Design of optimum recursive digital filters with double zeros on the unit circle leading to symmetric ladder wave digital filter structures

2004 ◽  
Author(s):  
B. Soltanian ◽  
T. Saramaki ◽  
H. Johansson
Keyword(s):  
Unit Circle ◽  
Digital Filter ◽  
Digital Filters ◽  
Wave Digital Filter ◽  
Filter Structures ◽  
Recursive Digital Filters

scholarly journals Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis

Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1488
Author(s):  
Damian Trofimowicz ◽  
Tomasz P. Stefański
Keyword(s):  
Unit Circle ◽  
Phase Analysis ◽  
Characteristic Equation ◽  
Digital Filter ◽  
Digital Filters ◽  
Computing Time ◽  
Complex Function ◽  
Stochastic Methods ◽  
Final Population ◽  
Candidate Regions

In this paper, novel methods for the evaluation of digital-filter stability are investigated. The methods are based on phase analysis of a complex function in the characteristic equation of a digital filter. It allows for evaluating stability when a characteristic equation is not based on a polynomial. The operation of these methods relies on sampling the unit circle on the complex plane and extracting the phase quadrant of a function value for each sample. By calculating function-phase quadrants, regions in the immediate vicinity of unstable roots (i.e., zeros), called candidate regions, are determined. In these regions, both real and imaginary parts of complex-function values change signs. Then, the candidate regions are explored. When the sizes of the candidate regions are reduced below an assumed accuracy, then filter instability is verified with the use of discrete Cauchy’s argument principle. Three different algorithms of the unit-circle sampling are benchmarked, i.e., global complex roots and poles finding (GRPF) algorithm, multimodal genetic algorithm with phase analysis (MGA-WPA), and multimodal particle swarm optimization with phase analysis (MPSO-WPA). The algorithms are compared in four benchmarks for integer- and fractional-order digital filters and systems. Each algorithm demonstrates slightly different properties. GRPF is very fast and efficient; however, it requires an initial number of nodes large enough to detect all the roots. MPSO-WPA prevents missing roots due to the usage of stochastic space exploration by subsequent swarms. MGA-WPA converges very effectively by generating a small number of individuals and by limiting the final population size. The conducted research leads to the conclusion that stochastic methods such as MGA-WPA and MPSO-WPA are more likely to detect system instability, especially when they are run multiple times. If the computing time is not vitally important for a user, MPSO-WPA is the right choice, because it significantly prevents missing roots.

Generalized Stability-Triangle for Guaranteeing the Stability-Margin of the Second-Order Digital Filter

2016 ◽  
Vol 25 (08) ◽  
pp. 1650094 ◽  
Author(s):  
Tian-Bo Deng
Keyword(s):  
Digital Filter ◽  
Upper Bound ◽  
Digital Filters ◽  
Stability Margin ◽  
Second Order ◽  
Recursive Digital Filter ◽  
The Stability ◽  
Special Case ◽  
Recursive Digital Filters ◽  
Generalized Stability

In the design of recursive digital filters, the stability of the recursive digital filters must be guaranteed. Furthermore, it is desirable to add a certain amount of margin to the stability so as to avoid the violation of stability due to some uncertain perturbations of the filter coefficients. This paper extends the well-known stability-triangle of the second-order digital filter into more general cases, which results in dented stability-triangles and generalized stability-triangle. The generalized stability-triangle can be viewed as a special case of the dented stability-triangles if the two upper bounds on the radii of the two poles are the same, which is a generalized version of the existing conventional stability-triangle and can guarantee the radii of the two poles of the second-order recursive digital filter below some prescribed upper bound. That is, it is able to provide a prescribed stability-margin in terms of the upper bound of the pole radii. As a result, the generalized stability-triangle increases the flexibility for guaranteeing a prescribed stability-margin. Since the generalized stability-triangle is parameterized by using the upper bound of pole radii, i.e., the stability-margin is parameterized as a function of the upper bound, the proposed generalized stability-triangle facilitates the stability-margin guarantee in the design of the second-order as well as high-order recursive digital filters.

Group Delay Equalization of Polynomial Recursive Digital Filters in Maximal Flat Sense

2019 ◽  
Vol 28 (10) ◽  
pp. 1950173 ◽  
Author(s):  
Negovan Stamenković ◽  
Nikola Stojanović ◽  
Goran Perinić
Keyword(s):  
Digital Filter ◽  
Digital Filters ◽  
Group Delay ◽  
Minimum Phase ◽  
Iir Filter ◽  
Delay Response ◽  
Cascade Connection ◽  
Allpass Filter ◽  
The Given ◽  
Recursive Digital Filters

The paper presents the development of an algorithm to obtain stable allpass filter, which acts as a group delay equalizer, with the aim to equalize group delay of the polynomial IIR filter in a maximal flat sense. The proposed method relies on a set of nonlinear equations, derived directly from the flatness conditions of the group delay response at the origin in the [Formula: see text]-plane, with the order to obtain the unknown values of the allpass filter coefficients. The algorithm implemented in the MATLAB platform returns the coefficients of allpass filter. In the given example, first we construct a minimum phase polynomial IIR digital filter with a maximally flat magnitude at origin, next we augment the system with cascade connection of nonminimum allpass digital filter with order to equalize the group delay response of the whole filter in a maximally flat sense.

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