# A simple method of computing the input quantization and multiplication roundoff errors in a digital filter

01 Oct 1974-IEEE Transactions on Acoustics, Speech, and Signal Processing (IEEE)-Vol. 22, Iss: 5, pp 326-329

TL;DR: A simple method of calculating the steady-state value of the variance of the output noise of a digital filter due to the input quantization noise or internally generated noise from product round-off is presented.

Abstract: A simple method of calculating the steady-state value of the variance of the output noise of a digital filter due to the input quantization noise or internally generated noise from product round-off is presented. The output noise is expressed as a sum of simpler terms belonging to one of four basic groups. Explicit expressions have been developed for rapid evaluation of these terms in the expansion. The method is illustrated by means of examples.

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TL;DR: An efficient algorithm for obtaining solutions is given and shown to be closely related to a well-known algorithm of Levinson and the Jury stability test, which suggests that they are fundamental in the numerical analysis of stable discrete-time linear systems.

Abstract: It is common practice to partially characterize a filter with a finite portion of its impulse response, with the objective of generating a recursive approximation. This paper discusses the use of mixed first and second information, in the form of a finite portion of the impulse response and autocorrelation sequences. The discussion encompasses a number of techniques and algorithms for this purpose. Two approximation problems are studied: an interpolation problem and a least squares problem. These are shown to be closely related. The linear systems which form the solutions to these problems are shown to be stable. An efficient algorithm for obtaining solutions is given and shown to be closely related to a well-known algorithm of Levinson and the Jury stability test. The close connection between these algorithms suggests that they are fundamental in the numerical analysis of stable discrete-time linear systems.

196 citations

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Philips

^{1}TL;DR: A classification is given of the various possible nonlinear effects that can occur in recursive digital filters due to signal quantization and adder overflow, which include limit cycles, overflow oscillations, and quantization noise.

Abstract: A classification is given of the various possible nonlinear effects that can occur in recursive digital filters due to signal quantization and adder overflow. The effects include limit cycles, overflow oscillations, and quantization noise. A review is given of recent literature on this subject. Alternative methods of avoiding some of these nonlinear phenomena are discussed.

181 citations

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TL;DR: In this paper, sufficient conditions are derived for a second-order statespace digital filter with L 2 scaling to be optimal with respect to output roundoff noise; and from these, a simple synthesis procedure is developed.

Abstract: Sufficient conditions are derived for a second-order statespace digital filter with L_2 scaling to be optimal with respect to output roundoff noise; and from these, a simple synthesis procedure is developed. Parallel-form designs produced by this method are equivalent to the block-optimal designs of Mullis and Roberts. The corresponding cascadeform designs are not equivalent, but they are shown, by example, to be quite close in performance. It is also shown that the coefficient sensitivities of this structure are closely related to its noise performance. Hence, the optimal design has low-coefficient sensitivity properties, and any other low-sensitivity design is a good candidate for near-optimal noise performance. The uniform-grid structure of Rader and Gold is an interesting and useful case in point.

158 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: In this paper, a systematic method is outlined to realize an m th-order all-pass digital transfer function using only m multipliers as a cascade of first-order and/or second-order sections.

Abstract: A systematic method is outlined to realize an m th-order all-pass digital transfer function using only m multipliers as a cascade of first-order and/or second-order all-pass sections. The realization is based on the multiplier extraction approach in which the n th-order filter section is considered as a digital (n + 1) -pair of which n pairs of input and output terminal variables are constrained by n multipliers. The transfer matrix parameters of the digital (n + 1) -pair, containing only delays and adders, are first identified from which the realization is obtained by inspection. Both canonic and noncanonic realizations are derived. All realizations are then compared with regard to the effect of multiplication roundoff and hardware requirements.

148 citations

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More filters

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TL;DR: In this paper, a systematic method is outlined to realize an m th-order all-pass digital transfer function using only m multipliers as a cascade of first-order and/or second-order sections.

Abstract: A systematic method is outlined to realize an m th-order all-pass digital transfer function using only m multipliers as a cascade of first-order and/or second-order all-pass sections. The realization is based on the multiplier extraction approach in which the n th-order filter section is considered as a digital (n + 1) -pair of which n pairs of input and output terminal variables are constrained by n multipliers. The transfer matrix parameters of the digital (n + 1) -pair, containing only delays and adders, are first identified from which the realization is obtained by inspection. Both canonic and noncanonic realizations are derived. All realizations are then compared with regard to the effect of multiplication roundoff and hardware requirements.

148 citations

### "A simple method of computing the in..." refers methods in this paper

...Example 3: As a last example, let us calculate analytically the output noise variance of the second-order digital filter [ 7 ] shown in Fig. 2 due to the product roundoff .caused by multipliers bz and - b3....

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TL;DR: In this paper, a numerical method for evaluating the complex integral I = \frac{1}{2\pij}, where I is a function of the number of polynomials in the unit circle in a positive direction.

Abstract: A numerical method for evaluating the complex integral I = \frac{1}{2\pij} \oint \frac{B(z)B(z^{-1})}{A(z)A(z^{-1})} \frac{dz}{z} along the unit circle in a positive direction (where A and B are polynomials with real coefficients), is presented in this paper. The method developed in this paper is shown to be obtainable as a FORTRAN program as well as a table form. The results achieved represent significant reduction of the computations compared to other existing methods.

66 citations

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22 citations