Roundoff Noise Gain Research Articles

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.

An input-balanced realization-based structure for efficient implementation of digital IIR filters

Based on an input-balanced state-space realization, this paper derives a novel structure for efficient implementation of digital IIR filters. Such a structure is very similar to a normalized lattice and, for an N th order IIR filter, it possesses 5 N -1 non-trivial multipliers: two multipliers less than in a normalized lattice. Expressions for transfer function sensitivity measures and roundoff noise gain for the proposed structure are derived. Interestingly, it is shown that, when the filter is all-pass, the proposed structure requires 4 N multiplications in order to compute one output sample. Moreover, the corresponding roundoff noise gain is 4 N , much smaller than that of ( N +1) 2 given by the classical minimum roundoff state-space realizations. Results of simulations performed show that the proposed structure outperforms many classical ones in terms of implementation efficiency and robustness against finite-word-length effects.

Generalised modal realisation as a practical and efficient tool for FWL implementation

Finite word length (FWL) effects have been a critical issue in digital filter implementation for almost four decades. Although some optimisations may be attempted to get an optimal realisation with regards to a particular effect, for instance the parametric sensitivity or the round-off noise gain, the purpose of this article is to propose an effective one, i.e. taking into account all the aspects. Based on the specialised implicit form, a new effective and sparse structure, named ρ -modal realisation, is proposed. This realisation meets simultaneously accuracy (low sensitivity, round-off noise gain and overflow risk), few and flexible computational efforts with a good readability (thanks to sparsity) and simplicity (no tricky optimisation is required to obtain it) as well. Two numerical examples are included to illustrate the ρ -modal realisation's interest.

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.

An improved orthogonal digital filter structure

A novel structure is derived for digital filter implementation. This structure is actually an improved version of an existing one in terms of implementation efficiency and reducing finite word length (FWL) effects. Expression of roundoff noise gain is obtained for the proposed structure. Design examples are given to demonstrate the performance of this structure and to compare it with the existing one and the classical minimum roundoff state-space realizations. Numerical examples show that the proposed structure outperforms the others in terms of minimizing roundoff noise as well as implementation efficiency.

Robust state-space realizations of digital filters against finite word length errors *

This article deals with two important issues in digital filter implementation: roundoff noise and limit cycles. A novel class of robust state-space realizations, called normal realizations, is derived and characterized. It is seen that these realizations are free of limit cycles. Another interesting property of the normal realizations is that they yield a minimal error propagation gain. The optimal realization problem, defined as to find those normal realizations that minimize roundoff noise gain, is formulated and solved analytically. A design example is presented to demonstrate the behavior of the optimal normal realizations and to compare them with several well-known digital filter realizations in terms of minimizing the roundoff noise and the error propagation.

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.

A Matrix Factorization-Based Structure for Digital Filters

In this correspondence, a novel digital filter structure is derived using matrix factorization. This structure, called the LCW-structure, is an improved version of the so-called LGS-structure proposed by Li, Gevers, and Sun [ldquoPerformance Analysis of a New Structure for Digital Filter Implementation,rdquo IEEE Transactions on Circuits and Systems I, vol. 47, no. 4, pp. 474-482, April 2000]. For a digital filter of order , this new structure requires 4N - 1 multiplications and 4N - 1 additions for computing one filter output sample, instead of 7N - 3 and 6N - 3, respectively, which are needed for the LGS-structure. An expression of the roundoff noise gain is derived for the LCW-structure. Design examples are presented to demonstrate the behavior of this structure and to compare with several well-known digital filter structures in terms of minimizing roundoff noise and implementation efficiency.

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 analysis of two efficient digital filter structures

In this correspondence, two efficient structures are analyzed for the implementation scheme of rounding before multiplication (RBM). The first one is the direct-form II transposed structure in /spl rho/-operator (/spl rho/DFIIt), recently proposed in . The expression of roundoff noise gain is derived, based on which a revised /spl rho/DFIIt structure, denoted as /spl rho/RDFIIt, is obtained. It is shown that the /spl rho/RDFIIt structure, having the same implementation complexity, yields a smaller roundoff noise gain than the /spl rho/DFIIt. The roundoff noise gain for the /spl rho/RDFIIt structure with error feedbacks is also derived. Each structure can be optimized with regard to the free parameters it possesses with a genetic algorithm. Numerical examples are presented, which show that the optimized structures are very competitive because they can even outperform the traditional optimal state-space realization in terms of the roundoff noise performance as well as the implementation simplicity.

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.

Two efficient structures for 2D digital filter implementation

In the paper, a new efficient structure for two-dimensional digital filters with separable denominator, denoted as 2D ZL, is derived based on a set of special polynomial operators. For a (M, N)th order filter, this structure possesses MN+3(M+N)+1 nontrivial parameters plus M+N free parameters, compared to the optimal state-space realisation which has (M+N+1)2 nontrivial parameters. The equivalent state-space realisation of the 2D ZL yields another interesting efficient structure. Performance analysis is carried out for both structures by deriving the expressions of roundoff noise gain and L2-sensitivity measure. A genetic algorithm is proposed to efficiently solve the optimal structure problems. Two examples are given to show that the optimised proposed structures can yield a better performance than the shift-operator and delta-operator based structures and even outperform the optimal fully parameterised state-space realisation.

On the Generalized DFIIt Structure and its State–Space Realization in Digital Filter Implementation

It is well known that for a digital filter of order p, the number of nontrivial parameters in the classical optimal state-space realizations is proportional to p/sup 2/, while the traditional shift operator z-based direct-form II transposed (zDFIIt) structure, though having poor numerical properties, is one of the most efficient structures, just possessing 3p+1 nontrivial parameters. In this paper, based on the concept of polynomial operators, a new structure is proposed for digital filter implementation, which is a generalization of the traditional zDFIIt and the prevailing /spl delta/DFIIt structures. This structure, denoted as /spl rho/DFIIt, possesses 3p+1 nontrivial parameters plus p parameters at choice. Expressions for evaluating the sensitivity measure and the roundoff noise gain are derived for the /spl rho/DFIIt structure and its equivalent state-space realization that has the same structure complexity. It is shown that the state-space realization always yields a smaller roundoff noise gain than the /spl rho/DFIIt structure. One of the nice properties of these two structures is that for a given digital filter, they can be optimized with the p free parameters. The optimal structure problems can be solved with exhaustive researching under practical considerations. Numerical examples are presented to illustrate the design procedure.

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.

A Polynomial-Operator-Based DFIIt Structure for IIR Filters

In this paper, based on a polynomial-operator approach, a new structure is derived. This structure is a generalization of the transposed direct-form II structures in the conventional shift operator and the prevailing delta operator. The use of polynomial operators provides more degrees of freedom to minimize the roundoff noise gain of the filter without increasing the implementation complexity. Numerical examples are presented to illustrate the design procedure.

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.

Optimum structures of digital controllers in sampled-data systems: a roundoff noise analysis

In this paper, the effect of roundoff noise in a digital controller is analyzed for a sampled-data system in which the digital controller is implemented in a state-space realization. A new measure, called averaged roundoff noise gain, is derived. Unlike the traditionally used measure, where the analysis is done based on an equivalent digital control system, this newly defined averaged roundoff noise gain allows us to take consideration of the inter-sample behavior. It is shown that this measure is a function of the state-space realization. Noting the fact that the state-space realizations of a digital controller are not unique, the problem of optimum controller structure is to identify those realizations that minimize the averaged roundoff noise gain subject to the $l_2$-scaling constraint which is for preventing the signals in the controller from overflow. An analytical solution to the problem is presented and a design example is given. Both theoretical analysis and simulation results show that the optimum controller realizations obtained with the proposed approach are superior to those obtained with the traditional analysis based on a digital control system.

A generalized direct-form delta operator-based IIR filter with minimum noise gain and sensitivity

This brief presents the derivation of an arbitrary order delta operator-based direct-form IIR filter with minimum roundoff noise gain and sensitivity. It utilizes the concept of different coupling coefficients at different branch nodes for better noise gain suppression. Two possible structures for realizing the inverse delta operator are considered and procedures for calculating the optimal filter coefficients are given. By means of state-space representation and matrix manipulation, it is also shown that expressions for sensitivity measures of different filter coefficients and their corresponding roundoff noise gain expressions are the same. This enables the simultaneous minimization of sensitivity and noise power for the proposed generalized filter structure.