Overflow Nonlinearities Research Articles

Relationship between input-scaling and stability of the forced response of recursive digital filters

The stability of forced response of a general recursive digital filter operating in the inherent presence of overflow nonlinearities is examined. A theorem is derived that relates forced-response stability to the degree of input-scaling. The technique is applied to the investigation of the stability properties of state-space digital filters, resulting in a criterion for the forced-input stability of Nth-order state-space filters. Subsequently, it is applied to the second-order case. The resultant stability criterion is applied to the investigation of optimal state-space and normal filters. Finally, the stability properties of wave digital filters are discussed. >

Decoupling of the overflow and quantization phenomena in orthogonal biquad recursive digital filters

In digital communication networks, a special class of complex biquad recursive digital filters called orthogonal filters is increasingly being used. The separate effects of the overflow and quantization nonlinearities on these orthogonal filters' responses have been investigated [5], [6]. In this paper we examine the zero-input stability properties of the actual orthogonal filter having both overflow and quantization nonlinearities. The overflow nonlinearities considered include saturation, bit-by-bit inversion, zeroing, and modulo 2 arithmetic. The quantization techniques used may be roundoff, magnitude, or value truncation. An example demonstrates the adverse coupling effect between the overflow and quantization nonlinearities. Two criteria are therefore derived to ensure asymptotic overflowstability of the filter in the presence of quantization. These criteria have been translated to the coefficient plane; various regions corresponding to different minimum wordlengths required to ensure decoupling of the overflow and quantization phenomena have been derived.

Conditions for overflow stability of a class of complex biquad digital filters

In digital communication networks, complex biquad recursive filters are increasingly being used in the processing of complex signals. This paper examines the stability properties of a class of complex biquad filters having different overflow nonlinearities. Two new sets of conditions have been derived to ensure asymptotic overflow stability of the complex biquad filter with respect to various admissible classes of input signals <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{\sim c}</tex> . The subscript <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> refers to the signal level of the complex input sequences; it is a measure of the degree of input scaling. When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c = 0</tex> , the formulation degenerates to the zero-input stability problem. Using the new stability criteria, various regions of stability in the coefficient plane have been derived as a function of the factor <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> . The overflow nonlinearities considered include saturation, bit-by-bit inversion, zeroing, and modulo 2 arithmetic.

Spatial-domain design of overflow-stable 2-D digital filters

In this paper, we present a constrained l_P optimization design procedure for local state-space (LSS) two-dimensional (2-D) digital filters. The designed filter is guaranteed to be free of oscillations due to the overflow nonlinearity. This approach eliminates the need for first finding the LSS realization from the transfer function and then applying transformations to obtain an overflow-stable realization. It is shown that for the general second-order LSS filter, the constraint does not limit the flexibility of the design. An example to illustrate this approach is also given.

Design criterion for zero-input asymptotic overflow-stability of recursive digital filters in the presence of quantization

This paper investigates the zero-input stability properties of the exact second-order recursive digital filter having both overflow and quantization non-linearities. Two examples demonstrate the adverse influence of quantization on the overflow-stability property of the filter. Three sets of conditions are presented to ensure asymptotic overflow-stability in the presence of quantization. Using these criteria, various regions in the coefficient plane corresponding to different minimum internal wordlengths required to ensure the non-interaction of overflow and quantization are derived. These results thus form a useful design criterion.

Effects of input-scaling on the asymptotic overflow-stability properties of second-order recursive digital filters

This paper examines the stability properties of a second-order recursive digital filter having different overflow nonlinearities. Three new sets of sufficiency conditions have been derived to ensure asymptotic stability of the filter with respect to various admissible classes of input signals under different input-scaling conditions. Using these criteria, various regions of stability in the coefficient plane have been computed as a function of the input scale factor <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> . Examples of a filter's responses illustrate the dependence of the stability property of the forced response on the input-scaling conditions.

Comments on "The Absolute Stability of High-Order Discrete-Time Systems Utilizing the Saturation Nonlinearity"

It is shown that passivity properties introduced by Mitra for saturation type overflow nonlinearities determine this nonlinear characteristic up to a scaling factor, and that a stability criterion based on these passivity properties can be derived in a different way. It is indicated that this alternative derivation allows a generalization of the criterion to systems with more than one saturation nonlinearity.

On the stability of the forced response of digital filters with overflow nonlinearities

The forced response of a general type of digital filter, implemented with overflow nonlinearities, is studied. A definition of stability of this forced response is given. It is indicated that this stability can be studied by means of the zero-input behavior of a system, with the same structure but with time-varying nonlinearities. A theorem is derived giving sufficient conditions for the stability of the forced response of a digital system. Some results of the application of this theorem to wave digital filters are given.