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I have many information i want to be clear very well .

The defining characteristic of FIR filters optimal in the Chebyshev sense is that they minimize the maximum frequency-response error-magnitude over the frequency axis. In other terms, an optimal Chebyshev FIR filter is optimal in the minimax sense: The filter coefficients are chosen to minimize the worst-case error (maximum weighted error-magnitude ripple) over all frequencies.In another word, minimizing the worst case squared error induces a minimum Chebyshev error problem in some formulations.

In order to achieve an optimal equiripple complex FIR filter in the complex Chebyshev sense ,we can use an evolutionary algorithm.

What is the relation which allowed using the complex Chebyshev sense for optimisation equiripple complex FIR filter? What is effective evolutionary algorithm in this case?

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  • $\begingroup$ Most filter designs are expressed as convex optimizations with efficient algorithm. Evolutionary computing is not common in filter design $\endgroup$ – Stanley Pawlukiewicz Jan 4 '18 at 3:56
  • $\begingroup$ @ Stanley Pawlukiewicz: what do you mean by ''Evolutionary computing is not common in filter design''.There is an iterative method is introduced to find the optimal solution of FIR filter design problem such as particle swarm optimization PSO,genetic algorithm GA,simulated annealing SA...Or do you mean something else?Can you explain more. $\endgroup$ – K.n90 Jan 8 '18 at 16:20
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The Chebyshev criterion minimizes the maximum weighted absolute error:

$$\epsilon=\max_{\omega\in\Omega}W(\omega)|H(\omega)-D(\omega)|\tag{1}$$

where $\Omega$ is the union of all frequency bands of interest, $W(\omega)$ is a positive weight function, $H(\omega)$ is the filter's complex frequency response, and $D(\omega)$ is the complex desired frequency response.

Note that in general, $(1)$ is a complex approximation problem, because $H(\omega)$ and $D(\omega)$ are generally complex-valued. However, in the special case of linear phase FIR filters, $(1)$ can be rewritten as a real approximation problem, for which a very efficient algorithm exists (the Remez exchange algorithm used in the Parks-McClellan method). In the general case of non-linear phase FIR filters or IIR filters, $(1)$ can be solved by (much) more complex general optimization algorithms. For FIR filters, the problem is still a lot easier than for IIR filters, and several algorithms have been suggested (cf. the thesis linked here and the references therein). One of the algorithms is implemented in Matlab's signal processing toolbox: cfirpm.m

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