Link to publicationCitation for published version (APA): Ritzerfeld, J. H. F. (1989). A condition for the overflow stability of second-order digital filters that is satisfied by all scaled state-space structures using saturation. IEEE Transactions on Circuits and Systems, 36(8), 1049-1057. DOI: 10.1109 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract -A set of conditions is derived that ensures overflow stability of second-order digital filters for different classes of overflow arithmetics, involving only the elements of the state-bansition matrix. The well-known arithmetic saturation, zeroing, and two's-complement lead to different stability conditions, the condition for saturation being the least restrictive.As a result, all properly scaled second-order state-space structures are zero-input overflow stable if saturation is used for overflow correction. Furthermore, conditions are derived for stable second-order digital filters in a nonzero input situation by introducing a weaker form of stability of the forced response. The presented analysis is based on determining the set of Lyapunov functions for a general second-order state-transition matrix given a certain overflow arithmetic.

Abstract-The quantization noise of a fixed-point digital filter is commonly expressed in terms of its noise gain, i.e., the factor by which the noise power 2 12 of a single quantizer is amplified to the output of the filter. In this brief, first a closed-form expression for the optimal second-order noise gain in terms of the coefficients of the numerator and denominator polynomials of the transfer function is derived. It is then shown, by deriving a similar expression for its noise gain, that the second-order direct form structure has an arbitrarily larger noise gain the closer the filter poles are to the unit circle. The main result, however, is that the wave digital form and the normal form structures have noise gains which are only marginally larger than the minimum gain. For these forms, the expressions for their noise gain in terms of the transfer function are given as well. The importance of these forms lies in the fact that they use less multipliers than the optimal structure and that they are much easier to design: properly scaled forms are given requiring no design tools.Index Terms-Filter structures, quantization noise, noise gain, second-order modes, wave digital filters.

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In this paper we demonstrate some new proofs of suppressing zero-input overflow oscillations in resive digital filters. These 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, spiralling towards the origin, as long as no overflow correction is applied. Following this state signal an energy function can be defined,which is a natural candidate for a Lyapunov function. For the second-order direct form digital filter with a saturation characteristic this energy function is a Lyapunov function. However, this function is not the only possible Lyapunov function of this filter. All energy functions with anenergy matrix that is diagonally dominant, guarantee zero-input stability, if a saturation characteristic is used for overflow correction. In this paper we determine the condition a general second-order digital filters has to fulfil so that there exists at least one energy function with a matrix, which is diagonally dominant.