### Nonlinear-Phase FIR Filter Design

Above, we considered only linear-phase (symmetric) FIR filters. The same methods also work for antisymmetric FIR filters having a purely imaginary frequency response, when zero-centered, such as differentiators and Hilbert transformers [224].We now look at extension to

*nonlinear-phase*FIR filters, managed by treating the real and imaginary parts separately in the frequency domain [218]. In the nonlinear-phase case, the frequency response is

*complex*in general. Therefore, in the formulation Eq. (4.35) both and are complex, but we still desire the FIR filter coefficients to be real. If we try to use ' ' or

`pinv`in matlab, we will generally get a complex result for .

#### Problem Formulation

(5.52) |

where , , and . Hence we have,

(5.53) |

which can be written as

(5.54) |

or

(5.55) |

which is written in terms of only

*real*variables. In summary, we can use the standard least-squares solvers in matlab and end up with a

*real*solution for the case of complex desired spectra and nonlinear-phase FIR filters.

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Matlab for General FIR Filter Design

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More General Real FIR Filters