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 ${\underline{d}}$ and $\mathbf {A}$ are complex, but we still desire the FIR filter coefficients ${\underline{h}}$ to be real. If we try to use ' $\backslash$ ' or pinv in matlab, we will generally get a complex result for ${\underline{\hat{h}}}$ .