For the class of digital filters described by the state equations, new expressions for the roundoff noise and the dynamic-range constraint equation have been established. Then, minimization of the roundoff noise subject to the l 2 norm dynamic-range constraint is considered. Using the polar decomposition of a nonsingular matrix, the effects of an equivalent transformation on the noise are analyzed, then, a lower bound and the global minimum of the unit noise, i.e., the noise generated-under the assumption that each state equation contains exactly one noise source, is obtained. A method of realizing the minimum unit noise filter is given, and possible further optimization of the filter is discussed. A numerical example is given to illustrate the computational procedure.

Minimum uncorrelated unit noise in state-space digital filtering

The family of filters {H(F(z)):F(z) a frequency transformation} generated from a prototype filter H(z) is shown to possess certain common properties. These are coordinate-free quantities (called second-order modes) which are invariant under frequency transformation. The invariance is significant in the design of low-noise fixed-point digital filter structures since the second-order modes characterize the minimum attainable noise. Filter structures (including parallel, cascade, and ladder configurations) are studied whose output noise is essentially independent of bandwidth and center frequency. An analysis of direct form structures (whether isolated or as one section within a cascade or parallel configuration) results in an expression giving the dominant term in the output noise as a function of the parameter in the low-pass-low-pass transformation. This noise term approaches infinity as bandwidth approaches zero. Thus, for narrowband filters, a difference of several orders of magnitude in the output noise can exist between a scaled direct form (having six multiplications per two-pole section) and the optimal form (having nine multiplications per two-pole section).

https://ieeexplore.ieee.org/iel6/29/26121/01162869.pdf

Roundoff noise in digital filters: Frequency transformations and invariants

A new noise expression is formulated for the class of fixed-point digital filters described by the state equations, and two methods of its computation are discussed. The effects of possible structure transformation and state-amplitude scalings are then incorporated in this expression, and the results have been analyzed. In particular, it is shown that the output noise and state amplitudes are inversely proportional, and that an elementary transformation is well suited for a step-by-step generation of a low-noise filter.

Roundoff noise in state-space digital filtering: A general analysis

The sensitivities of the transfer function of a digital filter with respect to its coefficients are utilized to derive lower bounds on the roundoff noise output in the cases of L_{\infty} and L_{\infty} scaling for fixed-point arithmetic. General bounds are produced which apply to any filter structure if rounding is performed after multiplication and the filter has already been scaled. For the parallel and cascade forms, alternate bounds are derived which apply to rounding after multiplication or summation and which do not require prior scaling. The alternate bounds arethus independent (or nearly so) of pairing, ordering, and transposition. Examples are presented which show that the bounds are reasonably tight.

Roundoff noise bounds derived from coefficient sensitivities for digital filters

Error spectrum shaping (ESS) can significantly reduce finite-wordlength error in recursive fixed-point digital filters. Most past work on ESS has focused on second-order filters. In this paper, expressions for the ESS coefficients minimizing the roundoff error are derived for high-order filters composed of cascaded second-order sections. A comparison of the noise gains is made between optimal and suboptimal ESS structures and the section-optimal structure; the ESS structures perform well in comparison. A heuristic section-ordering strategy, based on ideas underlying ESS, is proposed and deemed useful, particularly for elliptic filters.

Optimal and suboptimal error spectrum shaping for cascade-form digital filters

This paper treats the problem of minimizing the roundoff noise in digital filters using fixed-point arithmetic under sinusoidal input. A basic assumption made is that of representing the roundoff error as white noise that is independent from sample to sample and from source to source. The minimax noise principle is introduced to serve as a guide in the filter design to optimize the structure for minimum noise. One application is illustrated through the design of low-noise cascade digital filters under dynamic range constraints. Numerical examples demonstrate the lower noise possible in comparison to other known designs and serve to verify the effectiveness of a design procedure based on the minimax concept.

http://ieeexplore.ieee.org/iel5/31/23433/01083865.pdf

W. Lee

Papers

Optimization of digital filters for low roundoff noise