Minimum Roundoff Noise Research Articles

Synthesis of three–dimensional separable–denominator digital filters with minimum roundoff noise

AbstractA roundoff noise analysis is presented for three‐dimensional (3‐D) local state‐space digital filters with separable denominator. It is clarified that the output‐noise power and the signal scaling condition are closely related to three, and three positive‐definite matrices, respectively. A procedure is also developed for the synthesis of an optimal 3‐D local state‐space digital filter with separable denominator which minimizes the output‐noise due to roundoff subject to a scaling condition on the state variables. The optimal state‐space realization can be obtained through the 3‐D similarity transformation by individually solving three 1‐D optimization problems.

Minimum round‐off noise realizations of separable‐denominator two‐dimensional digital filters

AbstractIn real‐time two‐dimensional digital filters using a special‐purpose processor with fixed‐point arithmetic, the word length should be as small as possible to realize the desired processing accuracy. A digital filter which is affected least by quantization should be used. From such a viewpoint, this paper aims at the synthesis of the two‐dimensional digital filter structure with the separable denominator and with the minimum round‐off error. As the first step, the two‐dimensional digital filter with the separable denominator is represented by the local state‐space model. The result of analysis for the round‐off error already made by the authors is outlined. Based on these results, the synthesis of the structure with the minimum round‐off noise is formulated by equivalent transformation. A method of synthesis is presented. The two‐dimensional digital filter with the separable denominator and with the minimum roundoff noise, which is obtained by the synthesis method in this paper, is shown to be of a structure which does not produce an overflow oscillation for zero input. Finally, the effectiveness of the proposed method is shown by a numerical example.

Overflow and roundoff analysis of a very-low sensitivity digital filter structure†

Bhuyan and Chatterji (1981) have proposed a digital filter structure having very-low sensitivity for poles near z = 1. In this paper is shown that there is a range of values for the coefficient multipliers for which the structure is overflow stable. In addition, roundoff noise expressions are derived and it is shown that the structure also attains the minimum roundoff noise for values of the coefficients within this range..

Synthesis of 2-D state-space fixed-point digital-filter structures with minimum roundoff noise

Based on a roundoff-noise analysis, a general synthesis procedure is developed which leads to an optimal local state-space 2-D digital-filter realization that minimizes the output-noise power due to roundoff subject to a scaling condition on the state variables. The outputnoise power and the signal scaling condition are closely related to two positive-definite matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> . These matrices provide two sets of invariants, called the 2-D second-order modes of the filter, which play a crucial role in the minimization of the output-noise power. With the availability of matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> , the 2-D similarity transformation that yields an optimal state-space realization can be obtained by solving separately two 1-D optimization problems so that the well-developed techniques for minimizing roundoff noise in 1-D state-space digital filters can also be used for minimizing roundoff noise in 2-D state-space digital filters.

Roundoff noise minimization and pole-zero sensitivity in fixed-point digital filters using residue feedback

The problem of designing a finite wordlength fixed-point realization of an Nth-order digital filter, which uses residue feedback to minimize the output roundoff noise subject to l 2 -scaling, is considered. The new structures require N extra additions, but no more multiplications than the earlier low noise structures of Mullis and Roberts, and have lower roundoff noise for sufficiently narrow bandwidth filters. A new set of filter invariants, called the residue modes, are defined which characterize the new low noise structures and determine the output noise variance. If the sum of the residue modes is less than the sum of the second-order modes of Mullis and Roberts, then lower roundoff noise is achieved. Every filter structure is shown to define a unique (symmetrizing) matrix Q. Pole-zero sensitivities and a new noise measure are defined in terms of Q. Numerical results are included which compare the noise and pole-zero sensitivity characteristics of different filter structures.

A unified study on the roundoff noise in 2-D state space digital filters

This paper studies the roundoff noise and the scaling in 2-D state space digital filters. The noise matrix and the covariance matrix in the 2-D case are derived. They are expressed with double infinite sums or double integral. A formulation is presented to synthesize minimum roundoff noise 2-D state space digital filters. A practical synthesis procedure is also provided. A numerical example is given to show the analysis method and synthesis method presented here.

Synthesis of 2-D separable denominator digital filters with minimum roundoff noise and no overflow oscillations

This paper presents a new expression of the variance of roundoff noise in 2-D separable denominator digital filters described by Roesser's local state-space model. The covariance matrices and noise matrices necessary to analyze the variance of roundoff noise under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling are obtained as the solutions of Liapunov equations. The synthesis problem of 2-D separable denominator digital filters with minimum roundoff noise are formulated under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling. The synthesis method of 2-D minimum noise realizations under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling is proposed by applying the minimization technique of roundoff noise in the 1-D case. Furthermore, it is shown that 2-D minimum noise realizations are rotated and scaled balanced realizations. Using this property, 2-D minimum noise realizations are proved to be free of overflow oscillations under zero input conditions, if their second-order modes are distinct.

Optimal roundoff noise selection in class of minimal canonical nonproduct of multiplier digital filters

A study is presented for the roundoff performance of a class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> th-order digital filters known as minimal canonical nonproduct of multiplier in which a minimum amount of delay units <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">( \leq m)</tex> , multipliers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">( \leq 2 m)</tex> , and 2-input adders <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\leq 2 m)</tex> are used with the constraint that their transfer function expressions do not contain product of multiplier coefficient terms. An expression for roundoff noise gain is developed. This expression is useful in selecting structures with minimum roundoff noise among all existing equivalent structures of the previously mentioned class of digital filters.

Minimum roundoff noise digital filters with some power-of-two coefficients

A new technique and numerical algorithm are introduced for synthesizing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order minimum roundoff noise state-space structures for nth-order fixed-point recursive digital filters. The technique yields structures which employ <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n^{2} - n -1)</tex> trivial power-of-two multiplies and so require only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(3n + 2)</tex> nontrivial multiplies. This compares to the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)^{2}</tex> nontrivial multiplies required by other minimum-noise structures. Although the power-of-two structures do not satisfy theoretical conditions for roundoff noise optimality, their roundoff noise is found to be but negligibly higher than minimum. A numerical example illustrating the synthesis technique is provided.

Optimal design of state - space digital filters by simultaneous minimization of sensitivity and roundoff noise

The relation between a newly defined sensitivity measure and the noise power gain for scaled filters is given. By means of this relation it is shown that the necessary and sufficient condition to minimize the noise power gain is to minimize the sensitivity measure for unsealed filters. It is shown that the minimization of this sensitivity measure can be carried out without taking into account the dynamic range constraint. A procedure for the simultaneous minimization of the sensitivity measure and the noise power gain is also given.

On the design of optimal state-space realizations of second-order digital filters

Simple design equations are derived for optimal (minimum roundoff noise) state-space realizations of second-order digital filter sections with complex poles. These relations permit easy computation of filter coefficients and unit noise gain directly from the filter transfer function. New parameters are identified that are easily computed from the transfer function, are invariant under certain frequency-scaling transformations, and which can be used to simply describe all of the design variables of optimal second-order filters, including state covariance matrix, unit noise matrix and second-order modes. The design relations are illustrated by computing the unit noise gains and eigenvalue sensitivities of optimal parallel realizations of Chebyshev lowpass filters.

Design of sensitivity and round-off noise optimal state-space discrete systems

A previously defined sensitivity measure for state-space realizations is used, and the attainable minimum value and its necessary and sufficient conditions are given. An explicit optimization procedure which evaluates the complete class of realizations with minimum sensitivity is proposed. Conditions for minimizing the sensitivity under an 1, dynamic range constraint are established and an explicit method for obtaining this minimum is given. Since the relation between this sensitivity measure and the noise power gain for I, scaled digital filters can be given, and since the main part of the optimization can be carried out without taking into account the dynamic range constraint, a simple procedure for simultaneous minimization of sensitivity and round-off noise is obtained. A numerical example is given to exemplify the computational procedure.

FIR filter structures having low sensitivity and roundoff noise

A class of structures for FIR filters is presented, which exhibits reduced coefficient sensitivity and superior roundoff noise properties as compared to the direct form realization. It is shown that using fixed-point arithmetic, these structures achieve the same accuracy and about the same roundoff noise as those obtained in the floating-point implementationn. The optimum structure to achieve the minimum roundoff noise can be found; in most cases near optimum results are easily obtained by simple permutations and combinations of the impulse response coefficients. While a serial form realization of these structures requires a certain amount of software complexity, a parallel form, on the other hand, does not require additional complexity.

Minimum round-off noise realization of block-state recursive digital filters

The development of a block-state structure with minimum round-off noise subject to l2-norm dynamic range constraint is outlined. The pertinent equations for scaling and for round-off noise analysis of block-state structures implemented using fixed-point arithmetic are first derived. Next, a lower bound and the global minimum of the output noise due to the round-off of the block-state-variables are derived. A method of deriving the minimum round-off noise block state-structure is outlined. A numerical example is included. With regard to computational complexity and overall noise performance, the block-state realization of recursive digital filters is shown to be superior.

Roundoff noise and overflow in normal digital filters

Minimum-norm realizations of fixed-point digital filters provide guaranteed immunity from autonomous overflow limit cycles. This paper presents an analysis of the relationship between dynamic range and roundoff noise for a class of minimum-norm realizations called "normal." For normal realizations, the eigenvectors of the system matrix form an orthogonal basis for the system state space. An explicit expression for minimum roundoff noise, under an 12 dynamic range constraint, is derived, and means for achieving this minimum are given. The simple expression for minimum roundoff noise permits easy determination, by trial and error, of optimal subfilter structures. Explicit expressions for the state-space parameters of optimal secondorder normal filters are given.

Synthesis of minimum roundoff noise fixed point digital filters

Beginning with an external specification of a digital filter, structures which minimize roundoff noise are investigated. After fixing the probability of overflow through an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l_{2}</tex> scaling procedure, roundoff noise is studied via the internal structure of the filter using a state variable formulation. An output noise variance formula in terms of the internal structure is derived. Conditions for minimizing this output noise are established and realizations which meet these conditions are constructed. A new set of filter invariants called second-order modes are defined and shown to play a definitive role in minimal noise realizations. From these invariants, for example, one can calculate the minimal output noise variance of a given external specification. Numerical results are given which compare these new filter structures with the usual parallel and cascade connections of second-order filters, both theoretically and through simulations. For narrow-band filters, these new structures can be orders of magnitude better (in terms of output noise variance). One drawback of these new structures is a large increase in the number of multipliers needed to realize them. However, by applying the theory to subfilters connected in parallel and cascade, a good compromise between output noise and number of multipliers is obtained.