Design of interpolative sigma delta modulators via a semi- infinite programming approach

This paper considers the design of interpolative sigma delta modulators (SDMs). The design problem is formulated as two different optimization problems. The first optimization problem is to determine the denominator coefficients. The objective of the optimization problem is to minimize the energy of the error function in the passband of the loop filter in which the error function reflects the noise output transfer function and the ripple of the input output transfer function. The constraint of the optimization problem refers to the specification of the error 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 output and input output transfer functions. These two optimization problems are actually quadratic semi-infinite programming (SIP) problems. By employing our recently proposed dual parameterization method for solving the problems, global optimal solutions that satisfy the corresponding continuous constraint are guaranteed if the solutions exist. The advantages of this formulation are the guarantee of the stability of the noise output and input output transfer functions, applicability to design rational IIR filters without imposing specific filter structures such as Laguerre filter and Butterworth filter structures, and the avoidance of the iterative design of numerator and the denominator coefficients because the convergence of the iterative design is not guaranteed. Our simulation results show that this proposed design yields a significant improvement in the signal-to-noise ratio (SNR) compared to the existing designs

Noise analysis of modulated quantizer based on oversampled signals

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 13.0960dB to 21.4700dB improvements on SNR over the conventional scheme, depends on the types of bandlimited input signals

Global stability, limit cycles and chaotic behaviors of second order interpolative sigma delta modulators

It is well known that second order lowpass interpolative sigma delta modulators (SDMs) may suffer from instability and limit cycle problems when the magnitudes of the input signals are at large and at intermediate levels, respectively. In order to solve these problems, we propose to replace the second order lowpass interpolative SDMs to a specific class of second order bandpass interpolative SDMs with the natural frequencies of the loop filters very close to zero. The global stability property of this class of second order bandpass interpolative SDMs is characterized and some interesting phenomena are discussed. Besides, conditions for the occurrence of limit cycle and fractal behaviors are also derived, so that these unwanted behaviors will not happen or can be avoided. Moreover, it is found that these bandpass SDMs may exhibit irregular and conical-like chaotic patterns on the phase plane. By utilizing these chaotic behaviors, these bandpass SDMs can achieve higher signal-to-noise ratio (SNR) and tonal suppression than those of the original lowpass SDMs

Nonlinear behaviors of bandpass sigma delta modulators with stable system matrices

It has been established that a class of bandpass sigma delta modulators (SDMs) may exhibit state space dynamics which are represented by elliptical or fractal patterns confined within trapezoidal regions when the system matrices are marginally stable. In this paper, it is found that fractal patterns may also be exhibited in the phase plane when the system matrices are strictly stable. This occurs when the sets of initial conditions corresponding to convergent or limit cycle behavior do not cover the whole phase plane. Based on the derived analytical results, some interesting results are found. If the bandpass SDM exhibits periodic output, then the period of the symbolic sequence must equal the limiting period of the state space variables. Second, if the state vector converges to some fixed points on the phase portrait, these fixed points do not depend directly on the initial conditions

Stability of sinusoidal responses of marginally stable bandpass sigma delta modulators

In this paper, we analyze the stability of the sinusoidal responses of second order interpolative marginally stable bandpass sigma delta modulators (SDMs) with the sum of the numerator and denominator polynomials equal to one and explore new results on the more general second order interpolative marginally stable bandpass SDMs. These results can be further extended to the high order interpolative marginally stable bandpass SDMs

Estimation of an initial condition of sigma-delta modulators via projection onto convex sets

Abstract—In this paper, an initial condition of strictly causal rational interpolative sigma-delta modulators (SDMs) is estimated based on quantizer output bit streams and an input signal. A set of initial conditions generating bounded trajectories is characterized. It is found that a set of initial conditions generating bounded trajectories but not necessarily corresponding to quantizer output bit streams is convex. Also, it is found that a set of initial conditions corresponding to quantizer output bit streams but not necessarily generating bounded trajectories is convex too. Moreover, it is found that an initial condition both corresponding to quantizer output bit streams and generating bounded trajectories is uniquely defined if the loop filter is unstable (Here, an unstable loop filter refers to that with at least one of its poles being strictly outside the unit circle). To estimate that unique initial condition, a projection onto convex set approach is employed. Numerical computer simulations show that the employed method can estimate the initial condition effectively

Fuzzy impulsive control of high order interpolative lowpass sigma delta modulators

In this paper, a fuzzy impulsive control strategy is proposed. The state vectors that the impulsive controller resets to are determined so that the state vectors of interpolative low-pass sigma-delta modulators (SDMs) are bounded within any arbitrary nonempty region no matter what the input step size, the initial condition and the filter parameters are, the occurrence of limit cycle behaviors and the effect of audio clicks are minimized, as well as the state vectors are close to the invariant set if it exists. To work on this problem, first, the local stability criterion and the condition for the occurrence of limit cycle behaviors are derived. Second, based on the derived conditions, as well as a practical consideration based on the boundedness of the state variables and a heuristic measure on the strength of audio clicks, fuzzy membership functions and a fuzzy impulsive control law are formulated. The controlled state vectors are then determined by solving the fuzzy impulsive control law. One of the advantages of the fuzzy impulsive control strategy over the existing linear control strategies is the robustness to the input signal, the initial condition and the filter parameters, and that over the existing nonlinear control strategy are the efficiency and the effectiveness in terms of lower frequency of applying the control force and higher signal-to-noise ratio (SNR) performanc

Difference between irregular chaotic patterns of second-order double-loop ΣΔ modulators and second-order interpolative bandpass ΣΔ modulators

In this paper, we find that, by computing the difference between two consecutive state vectors of second-order double-loop sigma-delta modulators (SDMs) and plotting one component of the subtracted vectors against the other component, irregular chaotic patterns will become two vertical lines. By multiplying a matrix on the subtracted vectors, it can be further transformed to two fixed points. However, second-order interpolative bandpass SDMs still exhibit chaotic behaviors after applying the same transformations. Moreover, it is found that the Lyapunov exponent of state vectors of second-order double-loop SDMs is higher than that of second-order interpolative bandpass SDMs, whereas the Lyapunov exponent of transformed vectors becomes negative infinity for second-order double-loop SDMs and increases for second-order interpolative bandpass SDMs. Hence, by examining the occurrence of chaotic behaviors of the transformed vectors of these two SDMs, these two SDMs can be distinguished from their state vectors and their transformed vectors without solving the state equations and knowing the information of input signals