Can a second order bandpass sigma delta modulator achieve high signal-to-noise ratio for lowpass inputs

Institutively, second order SDMs usually achieve lower SNR than high order ones because high order loop filters can achieve better noise shaping characteristics. Moreover, the signal transfer function should be designed to have large values and the noise transfer function should be designed to have small values at the passband of loop filters in order to achieve good noise shaping characteristics, so SNR should be high if input signal bands match passbands of loop filters and low otherwise. Based on this argument, one may expect that SNR will be low when input signals have lowpass characteristics while loop filters have bandpass characteristics. However, since the above argument is based on the noise shaping theory which is formulated using a linear model, while quantizers in SDMs are nonlinear components, the linear model may not explain nonlinear system behaviors. In this letter, a counterexample is given to illustrate that a second order bandpass interpolative SDM may also give a very high SNR for lowpass inputs

Stability of sinusoidal responses of interpolative sigma delta modulators

In this paper, stability of sinusoidal responses of interpolative sigma delta modulators (SDMs) is investigated. It is found that interpolative SDMs may switch from unstable to stable behaviors even though the magnitude or the frequency of the input sinusoidal signals increase. Hence, the input magnitude stability margin and the input frequency stability margin are redefined as the minimum input magnitude and the minimum input frequency of the input sinusoidal signals such that the output of the loop filter is bounde

Stabilization of (L,M) shift invariant plant

In this paper, a lifting technique is employed to realize a single input single output linear (L,M) shift invariant plant as a filter bank system. Based on the filter bank structure, a controller is designed so that the aliasing components in the control loop are cancelled and the loop gain becomes a time invariant transfer function. Pole placement technique is applied to stabilize the overall system and ensure the causality of the filters in the controller. An example on the control of a linear (L,M) shift invariant plant with simulation result is illustrated. The result shows that our proposed algorithm is simple and effective

Step response of a second-order digital filter with two’s complement arithmetic

It is well known that the autonomous response of a second-order digital filter with two’s complement arithmetic may exhibit chaotic behaviors [1]. In this paper, results of the step response case are presented. Even though in the presence of the overflow nonlinearity, it is found that the step response behaviors can be related to some corresponding autonomous response behaviors by means of an appropriate affine transformation. Based on this method, some differences between the step response and the autonomous response are explored. The effects of the filter parameter and input step size on the trajectory behaviors are presented. Some previous necessary conditions for the trajectory behaviors, initial conditions and symbolic sequences are extended and strengthened to become necessary and sufficient conditions. Based on these necessary and sufficient conditions, some counter-intuitive results are reported. For example, it is found that for some sets of filter parameter values, the system may exhibit the type I trajectory even when a large input step size is applied and overflow occurs. On the other hand, for some sets of filter parameter values, the system will not give the type I trajectory for any small input step size, no matter what the initial conditions are

Periodic input response of a second-order digital filter with two’s complement arithmetic

The dynamic behaviors of a nonlinear second-order digital filter with two’s complement arithmetic under periodic inputs are explored. The conditions for avoiding overflow are derived. Various dynamic periodic responses are analyzed, accompanied by numerous simulation examples

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

Initiation of HIV therapy

In this paper, we numerically show that the dynamics of the HIV system is sensitive to both the initial condition and the system parameters. These phenomena imply that the system is chaotic and exhibits a bifurcation behavior. To control the system, we propose to initiate an HIV therapy based on both the concentration of the HIV-1 viral load and the ratio of the CD4 lymphocyte population to the CD8 lymphocyte population. If the concentration of the HIV-1 viral load is higher than a threshold, then the first type of therapy will be applied. If the concentration of the HIV-1 viral load is lower than or equal to the threshold and the ratio of the CD4 lymphocyte population to the CD8 lymphocyte population is greater than another threshold, then the second type of therapy will be applied. Otherwise, no therapy will be applied. The advantages of the proposed control strategy are that the therapy can be stopped under certain conditions, while the state variables of the overall system is asymptotically stable with fast convergent rate, the concentration of the controlled HIV-1 viral load is monotonic decreasing, as well as the positivity constraint of the system states and that of the dose concentration is guaranteed to be satisfied. Computer numerical simulation results are presented for an illustration

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

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

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