# A new statistical approach to the coefficient word length problem for digital filters

TL;DR: It is shown that by minimizing this statistical word length, the actual coefficient word length upon rounding of the coefficients can generally be reduced as well.

Abstract: A statistical approach is proposed for estimating the necessary coefficient word length of a digital filter. With this statistical word length definition, an optimization procedure is then proposed for minimizing the statistical word length for a given filter structure and a given set of maximum error constraints. It is shown that by minimizing this statistical word length, the actual coefficient word length upon rounding of the coefficients can generally be reduced as well. Several examples are given and improvements of one to three bits in the actual coefficient word length are observed. The procedure does not necessarily lead to the actual global minimum coefficient word length.

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TL;DR: In this article, the remaining unquantized coefficients of a FIR linear phase digital filter when one or more of the filter coefficients takes on discrete values are optimized using the least square response error.

Abstract: An efficient method optimizing (in the least square response error sense) the remaining unquantized coefficients of a FIR linear phase digital filter when one or more of the filter coefficients takes on discrete values is introduced. By incorporating this optimization method into a tree search algorithm and employing a suitable branching policy, an efficient algorithm for the design of high-order discrete coefficient FIR filters is produced. This approach can also be used to design FIR filters on a minimax basis. The minimax criterion is approximated by adjusting the least squares weighting. Results show that the least square criteria is capable of designing filters of order well beyond other approaches by a factor of three for the same computer time. The discrete coefficient spaces discussed include the evenly distributed finite wordlength space as well as the nonuniformly distributed powers-of-two space.

240 citations

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01 Apr 1975TL;DR: A framework is presented for the analysis, representation, and evaluation of digital filter structures Based on the notation of linear signal-flow graphs and their equivalent matrix representation, a set of general linear digital network properties are reviewed, including precedence relations computability, Tellegen's theorem, interreciprocity, and network transposition.

Abstract: A framework is presented for the analysis, representation, and evaluation of digital filter structures. Based on the notation of linear signal-flow graphs and their equivalent matrix representation, a set of general linear digital network properties are reviewed, including precedence relations computability, Tellegen's theorem, interreciprocity, and network transposition. These properties are then utilized in developing time and frequency domain analysis techniques and sensitivity analysis techniques. These techniques, in turn, are applied to the comparison of several basic digital filter structures.

207 citations

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TL;DR: The application of a general-purpose integer-programming computer program to the design of optimal finite wordlength FIR digital filters is described and an analysis of the approach based on the results of more than 50 design cases is presented.

Abstract: The application of a general-purpose integer-programming computer program to the design of optimal finite wordlength FIR digital filters is described. Examples of two optimal low-pass FIR finite wordlength filters are given and the results are compared with the results obtained by rounding the infinite wordlength coefficients. An analysis of the approach based on the results of more than 50 design cases is presented and the problem of optimal wordlength choice is discussed.

164 citations

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TL;DR: This paper reviews the issues of digital control implementation, from algorithms through current hardware up to the various problems arising with non-ideal behaviour of digital controllers.

Abstract: Stimulated by microprocessor technology there is increasing interest in the issues of digital control implementation. This paper reviews these issues, from algorithms through current hardware up to the various problems arising with non-ideal behaviour of digital controllers.

158 citations

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TL;DR: The statistical coefficient sensitivity introduced by the statistical approach is shown to be equivalent to the round off noise power gain and state-space digital filters which are optimal with respect to both coefficient sensitivity and roundoffs noise can be synthesized by the method of minimization of the roundoff noise.

Abstract: This paper proposes two expressions of the output error variance due to coefficient quantization of fixed-point state-space digital filters in the time domain. One is the deterministic approach, which gives precisely the output error variance of state-space digital filters. The other is the statistical approach, where the errors of coefficient quantization are assumed to be independent random variables. The statistical approach gives a simple and easy way to analyze the output error variance due to coefficient quantization of state-space digital filters. The statistical coefficient sensitivity introduced by the statistical approach is shown to be equivalent to the round off noise power gain. Thus, state-space digital filters which are optimal with respect to both coefficient sensitivity and roundoff noise can be synthesized by the method of minimization of the roundoff noise. Such optimal state-space digital filters which could be of any order are proved to be free of autonomous overflow limit cycles. A numerical example is given to illustrate the effectiveness of the analysis of the output error variance due to coefficient quantization and the synthesis method proposed here.

96 citations

##### References

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01 Jan 1962

TL;DR: This inexpensive paperback edition of a groundbreaking classic is unique in its emphasis on the frequency approach and its use in the solution of problems.

Abstract: From the Publisher:
For this inexpensive paperback edition of a groundbreaking classic, the author has extensively rearranged, rewritten and enlarged the material. Book is unique in its emphasis on the frequency approach and its use in the solution of problems. Contents include: Fundamentals and Algorithms; Polynomial Approximation Classical Theory; Fourier ApproximationModern Therory; Exponential Approximation.

1,612 citations

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1,322 citations