Saturation Arithmetic Research Articles

A new criterion for the overflow stability of second-order state-space digital filters using saturation arithmetic

Two recent approaches (one due to Singh [1990] and the other due to Liu and Michel [1992]) for the elimination of zero-input overflow oscillations in state-space digital filters designed with saturation arithmetic are compared. It is demonstrated that Singh's approach leads to a relatively less stringent condition for the nonexistence of overflow oscillations. Using Singh's approach, an improved version of Ritzerfeld-Werter's criterion for the nonexistence of overflow oscillations in second-order state-space digital filters is made available. Finally, a new zero-input limit cycle-free realizability condition for a generalized overflow characteristic is presented.

Stability of 2-D discrete systems described by the Fornasini-Marchesini second model

Based on the Fornasini-Marchesini second local state-space (LSS) model, criteria that sufficiently guarantee the asymptotic stability of 2-D discrete systems are given. A sufficient condition for a 2-D nonlinear discrete system to be free of overflow oscillations is then shown in the case when a 2-D discrete system is employed by saturation arithmetic. Finally, an upper bound on parameter variations which guarantees the asymptotic stability of a perturbed 2-D discrete system is considered. It is shown that the upper bound stated here is less conservative than the existing ones.

A separation theorem for finite-precision digital systems

For models of zero-input direct-form digital filters implemented in fixed-point digital hardware it is known that if saturation arithmetic is used and stability holds when the quantization is ignored, then the amplitude of all limit cycles can be made arbitrarily small by making the bound on the magnitude of the quantization sufficiently small. In that sense the effects of quantization and overflow can be considered separately. Recently, this proposition was extended to the case in which the input need not be zero. Here we give a result to the effect that quantization and overflow can be considered separately for a much more general class of discrete-time systems, and we give as an example an application of this result to systems governed by state equations.

A separation theorem for finite precision digital filters

For models of zero-input direct-form digital filters implemented in fixed-point digital hardware, it is known that if saturation arithmetic is used, and stability holds when the quantization is ignored, then the amplitude of all limit cycles can be made arbitrarily small by making the bound on the magnitude of the quantization sufficiently small. In that sense the effects of quantization and overflow can be considered separately. Here we extend this proposition to the important case in which the input need not be zero and a certain more general type of arithmetic is used.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

COMPLEX BEHAVIOR IN DIGITAL FILTERS

This paper presents recent results concerning possibilities of complex behavior in extremely simple second-order digital filter structures. Two types of such behavior have been confirmed by simulation experiments and rigorous mathematical analysis: 1) chaotic behavior defined here as the existence of aperiodic, bounded trajectories displaying sensitive dependence to the initial states. Such trajectories often form very complex, self-similar patterns in the state-space. 2) an abundance of oscillatory solutions and final state sensitivity with respect to system parameters. This type of behavior can be identified by an extremely complicated structure of the parameter space (existence of Arnold tongues) and fine structure of changes of dynamic behavior when varying filter parameters (devil’s staircase). The first type of behavior has been discovered in filter sections employing two’s complement adder overflow characteristics and confirmed via symbolic dynamics technique. Properties of the second type have been found in extensive numerical experiments carried out for filter sections employing saturation arithmetic and in which a full confirmation via the mathematical analysis of an associated one-dimensional model of the system has been made.

Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic

An unconventional approach for guaranteeing the nonexistence of zero-input limit cycles in fixed-point state-space digital filters using saturation arithmetic is presented. The approach is based on a 'passivity' property associated with the multiple saturation nonlinearities. Under certain conditions, the approach specializes to the known approach based on the sector information of the nonlinearities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new realizability condition for limit cycle-free state-space digital filters employing saturation arithmetic

A criterion for the absence of zero-input limit cycles in state-space digital filters employing saturation arithmetic is presented. An example is given, which brings out the novelity of the present approach.

A Microprocessor for Signal Processing, the RSP

Signal processing is a data processing domain that contains a diversity of applications, including speech processing, image processing, radar, sonar, medical imaging, data communications, seismic processing, and many others. Despite the diversity of the applications, this processing domain has a very structured set of characteristics. These include real-time operation, dominance of arithmetic operations, and well-structured data flows. The Real-Time Signal Processor (RSP) is a microprocessor architecture that was created to exploit these characteristics in order to provide an expeditious and economical way to implement signal processing applications. In this paper, the organization and architecture of the RSP are described. Features of the RSP, such as the instruction pipeline and the fractional fixed-point arithmetic, which exploit the characteristics of signal processing to provide additional computational power, are emphasized. Other features, such as the powerful indexing, the saturation arithmetic, the guard bits, and the double-word-width accumulator, which add much to the processor's versatility and programmability, are also highlighted. The performance of the RSP is illustrated through examples.

The zero-input response of digital filters using saturation arithmetic

It is known that second-order digital filters realized in direct form using saturation arithmetic have the property that the amplitudes of all self-sustained limit cycles can be made arbitrarily small by increasing the numbers of bits associated with the representation of the data samples. In this paper we observe that a recent extension to certain higher order cases of an idealized version of that result (quantization is ignored in the Idealized model) is a direct corollary of a theorem in the theory of feedback systems, and that the extension (and related more general propositions) can be proved without the earlier theorem by using simple operator-theoretic methods. The main result given in the paper is an extension of the result for nonidealized second-order sections to a class of filters of arbitrary order.

The absolute stability of high-order discrete-time systems utilizing the saturation nonlinearity

Previously, we investigated the stability of a class of discrete-time filters involving a restricted class of linear systems, the all-pole systems, and a single, specific, memoryless nonlinearity-the saturation nonlinearity. A rather effective criterion was derived for determining if a given system is free of all periodic oscillations except the trivial null solution. The derivation relied on the observation that certain rather exclusive properties, called passivity properties, are associated with the saturation nonlinearity. Here, we generalize the results in two directions. First, the linear part of the system is allowed to be quite general, i.e., both poles and zeros may occur in the transfer function. Thus in the context of digital filters, the restriction to the direct-form realization is lifted. Secondly, the criterion obtained here guarantees the absolute stability of the system and not just the absence of nontrivial periodic oscillations. Thus the possibility of aperiodic or chaotic behavior is eliminated. The broadened formulation enhances the utility of the criterion, making it applicable to the large class of recursive digital processing systems, including digital filters, which use the saturation arithmetic to cope with overflow in the course of forming the sum of the results of many multiplicative operations. Also, the methods developed indicate an approach for the stability investigation of large digital transmission networks containing nonlinearities. The technique used for deriving the new result is entirely different. The new element is the demonstration that the hypothesis which is in the form of a condition in the frequency domain implies the existence of a Lyapunov function.

On the Stability of Higher Order Digital Filters Which Use Saturation Arithmetic

The device of using “saturation arithmetic” to cope with adder overflow in recursive digital filters has, for a number of years now, been known to yield stable operation when the filter is of second order and is linearly stable. Mitra has recently given examples to show that this happy situation does not prevail for higher order filters. Here we investigate conditions on the filter coefficients which would guarantee stability for higher order filters using saturation arithmetic. We are only able to give sufficient conditions for stability. These conditions in their simplest form can be written as linear inequalities involving the coefficients of the filter.

Criteria for Determining if a High-Order Digital Filter Using Saturation Arithmetic Is Free of Overflow Oscillations

Recently we found that, among recursive digital filters using saturation arithmetic to contend with overflow, a fundamental difference exists between second and higher order filters: the latter may sustain large-amplitude overflow oscillations. In this paper we have derived a new criterion expressly designed for determining when a given high-order recursive system using saturation arithmetic is free of overflow oscillations. The new criterion, which is easy to use, follows from this result: we associate with the given system two trigonometric polynomials in θ of degree equal to the order of the given system; if any linear combination of the polynomials with nonnegative weights is positive for all θ in [0, π] then the system is free of all nontrivial periodic oscillations. We prove that the new criterion subsumes certain well-known criteria, such as Tsypkin's criterion, from the literature on nonlinear systems. To illustrate, three classes of special systems are investigated, and in each case the new criterion gives substantial improvements. Finally, the new test is applied in the synthesis of high order sections for a realistic eighth-order system.

Large amplitude, self-sustained oscillations in difference equations which describe digital filter sections using saturation arithmetic

The possibility of oscillations due to adder overflow in digital filter sections of order exceeding two is investigated. Specifically, the following problem is posed: is the property that large amplitude limit cycles cannot be self-sustaining in saturation arithmetic special to second-order sections, or is it also true for some or all higher order sections? This issue is resolved and, beyond that, some new and rather interesting results are presented on the stability implications of approximating a nonlinear element in standard filter structures by a range of linear gains. For every order of the section beyond two we are able to identify a set of coefficients corresponding to an absolutely stable underlying linear system, and a set of initial conditions for which the solution (to the nonlinear recursion) is a limit cycle. As an aid to understanding this behavior, we show that a particular natural class of associated linear recursions is always stable if and only if their order does not exceed two. Finally, motivated by the above result and by the viewpoint of describing functions, we make a conjecture according to which these oscillations do not exist if every member of the above stated class of linear systems is absolutely stable. We show that even this conjecture is false for recursions of order three. Thus, to ensure that overflow oscillations do not occur in high-order sections in which saturation arithmetic is used particular care has to be exercised on a case by case basis.