### 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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