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.     

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$L_2$-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.     

Minimization of L/sub 2/-sensitivity for state-space digital filters subject to L/sub 2/-dynamic-range scaling constraints

The problem of minimizing an L/sub 2/-sensitivity measure subject to L/sub 2/-norm dynamic-range scaling constraints for state-space digital filters is formulated. It is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques. Next, the unconstrained optimization problem is solved by applying an efficient quasi-Newton algorithm with closed-form formula for gradient evaluation. The coordinate transformation matrix obtained is then used to construct the optimal state-space filter structure that minimizes the L/sub 2/-sensitivity measure subject to the scaling constraints. A numerical example is presented to illustrate the utility of the proposed technique.     

L/sub 2/-sensitivity minimization of one- and two-dimensional state-space digital filters subject to L/sub 2/-scaling constraints

This paper investigates the problem of minimizing an L/sub 2/-sensitivity measure subject to L/sub 2/-norm dynamic-range scaling constraints for one-dimensional (1-D) as well as a class of two-dimensional (2-D) state-space digital filters, where the 2-D digital filters are described by a transposed structure of the Fornasini-Marchesini second local state-space model. In each case, a novel iterative technique is developed to solve the constraint optimization problem directly. The proposed solution methods are largely based on the use of a Lagrange function and some matrix-theoretic techniques. Numerical examples are presented to demonstrate the effectiveness of the proposed techniques.     

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.     

一般二维数字系统的低噪声优化实现

本文给出了一般二维数字系统的两种高效率状态空间结构;导出了这两种结构的能控性和能观性格拉姆矩阵的有关性质;建立了二维数字系统定点运算的全局噪声模型;据此获得了一般二维数字系统低噪声、高效率的优化实现,文中举例说明了本文的两种结构的噪声性能和计算效率各不相同。     

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     

L/sub 2/-sensitivity analysis and minimization of 2-D separable-denominator state-space digital filters

For two-dimensional (2-D) state-space digital filters that are separable in the denominator, the coefficient sensitivity is analyzed by using a pure L/sub 2/-norm, and then, the problem of minimizing the L/sub 2/-sensitivity is considered. First, a novel expression is developed in closed form for the evaluation of the L/sub 2/-sensitivity. Next, an iterative procedure is presented for synthesizing the optimal filter structures that minimize the L/sub 2/-sensitivity. Finally, a numerical example is given to illustrate the utility of the proposed technique.     

Implementation of 2-D pseudo-rotated digital filters using a forward lattice structure

The 1-D FDLS shows the localized feedback property and is suitable for modular and concurrent implementation. It is known that the 1-D FDLS shows interesting properties with respect to finite word-length effects. In this paper, a new result is given for the estimation of the lower and upper bound of the variance of the roundoff noise. It is presented how the FDLS can be incorporated to implement 2-D pseudo-rotated digital filters. The 1-D roundoff noise analysis is extended to the 2-D case. It is indicated how 2-D filter banks can be derived from the FDLS.     

并行计算的全通数字滤波器结构

本文提出了两类并行计算的全通数字滤波器结构,并通过状态空间分析方法论述了并行处理的原理.通过舍入噪声分析,给出了噪声增益的表达式,对于一个N阶全通滤波器,其舍入噪声增益为4N.数值算例验证了所提结构的性能,同时,由于其具备并行处理的能力,更加适合高吞吐量的系统实现.     

Noise Reduction in Two-Dimensional Recursive Digital Filters with Discrete Optimized Error Feedback Coefficients

This paper treats the minimization problem for roundoff noise in two-dimensional (2-D) recursive digital filters with an error feedback (EF). The EF is known as an available scheme for the reduction of roundoff noise, and this can be achieved by extracting a roundoff error from the quantizer and feeding it back to the adder. Since the EF circuit is an additional device of the digital filter, the bit length of the coefficients should be as short as possible for the fast arithmetic and low-cost realization. In this paper, we propose an algorithm to optimize the EF coefficients in a discrete space. The proposed algorithm is based on the branch and bound method by estimating the lower bound of the optimal solution, and the optimality of the obtained solution can be guaranteed. Finally, we?give?a?numerical example to demonstrate the effectiveness of the proposed method, and show the optimal solution can be found without using the round-robin algorithm.     

Robust H/sub /spl infin// filtering for Uncertain2-D continuous systems

This paper considers the problem of robust H/sub /spl infin// filtering for uncertain two-dimensional (2-D) continuous systems described by the Roesser state-space model. The parameter uncertainties are assumed to be norm-bounded in both the state and measurement equations. The purpose is the design of a 2-D continuous filter such that for all admissible uncertainties, the error system is asymptotically stable, and the H/sub /spl infin// norm of the transfer function, from the noise signal to the estimation error, is below a prespecified level. A sufficient condition for the existence of such filters is obtained in terms of a set of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired H/sub /spl infin// filter is given. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed method.     

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.     

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.     

An improved LMI-based criterion for the design of optimal guaranteed cost controller for 2-D discrete uncertain systems

An improved criterion for the design of optimal static-state feedback guaranteed cost controller for two-dimensional (2-D) discrete uncertain systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model is proposed. The design problem of the optimal guaranteed cost controller is formulated as a convex optimization problem with linear matrix inequality (LMI) constraints. The proposed method yields tighter upper bound of the closed-loop cost function as compared to a recently reported method.     

Roundoff noise in multirate digital filters

The roundoff noise problem for multirate digital filters is difficult because rate changing inside the filter makes both scaling and roundoff noise calculation complicated. With a new multirate filter analysis technique, the transfer function between any two points inside a multirate filter can be easily found, thus, simplifies the scaling and noise calculation. This paper finds theL _P scaling and derives the roundoff noise expressions for fixed point implementations of the multistage decimator, interpolator, and multirate narrow-band low-pass filter. It is shown that the noise source at a low sampling rate stage is more important than that at a higher sampling rate stage. Methods to reduce output roundoff noise are discussed.This research was supported by NSF grant ECS 81-00453.     

Low-roundoff-noise limit-cycle-free implementation of recursivetransfer functions on a fixed-point digital signal processor

A method is presented for realizing recursive digital transfer functions on a fixed-point digital signal processor. The method is based on the parallel connection of L∞-norm scaled first- and second-order state-space structures. Magnitude truncation of the state update equations is employed to render the realization free of both overflow oscillations and constant-input limit cycles. The roundoff noise and coefficient sensitivity of the realization are also near minimum, giving a realization with outstanding performance in terms of all finite wordlength effects. An implementation on the DSP56000 family of digital signal processors demonstrates that the realization is efficient enough to achieve high sample rates     

Low roundoff noise augmented realizations of IIR filters

The scaling factor in a direct form II infinite impulse response (IIR) digital filter depends on only the system poles, not its zeros. Adding pole-zero cancellation pairs changes the scaling factor and the associated roundoff noise without affecting the transfer function. It is shown that with l₂ scaling, the upper bound of the noise reduction for direct form II IIR digital filters using K pole-zero cancellation pairs is about 6K dB. The authors consider parallel form 3P to be the basic structure, in which each subfilter is augmented with one or two pole-zero cancellation pairs and realized in direct form II. They seek an iterative optimization process for each subfilter to find the optimal pole-zero cancellation pairs that minimize the roundoff noise. Experimental results indicate the global optimal solution for these augmented subfilters can be obtained by using simple criteria to determine the starting points of the iterative processes. Alternative structures, illustrative examples, and overflow limit cycles are examined     

一种基于结构优化的FIR滤波器舍入噪声性能改进方案

针对滤波器组系统硬件实现时原型滤波器的有限字长效应问题,该文研究如何改善FIR原型滤波器由信号量化引起的舍入噪声,即降低舍入噪声增益,提出一种FIR滤波器优化结构。通过分析舍入噪声来源,利用多项式参数化方法对舍入噪声增益表达式进行推导。仿真实例证明,在不同字长约束条件下所提结构滤波器的幅频相频响应与理想状态基本吻合;通过与现有算法对比,所提结构具有较小的舍入噪声增益。     

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 \(5N+1\) 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.