On the Design of Filters by Synthesis

A Geometric Approach to Group Delay Network Synthesis

“…The methods in [1], [5], [6], [13], [16], require as input the desired number of poles as well as their starting locations. Here the starting values of the ω x components are evenly sub-divided across the bandwidth of interest while the starting  x components are all set such as to create a peak delay equal to the maximum initial delay variation of the cost function.…”

“…Numerical approaches aimed at finding optimal solutions in a large search space have gained prevalence in the modern group-delay synthesis literature [1], [5], [6], [14], [15], [16]. Here we will investigate two such approaches which are well suited to the synthesis problem, namely the genetic algorithm and simulated annealing technique.…”

“…Equation (15) , (16) allowing the relationship between  and the geometrical representation of Fig. 25.…”

“…Due to these problems with state-of-the-art analytical synthesis methods, numerical approaches relying on optimization algorithms have been widely sought in the modern group-delay synthesis literature [1], [5], [6], [14][15][16][17]. The most popular approach [14] is based on approximating a desired group delay curve with a summation of secondorder all-pass networks to within an arbitrary additive constant.…”

“…As with the method in [14], it is impossible to analytically determine the initial solution set which guarantees convergence. Other approaches using the differential evolution and genetic algorithms [1], [6], [16] have also been proposed. Such approaches tend to be computationally intensive and are prone to converge to local minima, as opposed to a global optimum.…”