### 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.

**Next Section:**

Matlab for General FIR Filter Design

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More General Real FIR Filters