Novel approach to designing digital differentiators

A novel approach to designing recursive stable digital differentiators is introduced. A four-step design procedure is presented. The procedure consists of obtaining or designing an integrator and then modifying its transfer function appropriately to obtain a stable differentiator. As an example a second order recursive differentiator is developed.<>     

Adaptive derivative estimation for delay-constrained accelerationmeasurement

The well-known noise problem in digital differentiation is addressed by means of using adaptive digital filtering for signal pre-processing. Rapidly responding differentiators with low-noise output can be constructed by using the adaptive filter in a predictor configuration. As the prefilter is adaptive, the approximation is more flexible than polynomial fitting. The recursive least-squares adaptive algorithm is used for prediction     

Design of digital differentiators for low frequencies

Optimal, maximally accurate digital differentiators (DDS) are derived for the low-frequency range. Exact coefficients used in the proposed DDs can be readily computed from explicit formulas, whereas the optimal (minimas RE) DDs require and optimization program to derive the coefficients. The lower the frequency of differentiation, the better is the performance of the proposed differentiators, making them suitable for many typical applications     

Compact formulas for least-squares design of digitaldifferentiators

New compact and simple formulas for the design of digital differentiators based on the least squares method are proposed. The new expressions allow a very fast and precise computation of the filter coefficients. Furthermore, using the proposed analytical solution, the need to solve the system of linear equations for the case of fullband differentiators is avoided     

Maximally linear FIR digital differentiators in frequencies ofπ/p-simplified formulas of weighting coefficients

Simplified formulas of weighting coefficients of maximally linear FIR digital differentiators have been derived     

Taylor series based two-dimensional digital differentiators

A new type of Taylor series based 2-D finite difference approximation is presented, and it is shown that the coefficients of these approximations are not unique. Explicit formulas are presented for one of the possible sets of coefficients for an arbitrary order, by extending the previously presented 1-D approximations. These coefficients are implemented as maximally linear 2-D FIR digital differentiators, and their formulas are modified to narrow the inaccuracy regions on the resultant frequency responses, close to the Nyquist frequencies     

Design of universal,variable frequency range FIR digital differentiators

An efficient design of digital differentiators (DD), with a variable frequency range of operation at low and midband frequencies, has been proposed. Implemented as a Taylor structure, a DD designed for orderN can be made to function as a universal configuration giving optimal transfer functions for all possible ordersN, without changing the coefficients.     

Least-squares design of higher order nonrecursive differentiators

A method is described that can be used to design non-recursive linear-phase higher order differentiators that can perform differentiation over any frequency range. The method is based on formulating the absolute mean-square error between the amplitude responses of the practical and ideal differentiator as a quadratic function. The coefficients of the differentiators are obtained by solving a set of linear equations. This method leads to a lower mean-square error and is computationally more efficient than both the eigenfilter method and the method based on the Remez exchange algorithm. Design of differentiators based on minimization of the relative mean-square error is also carried out. Finally, the method is extended to the design of frequency selective higher order differentiators     

Limit-cycle constraints for recursive-digital-filter design

Quantisation errors can cause nonlinear oscillations in recursive digital filters if the input signal is low or constant. Limit-cycle constraints are derived in terms of the feedback coefficients, the gain and the sampling ratio. These represent convenient ways of reducing nonlinear effects in the realisation of the recursive digital filter.     

A New Type of Digital Filter for Data Transmission

A digital filter is introduced consisting of a transversal part and a simple recursive network. The coefficients of the transversal part are equal to integer powers of two or zero; thus complicated multipliers are avoided and instead a simple routing circuit is used. Use of several types of the recursive network makes the filter applicable in different frequency ranges. The coefficients of the transversal part can be interpreted as differences of successive values of the impulse response of the filter. It is shown that this difference routing digital filter (DRDF) is especially suited for application in data transmission.     

分数导数与数字微分器设计

从信号处理角度来考察分数阶导数和微分问题.首先论述分数阶微积分的基本概念,评论分数阶导数的几种典型定义,并重点探讨分数阶导数的频域定义及其基本性质.紧接着详细研究分数阶微分的实现:理想分数阶数字微分滤波器和FIR分数阶数字微分滤波的基本性质和设计问题,并提出分数阶微分滤波器设计中遇到的新问题以及奇对称微分滤波器、偶对称微分滤波器等新概念.最后给出一些值得进一步探讨的有趣问题.     

Design of FIR first order digital differentiators of variablefractional sample delay using maximally flat error criterion

Optimum (in a maximally flat frequency response error sense) FIR digital differentiators of variable fractional sample delay are derived that calculate the derivative of an input uniformly sampled discrete-time signal with arbitrary centre frequency at an arbitrarily chosen instant of time within each sampling interval. The proposed class of differentiators includes maximally linear differentiators for low frequencies     

Design of digital FIR variable fractional order integrator and differentiator

This paper presents an easy and simple method to design variable fractional order digital FIR integrators and differentiators based on fractional order systems. First, closed-form digital IIR fractional order integrators and differentiators have been obtained from the analog rational functions approximations, in a given frequency band, of the fractional order integrator s ^?m and differentiator s ^m (0?<?m?<?1) through the Tustin generating function. Then, closed-form digital FIR fractional order integrators and differentiators by truncation of the digital IIR ones have been derived. Next, polynomial interpolation has been used to design digital FIR variable fractional order integrators and differentiators that can be implemented by the Farrow structure. The main feature of variable fractional order operator is that its order can be changed without re-designing a new fractional order operator. Some examples have been presented through the paper to illustrate the performance and the effectiveness of the proposed design method. The results obtained have been discussed and have been compared to one of the most recent works in the literature using the same design parameters.     

Design Of 2-D Recursive Digital Filters Using Nonsymmetric Half-Plane Allpass Filters

A novel structure using recursive nonsymmetric half-plane (NSHP) digital allpass filters (DAFs) is presented for designing 2-D recursive digital filters. First, several important properties of 2-D recursive DAFs with NSHP support regions for filter coefficients are investigated. The stability of the 2-D recursive NSHP DAFs is guaranteed by using a spectral factorization-based algorithm. Then, the important characteristics regarding the proposed novel structure are discussed. The design problem of 2-D recursive digital filters using the novel structure is considered. We formulate the problem by forming an objective function consisting of the weighted sum of magnitude, group delay, and stability-related errors. A design technique using a trust-region Newton-conjugate gradient method in conjunction with the analytic derivatives of the objective function is presented to efficiently solve the resulting optimization problem. The novelty of the presented 2-D structure is that it possesses the advantage of better performance in designing a variety of 2-D recursive digital filters over existing 2-D filter structures. Finally, several design examples are provided for conducting illustration and comparison.     

Economic linear-phase recursive digital filters

This letter describes the design by z plane methods of an attractive class of recursive digital filters. The computational economy achieved by recursive operation is further enhanced by the use of integer weighting coefficients, making the filters particularly suitable for online processing of sampled-data signals.     

Closed-form approach to design of all-pass digital filters using cepstral coefficients

A simple method for the design of all-pass digital filters is described. In this method, given a desired group delay, the cepstral coefficients corresponding to the denominator of a stable all-pass filter are determined using a least-squares approach. The coefficients of the all-pass filter are then obtained from the cepstral coefficients using a recursive relation.     

Minimax design of recursive digital filters with a latticedenominator

The authors deal with the problem of minimax recursive digital filter design with a lattice structure for the denominator. The design problem is formulated so that the coefficients for the numerator and denominator of a recursive filter can be found by solving the best linear complex Chebyshev approximation (LCCA). A design technique based on the weighted least-squares algorithm previously proposed by one of the authors is then developed for solving the resulting LCCA problem. During the design process, this technique finds the tap coefficients for the numerator and the reflection coefficients for the denominator simultaneously. The stability of the designed recursive filter is ensured by incorporating an efficient stabilisation procedure to make all of the reflection coefficient values fall between -1 and +1. Computer simulations show that the proposed technique provides better design results than existing techniques     

New design of full band differentiators based on Taylor series

The classical central difference approximations of the derivative of a function based on Taylor series are the same as type III maximally linear digital differentiators for low frequencies. A new finite difference formula is derived which can be implemented as a full band type IV maximally linear differentiator. The differentiator is compared with type III maximally linear and type IV equiripple minimax differentiators. A modification is proposed in the design to minimise the region of inaccuracy near the Nyquist frequency edge     

Realization of two-dimensional recursive digital filters using the mixed Cauer form†

The application of the mixed cauer continued fraction expansion to synthesis of two-dimensional recursive digital filters is presented. Examples are provided to show how to perform a mixed cauer expansion on two-dimensional transfer functions and eventually obtain canonic digital ladder filter structures. Furthermore, a test for two-dimensional mixed-cauer continued fraction expansion which at the same time determines the expansion coefficients is included.     

Recursive formulas for the partial fraction expansion of a rational function with multiple poles

Two simple recursive formulas are derived for obtaining the coefficients of the partial fraction expansion with multiple poles. The entire process is pure algebraic operation without any differential calculus and is readily adapted to a digital computer.