wing-kuen Ling
    Abstract:

    This chapter analyzes the quantization and two's Complement Arithmetic in digital filters. The second-order digital filter associated with both quantization and two's Complement Arithmetic is described. It is found that if the number of states of the machine is large enough and the period is very long, then the system may exhibit near-chaotic behaviors and the phase portrait may visually exhibit a near-fractal pattern if the chaotic behavior is exhibited in the corresponding infinite state machine for the same filter parameters and initial conditions. On the other hand, if linear or limit cycle behaviors are exhibited in the corresponding infinite state machine, one would intuitively expect that for the same filter parameters and initial conditions the finite state machine would also exhibit linear or limit cycle behaviors. However, this intuitive expectation is not true and a counter intuitive phenomenon is stated in a later observation. A finite state machine may exhibit a near-chaotic behavior even when its corresponding infinite state machine does not exhibit any chaotic behavior. The nonlinear behavior of unstable second-order digital filters is also elaborated in the chapter.

  • 5 autonomous response of digital filters with two s Complement Arithmetic
    Nonlinear Digital Filters#R##N#Analysis and Applications, 2007
    Co-Authors: Wing-kuen Ling
    Abstract:

    This chapter discusses autonomous response of digital filters with two's Complement Arithmetic. The two's Complement Arithmetic involves a periodic discontinuous nonlinear function, and the dynamics of digital filters could be very complicated even when no input signal is applied. Limit cycle and chaotic behaviors could occur even for second-order digital filters. The second-order real digital filters are represented by a state space model. The necessary and sufficient conditions for the nonlinear system to behave as a linear system after a number of iterations are given. It is found that once the initial conditions—the filter parameters and the input signal—are given, the state variables and the symbolic sequences can be uniquely defined by equations. It is found that the system would behave as a linear system after a number of iterations if and only if the state vector toggles between two points on a particular straight line of the phase plane. It is observed that the state vector toggles between two states in a steady state on a particular straight line. Several conditions for exhibiting eventually periodic state vector are also presented in the chapter.