Minimum Roundoff Noise Research Articles

Analysis and Minimization of Roundoff Noise for Generalized Direct-Form II Realization of 2-D Separable-Denominator Filters

Analysis and Minimization of Roundoff Noise for Generalized Direct-Form II Realization of 2-D Separable-Denominator Filters

Weighted Minimization of Roundoff Noise and Pole Sensitivity Subject to <i>l</i><sub>2</sub>-Scaling Constraints for State-Space Digital Filters

Weighted Minimization of Roundoff Noise and Pole Sensitivity Subject to <i>l</i><sub>2</sub>-Scaling Constraints for State-Space Digital Filters

An Improved Lattice Filter Structure with Minimum Roundoff Noise Gain

In this paper, a new filter structure is derived by combining the lattice structure and error feedback technique, which contains a free parameter used for structure optimization. The roundoff noise of the proposed structure is analyzed, and the corresponding mathematical expression is deduced. The problem of how to optimize the free parameter is investigated in terms of minimizing the roundoff noise gain. When the free parameter is expressed as an integer power-of-two, for an Nth-order digital filter, the proposed structure requires $$5N+1$$5N+1 multipliers, which yields the same implementation complexity as the normalized lattice structure. A numerical example is given to show the finite wordlength performance of the proposed structure.

Roundoff Noise Minimization in State-Space Discrete-Time Systems Using Joint Optimization of High-Order Error Feedback and Realization

This paper deals with the joint optimization problem of high-order error-feedback and realization for minimizing roundoff noise at the output of a closed-loop system with a full-order observer feedback system subject to l2-scaling constraints. It is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques. The unconstrained optimization problem at hand is then solved iteratively by employing an efficient quasi-Newton algorithm with closed-form formulas for key gradient evaluation. A case study is presented to demonstrate the validity and effectiveness of the proposed technique.

Roundoff Noise Minimization for a Class of 2-D State-Space Digital Filters Using Joint Optimization of High-Order Error Feedback and Realization

Roundoff Noise Minimization for a Class of 2-D State-Space Digital Filters Using Joint Optimization of High-Order Error Feedback and Realization

Jointly Optimal High-Order Error Feedback and Realization for Roundoff Noise Minimization in 1-D and 2-D State-Space Digital Filters

Joint optimization of high-order error feedback and state-space realization for minimizing roundoff noise at filter output subject to l2-scaling constraints is investigated for one- and two-dimensional state-space digital filters. Linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored, and an efficient quasi-Newton algorithm is then applied to solve the unconstrained optimization problem iteratively. In this connection, closed-form formulas are derived for fast and accurate gradient evaluation. Finally, case studies are presented to demonstrate that the high-order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state-space coordinate transformation and high-order error feedback matrices are optimized separately.

An Efficient and Robust Orthogonal Structure for Linear Discrete-Time Systems

An orthogonal structure is derived for system implementation. The expression of roundoff noise propagation gain for this structure is obtained. The proposed structure is an improved version of that reported in the paper by G. Li, M. Gevers, and Y.X. Sun (Performance analysis of a new structure for digital filter implementation, IEEE Trans. Circuits Syst. I 47:474–482, 2000), but is more efficient and robust against the quantization errors. For comparison, an alternative expression of roundoff noise gain for the normalized lattice structure is derived, based on which it is shown that the roundoff noise gain of an Nth order all-pass system, when implemented using the normalized lattice structure, is 4N and that an Nth order system implemented with the structure proposed in the paper by P.P. Vaidyanathan, S.K. Mitra, and Y. Neuvo (A new approach to the realization of low sensitivity IIR digital filters, IEEE Trans. Acoust. Speech Signal Process. ASSP-34(2):350–361, 1986), yields a roundoff noise gain of N+1, smaller than that of the classical optimal roundoff noise state-space realizations. Design examples are presented to illustrate the behavior of the proposed structure and to compare it with a class of existing orthogonal structures and the classical optimal roundoff noise realizations. It is shown that the proposed structure outperforms the others in terms of minimization of roundoff noise as well as implementation efficiency.

A novel digital filter structure with minimum roundoff noise

In this paper, a novel structure is derived for efficient implementation of digital filters as well as minimizing the finite word length (FWL) errors. Such a structure is actually an improved version of that reported previously. The performance of this new structure and the famous normalized lattice structure are analyzed by deriving the corresponding expression for the roundoff noise gain. Design examples are presented to illustrate the behavior of the proposed structure and to compare it with some existing ones. It is shown that the proposed structure outperforms the others in terms of minimizing roundoff noise as well as implementation efficiency.

On normal realizations of digital filters with minimum roundoff noise gain

It is well known that normal realizations are free of limit cycles and that a digital filter implemented with a state-space realization ( A , B , C , d ) has no limit cycles if there exists some diagonal matrix D > 0 such that D - A T DA ⩾ 0 . In this brief, a method is proposed to check the existence of such a D for any given realization. It is also shown that the normal realizations have a minimal error propagation gain. More interestingly, the normal realizations are characterized, the minimum roundoff noise normal realization problem is formulated and solved analytically. An example is presented to test the efficiency of the proposed method and to demonstrate the performance of the proposed optimal normal realizations.

Separate/joint optimization of error feedback and realization for roundoff noise minimization in a class of 2-D state-space digital filters

Techniques for the separate/joint optimization of error-feedback and realization are developed to minimize the roundoff noise subject to l 2-norm dynamic-range scaling constraints for a class of 2-D state-space digital filters. In the joint optimization, the problem at hand is converted into an unconstrained optimization problem by using linear-algebraic techniques. The unconstrained problem obtained is then solved by applying an efficient quasi-Newton algorithm. A numerical example is presented to illustrate the utility of the proposed techniques.

An efficient controller structure with minimum roundoff noise gain

Based on a polynomial operator approach, a new sparse controller structure is derived, which is actually an improved version of the recently proposed structure [Li, G. (2004). A polynomial-operator-based DFIIt structure for IIR filters. IEEE Transactions on Circuits and Systems II, 51, 147–151]. The performance of the proposed structure is analyzed by deriving the corresponding expression of roundoff noise gain and the problem of finding optimized sparse structures is solved efficiently with a genetic algorithm (GA). A numerical example is given, which shows that the newly developed structure can achieve much better performance than some well-known structures and particularly outperforms the traditional optimal fully parametrized realization greatly in terms of reducing roundoff noise and implementation complexity.

An Efficient State-Space Realization With Minimum Roundoff Noise Gain

This paper deals with efficient digital filter structures with roundoff noise consideration. Motivated by the direct-form II transposed (DFIIt) structure in rho-operator (rhoDFIIt) an alternative structure is obtained [Li in 2005, where, instead of the first-order rho-operators in rhoDFIIt, a set of second-order polynomial operators is used. In this paper, with the rounding before multiplication implementation taken into account, the equivalent state-space realization of the proposed structure by Li is derived and its roundoff noise performance is analyzed by deriving the roundoff noise expressions without/with error feedback consideration. This state-space realization can be efficiently implemented and has more degrees of freedom than its counterpart rhoDFIIt, which can be utilized to minimize the roundoff noise gain. A genetic algorithm is proposed to efficiently solve the optimal structure problem. Extensive examples are given to illustrate the advantage of this state-space realization and support the theoretical analysis

Roundoff Noise Minimization for 2-D State-Space Digital Filters Using Joint Optimization of Error Feedback and Realization

The joint optimization problem of error feedback and realization for two-dimensional (2-D) state-space digital filters to minimize the effects of roundoff noise at the filter output subject to L2-norm dynamic-range scaling constraints is investigated. It is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques. The unconstrained optimization problem at hand is then solved iteratively by applying an efficient quasi-Newton algorithm with closed-form formulas for key gradient evaluation. Analytical details are given as to how the proposed technique can be applied to the cases where the error-feedback matrix is a general, block-diagonal, diagonal, or block-scalar matrix. A case study is presented to illustrate the utility of the proposed technique

Jointly optimized error-feedback and realization for roundoff noise minimization in state-space digital filters

Roundoff noise (RN) is known to exist in digital filters and systems under finite-precision operations and can become a critical factor for severe performance degradation in infinite impulse response (IIR) filters and systems. In the literature, two classes of methods are available for RN reduction or minimization-one uses state-space coordinate transformation, the other uses error feedback/feed-forward of state variables. In this paper, we propose a method for the joint optimization of error feedback/feed-forward and state-space realization. It is shown that the problem at hand can be solved in an unconstrained optimization setting. With a closed-form formula for gradient evaluation and an efficient quasi-Newton solver, the unconstrained minimization problem can be solved efficiently. With the infinite-precision solution as a reference point, we then move on to derive a semidefinite programming (SDP) relaxation method for an approximate solution of optimal error-feedback matrix with sum-of-power-of-two entries under a given state-space realization. Simulations are presented to illustrate the proposed algorithms and demonstrate the performance of optimized systems.

An improved /spl rho/DFIIt structure for digital filters with minimum roundoff noise

In this brief, a new efficient digital filter structure with minimum roundoff noise is derived. This structure can be block-diagrammed as the recently proposed direct-form II transposed (DFIIt) structure in /spl rho/-operator, denoted as /spl rho/DFIIt, in which the first-order /spl rho/ operators are replaced with a set of second-order operators. The corresponding expression for the roundoff noise gain is obtained. The problem of how to choose these second-order operators is investigated in terms of minimizing the roundoff noise gain. Two design examples are given to illustrate the excellent performance of the proposed structure and to confirm the theoretical analysis.

Sparse controller realisation with small roundoff noise

The effect of roundoff noise in a digital controller is analysed for a digital feedback control system. An analytical expression for the roundoff noise gain, defined as the ratio between the variances of the output error and the rounding error, is obtained. The problem of identifying the minimum roundoff noise realisations can be solved using an existing procedure. Noting that the optimal realisations are fully parametrised, based on a polynomial operator approach a new sparse controller realisation is derived. This realisation is a generalisation of the direct forms in the classical shift operator and the prevailing delta-operator. It provides more degrees of freedom with which to reduce the roundoff noise. The problem of finding optimal polynomial operators can be solved by an exhaustive search, and a design example is given. It is shown that with the proposed sparse realisation the optimal polynomial operators can outperform the shift- and delta-operators.

状態推定フィードバックディジタル制御器の丸め誤差を最小化する誤差フィードバックと座標変換の同時最適化

状態推定フィードバックディジタル制御器の丸め誤差を最小化する誤差フィードバックと座標変換の同時最適化

Efficient structure for 2-D digital filters with minimum roundoff noise gain

Based on a polynomial operator approach a new structure for two-dimensional (2-D) digital filters is derived. This is a sparse structure and has the degrees of freedom to reduce the roundoff noise gain. A numerical example is given, which shows that the optimised structure can even surpass the classical optimal roundoff noise realisation.

Separate/joint optimization of error feedback and coordinate transformation for roundoff noise minimization in two-dimensional state-space digital filters

This paper is concerned with the minimization of roundoff noise subject to l/sub 2/-norm dynamic-range scaling constraints in two-dimensional (2-D) state-space digital filters. Two methods are proposed, with the first one using error feedback alone and the second one using joint error feedback and coordinate transformation optimization. In the first method, several techniques for the determination of optimal full-scale, block-diagonal, diagonal, and scalar error-feedback matrices for a given 2-D state-space digital filter are proposed. In the second method, an iterative approach for minimizing the roundoff noise under l/sub 2/-norm dynamic-range scaling constraints is developed by jointly optimizing a scalar error-feedback matrix and a coordinate transformation matrix, which may be regarded as an alternative approach to the conventional method for synthesizing the optimal 2-D filter structure with minimum roundoff noise. An analytical method for the joint optimization of a general error-feedback matrix and a coordinate transformation matrix under the scaling constraints is also proposed. A numerical example is presented to illustrate the utility of the proposed techniques.

Roundoff noise minimization of state-space digital filters using separate and joint error feedback/coordinate transformation optimization

This paper investigates the problem of minimizing roundoff noise under l/sub 2/-norm dynamic-range scaling constraints in state-space digital fibers by means of error feedback as well as joint error feedback/coordinate transformation optimization. First, several techniques, for the determination of optimal full-scale, diagonal, and scalar error-feedback matrices for a given state-space digital filter are proposed, where three realization schemes, namely, the general state-space realization, input-balanced realization, and optimal realization in the sense of Hwang-Mullis-Roberts are examined. Furthermore, an iterative approach is developed for jointly optimizing a scalar error-feedback matrix and a coordinate transformation matrix so as to minimize the roundoff noise subject to the l/sub 2/-norm dynamic-range scaling constraints. The proposed method may be regarded as an alternative, but much simpler and more general, approach to Hwang's method for synthesizing the optimal filter structure with minimum roundoff noise. A case study is included to illustrate the utility of the proposed techniques.