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In this paper, a noise analysis of a modulated quantizer is performed. If input signals are oversampled, then the quantization error could be reduced by modulating both the input and the output of the quantizer. The working principle is based on the fact that convolutions of bandpass signals would spread wider in the frequency spectrum than that of lowpass signals. Hence, by filtering the high frequency components, the signal-to-noise ratio (SNR) could be increased. Numerical simulation results show that the modulated quantization scheme could achieve an average of 13.0960 dB to 21.4700 dB improvements on SNR over the conventional scheme, depends on the types of bandlimited input signals

This correspondence considers the optimized design of interpolative sigma delta modulators (SDMs). The first optimization problem is to determine the denominator coefficients. The objective of the optimization problem is to minimize the passband energy of the denominator of the loop filter transfer function (excluding the dc poles) subject to the continuous constraint of this function defined in the frequency domain. The second optimization problem is to determine the numerator coefficients in which the cost function is to minimize the stopband ripple energy of the loop filter subject to the stability condition of the noise transfer function (NTF) and signal transfer function (STF). These two optimization problems are actually quadratic semi-infinite programming (SIP) problems. By employing the dual-parameterization method, global optimal solutions that satisfy the corresponding continuous constraints are guaranteed if the filter length is long enough. The advantages of this formulation are the guarantee of the stability of the transfer functions, applicability to design of rational infinite-impulse-response (IIR) filters without imposing specific filter structures, and the avoidance of iterative design of numerator and denominator coefficients. Our simulation results show that this design yields a significant improvement in the signal-to-noise ratio (SNR) and have a larger stability range, compared with the existing designs

In this correspondence, we consider the problem of designing a set of nonuniform near allpass complementary finite impulse response (FIR) filters. This problem can be formulated as a quadratic semi-infinite programming problem, where the objective is to minimize the sum of the ripple energy for the individual filters, subject to the passband and stopband specifications as well as to the allpass complementary specification. The dual parameterization method is used for solving the linear quadratic semi-infinite programming problem.

This letter shows some counter-intuitive simulation results that the symbolic sequences and the state variables of a digital filter with two's complement arithmetic and arbitrary initial conditions and order will be eventually zero when all the filter parameters are even numbers, no matter the system matrix of the filter is stable or not.

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.

In this paper, the behaviors of stable second-order digital filters with two’s complement arithmetic are investigated. It is found that even though the poles are inside the unit circle and the trajectory converges to a fixed point on the phase plane, that fixed point is not necessarily the origin. That fixed point is found and the set of initial conditions corresponding to such trajectories is determined. This set of initial conditions is a set of polygons inside the unit square, whereas it is an ellipse for the marginally stable case. Also, it is found that the occurrence of limit cycles and chaotic fractal pattern on the phase plane can be characterized by the periodic and aperiodic behaviors of the symbolic sequences, respectively. The fractal pattern is polygonal, whereas it is elliptical for the marginally stable case.

In this article, the second order digital filter with two’s complement arithmetic in [1] is considered. Necessary conditions for the symbolic sequences to be periodic after a number of iterations are given when the filter parameters are at b=a+1 and b=-a+1. Furthermore, for some particular values of a, even when one of the eigenvalues is outside the unit circle, the system may behave as a linear system after a number of iterations and the state vector may toggle between two states or converge to a fixed point at the steady state. The necessary and sufficient conditions for these phenomena are given in this article.

This paper considers an optimum nonuniform FIR transmultiplexer design subject to specifications in the frequency domain. Our objective is to minimize the sum of the ripple energy for all the individual filters, subject to the specifications on amplitude and aliasing distortions, and to the passband and stopband specifications for the individual filters. This optimum nonuniform transmultiplexer design problem can be formulated as a quadratic semi-infinite programming problem. The dual parametrization algorithm is extended to the design of this nonuniform transmultiplexer problem. If the lengths of the filters are sufficiently long and the set of decimation integers is compatible, then our algorithm guarantees that the solution obtained will give rise to the global minimum, and the required specifications are satisfied.

Since different multiwavelets, pre- and post-filters have different impulse and frequency responses characteristics, different multiwavelets, pre- and post-filters should be selected, integrated and applied at different noise levels if a signal is corrupted by an additive white Gaussian noise (AWGN). In this letter, some fuzzy rules on selecting and integrating different multiwavelets, pre- and post-filters together are proposed. These fuzzy rules are setup based on the training results of the denoising performances of applying different multiwavelets, pre- and post-filters at different noise levels. When a new electrocardiogram (ECG) signal is applied, the appropriate multiwavelets, pre- and post-filters are selected and integrated based on fuzzy rules and the noise level of the signal. A hard thresholding is applied on the multiwavelet coefficients. According to an extensive simulation, we found that our proposed fuzzy rule-based multiwavelet denoising algorithm achieves 30% improvement compared to the traditional multiwavelet denoising algorithms.

Wavelet denoising by singularity detection was proposed as an algorithm that combines Mallat and Donoho's denoising approaches. With wavelet transform modulus sum, we can avoid the error and ambiguities of tracing the modulus maxima across scales and the complicated and computationally demanding reconstruction process. We can also avoid the visual artifacts produced by shrinkage. In this paper, we investigate a multiwavelet denoising algorithm based on a modified singularity detection approach. Improved signal denoising results are obtained in comparison to the single wavelet case.

Some of the nodes of complex social networks may support for a given proposal, while the rest of the nodes may be against the given proposal. Even though all the nodes support for or are against the given proposal, the decision certitudes of individual nodes may be different. In this case, the steady state values of the decision certitudes of the majority of the nodes are either higher than or lower than a threshold value. Deriving the near consensus property is a key to the analysis of the behaviors of complex social networks. So far, no result on the behaviors of the complex social networks satisfying the near consensus property has been reported. Hence, it is useful to extend the definition of the exact consensus property to that of a near consensus property and investigate the behaviors of the complex social networks satisfying the near consensus property. This paper extends the definition of exact consensus complex social networks to that of near consensus complex social networks. For complex linear social networks, this paper investigates the relationships among the vectors representing the steady state values of the decision certitudes of the nodes, the influence weight matrix and the set of vectors representing the initial state values of the decision certitudes of the nodes under a given near consensus specification. The above analysis is based on the Eigen theory. For complex nonlinear social networks with certain types of nonlinearities, the relationship between the influence weight matrix and the vectors representing the steady state values of the decision certitudes of the nodes is studied. When a complex nonlinear social network does not achieve the exact consensus property, the optimal near consensus condition that the complex social network can achieve is derived. This problem is formulated as an optimization problem. The total number of nodes that the decision certitudes of the nodes are either higher than or lower than a threshold value is maximized subject to the corresponding near consensus specification. The optimization problem is a nonsmooth optimization problem. The nonsmooth constraints are first approximated by smooth constraints. Then, the approximated optimization problem is solved via a conventional smooth optimization approach. Computer numerical simulation results as well as the comparisons of the behaviors of complex nonlinear social networks to those of the complex linear social networks are presented. The obtained results demonstrate that some complex social networks can satisfy the near consensus property but not the exact consensus property. Also, the conditions for the near consensus property are dependent on the types of nonlinearities, the influence weight matrix and the vectors representing the initial state values of the decision certitudes of the nodes.

Applying gear shifting algorithms to the implementation of Phase Locked Loops (PLLs) can significantly improve their performances. However, the behaviors of gear shifting digital PLLs (GSDPLLs) have not been fully studied due to the existence of newly adaptive control parameters. These parameters play a very important role in the design of GSDPLLs. In this paper, various nonlinear behaviors of GSDPLLs including the steady state periodic behaviors, divergent behaviors and chaotic behaviors, are studied. In particular, the effects of the initial conditions of GSDPLLs on their dynamical behaviors are investigated. The obtained results are useful for the design of GSDPLLs. Numerical simulation results are presented for illustrations.