Low Roundoff Noise Research Articles

New second-order state-space structures for realizing low roundoff noise digital filters

Several new second-order state-space digital filter structures are introduced. Each of the new structures requires one or two less multiplies than the minimum-noise structure, while achieving a round-off noise performance which is nearly as good. Thus, they provide an excellent means of realizing second-order subfilters. When such subfilters are used to construct nth-order filters, the reduction in multiplies, as compared to using minimum-noise subfilters, is from about 4n to about 3n. An approach to developing simple, algebraic, real-arithmetic design equations for L 2 -norm scaled second-order structures is presented and then applied to obtaining design equations for the new structures and for the minimum-noise structure.

Computationally efficient second-order digital filter sections with low roundoff noise gain

A class of second-order recursive digital filter realizations is described that provides near-optimal roundoff noise performance, low-coefficient sensitivity, and freedom from overflow limit cycles. The number of multiplications required, per output sample, is one more than the canonical direct form realization. This class of realizations provides an attractive compromise between the computationally intensive optimal realization and the finite-word-sensitive direct form. Explicit design procedures are provided, and a numerical design example is presented that illustrates the near-optimal performance of this new class of realizations.

Alternative low-noise realizations for narrow-band recursive digital filters

Alternatives to Munson and Liu's structures [1] are given, which also realize a second-order section having both poles and zeros, with low sensitivity and roundoff noise. Instead of realizing zeros separately from poles, zeros are combined with poles in poles preceding zeros form. An illustration is worked out for comparison, and it is shown that such a realization could have better signal-to-noise ratio without any additional hardware.

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.

Extended state-space model of discrete-time dynamical systems

A model for discrete-time dynamical systems is discussed in which the future values of the internal variables depend on the present and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K - 1</tex> previous time instants, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> is the order of the model. We outline transition matrix and impulse response calculations, equivalences between model and conventional state variable models, methods of constructing such a model, and means for calculating certain quantities relevant to the design of cost-optimal low-roundoff noise digital filters.

The Normal Lattice-A Cascade Digital Filter Structure

A normal lattice is defined as a cascade of two-input two-output second-order normal form digital filter modules. An analysis of this structure is presented along with an iterative algorithm for constructing the system eigenvectors, a scaling algorithm, and an explicit expression for roundoff noise gain. These expressions and algorithms provide an explicit design procedure for realizing arbitrary rational transfer functions for single-input single-output filters. Comparisons with other digital filter structures show that the normal lattice has very low roundoff noise and low coefficient sensitivity in realizations of narrow-band filters. This structure is also well adapted to the realization of complex filters.

A new class of very low sensitivity and low roundoff noise recursive digital filter structures

A class of new structures is proposed, for the realization of the recursive digital filter with poles near the unit circle, by introducing the particular type of delay replacement scheme in the existing digital filter structure and then, inserting, delays at the. proper places to avoid the delay-free loops. By applying the proposed method to the second-order direct form structures, new structures are found with very low magnitude transfer function sensitivities with respect to multiplier coefficients and low roundoff noises. Some numerical results are also presented.

On low-roundoff noise and low-sensitivity digital filter structures

This note presents a simple alternative form for the error-feedback digital filter structure which, with some minor modification of assumptions, can also represent the low-sensitivity and low-noise structures of Agarwal and Burrus. This form can thus explain why the Agarwal-Burrus structures have low-roundoff noise and can be shown to have low sensitivity. Finally, this form is generalized to define a new class of low-noise and low-sensitivity realization structures for digital filters.

Fast state-space decimator with very low round-off noise

A method for the efficient realization of high-order Infinite Impulse Response (IIR) output-decimating State-Space Digital Filters (SSDF) is presented. By applying the state decimation and block processing to the nth-order narrow-band SSDFs, computational efficiency as well as low round-off noise levels are achieved. Further, the favorable behavior of certain SSDF forms with respect to the overflow limit cycles is retained. Such Fast State-Space IIR Decimators (FSSD) are simple to design, have multiplication rates comparable to Finite Impulse Response (FIR) decimators, have a highly parallel processing algorithm, and are very easy to implement on programmable signal processors. In the digital filter applications which do not require the linear-phase response, the FSSD may be considered as an alternative to a single-stage FIR decimator for moderate decimation factors.

Low roundoff noise and normal realizations of fixed point IIR digital filters

Explicit design and performance equations are given in terms of the parameters of the desired transfer function H(z) for two classes of realizations. These realizations offer the digital circuit designer an alternative to the usual designs based on direct forms. In addition to low roundoff noise, these new realizations offer other desirable properties such as low coefficient sensitivities and freedom from overflow oscillations. Numerical results are presented which detail the design process and verify the theory.

Fast Digital Filters with Low Round-Off Noise

A novel approach to the realization of fast and efficient IIR digital filters is presented. The realization is based on recently introduced state-space structures and retains and enhances their low noise and sensitivity properties while reducing the number-of-required multiplications. This permits the implementation of higher order optimal forms requiring only 1.34 to 1.65 times the number of multiplications used in the direct form (well known for 'its poor coefficient sensitivity and high output noise levels). The high inherent parallelism of the state-space structure is also further enhanced by the proposed fast digital filter (FDF). The resulting block-processing algorithm is suitable for high speed implementation of digital filters on parallel processing systems. These systems employ fast and efficient techniques, such as distributed arithmetic or multimicroprocessor techniques, to perform the required computation of the long-independent inner products. In addition to low quantization noise, the FDF structure has other desirable properties including low sensitivity and improved limit cycle behavior. For output decimating IIR filters additional savings in the number of multiplications is achieved with the proposed structure.

Intuitive explanation for low roundoff noise in normal realizations of narrow-band digital filters

The low roundoff noise performance of narrow-band low-pass digital filters realized in normal form is explained in terms of the zeros of the internal transfer functions that control the dynamic range-roundoff noise interaction. For narrow-band filters that have pole pairs close to the real axis, the internal transfer functions of normal realizations have a zero between each of these pole pairs. The effect of this zero is to reduce the norm of the transfer function, and produce a more favorable tradeoff between dynamic range and roundoff noise.

Correction to "Comparison of round-off noise variances in several low round-off noise digital filter structures"

Correction to "Comparison of round-off noise variances in several low round-off noise digital filter structures"

Comparison of round-off noise variances in several low round-off noise digital filter structures

This note presents round-off noise expressions for fixed-point digital filters realized in the error-feedback structure, and a comparison of the error-feedback structure with other low round-off noise digital filter structures in terms of output round-off noise variance.

Recursive digital filters with low roundoff noise

AbstractA novel method is introduced for designing recursive digital filters with very low roundoff noise. The proposed method is based on reducing the magnitudes of the noise transfer functions on the unit circle by ensuring the presence of zeros, while simultaneously synthetizing the prescribed transfer function. The final structure, which exhibits steady‐state roundoff noise with low variance, is simple and economical, and canonic with respect to delay elements, although not always canonical with respect to multipliers. An illustrative example is included. The realization method is also shown to reduce the multiplier coefficient sensitivities.

Optimization of digital filters for low roundoff noise

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.