Circulant and skew-circulant matrices as new normal-form realization of IIR digital filters

Normal-form fixed-point state-space realization of IIR (infinite-impulse response) filters are known to be free from both overflow oscillations and roundoff limit cycles, provided magnitude truncation arithmetic is used together with two's-complement overflow features. Two normal-form realizations are derived that utilize circulant and skew-circulant matrices as their state transition matrices. The advantage of these realizations is that the A-matrix has only N (rather than N2) distinct elements and is amenable to efficient memory-oriented implementation. The problem of scaling the internal signals in these structures is addressed, and it is shown that an approximate solution can be obtained through a numerical optimization method. Several numerical examples are included

SMT-Based Bounded Model Checking of Fixed-Point Digital Controllers

Digital controllers have several advantages with respect to their flexibility and design's simplicity. However, they are subject to problems that are not faced by analog controllers. In particular, these problems are related to the finite word-length implementation that might lead to overflows, limit cycles, and time constraints in fixed-point processors. This paper proposes a new method to detect design's errors in digital controllers using a state-of-the art bounded model checker based on satisfiability modulo theories. The experiments with digital controllers for a ball and beam plant demonstrate that the proposed method can be very effective in finding errors in digital controllers than other existing approaches based on traditional simulations tools

Chaotic behaviors of stable second-order digital filters with two’s complement arithmetic

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

A new approach to the realization of low-sensitivity IIR digital filters

A new implementation of an IIR digital filter transfer function is presented that is structurally passive and, hence, has extremely low pass-band sensitivity. The structure is based on a simple parallel interconnection of two all-pass sections, with each section implemented in a structurally lossless manner. The structure shares a number of properties in common with wave lattice digital filters. Computer simulation results verifying the low-sensitivity feature are included, along with results on roundoff noise/dynamic range interaction. A large number of alternatives is available for the implementation of the all-pass sections, giving rise to the well-known wave lattice digital filters as a specific instance of the implementation

New zero-input overflow stability proofs based on Lyapunov theory

The authors demonstrate some proofs of zero-input overflow-oscillation suppression in recursive digital filters. The proofs are based on the second method of Lyapunov. For second-order digital filters with complex conjugated poles, the state describes a trajectory in the phase plane, spiraling toward the origin, as long as no overflow correction is applied. Following this state signal, an energy function that is a natural candidate for a Lyapunov function can be defined. For the second-order direct-form digital filter with a saturation characteristic, this energy function is a Lyapunov function. However, it is not the only possible Lyapunov function of this filter. All energy functions with an energy matrix that is diagonally dominant guarantee zero-input stability if a saturation characteristic is used for overflow correction. The authors determine the condition that a general second-order digital filter has to fulfil so that there exists at least one energy function with a matrix that is diagonally dominan