Optimal design of continuous time irrational filter with a set of fractional order gammatone components via norm relaxed sequential quadratic programming approach

This paper proposes a continuous time irrational filter structure via a set of the fractional order Gammatone components instead of via a set of integer order Gammatone components. The filter design problem is formulated as a nonsmooth and nonconvex infinite constrained optimization problem. The nonsmooth function is approximated by a smooth operator. The domain of the constraint functions is sampled into a set of finite discrete points so the infinite constrained optimization problem is approximated by a finite constrained optimization problem. To find a near globally optimal solution, the norm relaxed sequential quadratic programming approach is applied to find the locally optimal solutions of this nonconvex optimization problem. The current or the previous locally optimal solutions are kicked out by adding the random vectors to them. The locally optimal solutions with the lower objective functional values are retained and the locally optimal solutions with the higher objective functional values are discarded. By iterating the above procedures, a near globally optimal solution is found. The designed filter is applied to perform the denoising. It is found that the signal to noise ratio of the designed filter is higher than those of the filters designed by the conventional gradient descent approach and the genetic algorithm method, while the required computational power of our proposed method is lower than those of the conventional gradient descent approach and the genetic algorithm method. Also, the signal to noise ratio of the filter with the fractional order Gammatone components is higher than those of the filter with the integer order Gammatone components and the conventional rational infinite impulse response filters.