Multirate algorithm for updating the coefficients

A large number of methods for designing all-pass filter have been developed in recent years.

In [2], for example, the author proposed a new design method for an all-pass filter where it has a least squares or an equiripple phase-error response.

Like most swarm intelligence algorithms, PSO is also a population-based search algorithm.

It simulates the social behavior of organisms, such as fish schooling and bird flocking.

is called the particle or individual of the PSO algorithm and many such particles then form a population.

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Kennedy and Eberhart initially proposed the PSO algorithm in 1995 and recently it became one of the popular and efficient optimization algorithms [9].

In the proposed method, all of the designed filter coefficients are firstly collected to be a parameter vector and this vector is regarded as a particle of the algorithm.

The MPSO with a modified velocity formula will force all particles into moving toward the optimal or near optimal solution by minimizing some defined objective function of the optimization problem.

In [3], an IIR all-pass filter with equiripple phase response was designed based on the eigenvalue problem and this design problem can be formulated as the representation of an eigenvalue problem via the Remez exchange algorithm.

A Hopfield neural network was combined to the design of IIR all-pass digital filters [5].

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