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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}}}$ .

Problem Formulation

$\displaystyle \min_{\underline{h}}\left\Vert\,\mathbf{A}{\underline{h}}-{\underline{d}}\,\right\Vert _2$ (5.52)

where $ \mathbf{A}\in {\bf C}^{N\times M}$ , $ {\underline{d}}\in {\bf C}^{N\times
1}$ , and $ {\underline{h}}\in {\bf R}^{M\times 1}$ . Hence we have,

$\displaystyle \min_{\underline{h}}\left\Vert \left[{\cal{R}}(\mathbf{A})+j{\cal{I}}(\mathbf{A})\right]{\underline{h}} - \left[ {\cal{R}}({\underline{d}})+j{\cal{I}}({\underline{d}}) \right] \right\Vert _2^2$ (5.53)

which can be written as

$\displaystyle \min_{\underline{h}}\left\Vert\, {\cal{R}}(\mathbf{A}){\underline{h}}- {\cal{R}}({\underline{d}}) +j \left[ {\cal{I}}(\mathbf{A}){\underline{h}}+{\cal{I}}({\underline{d}}) \right] \,\right\Vert _2^2$ (5.54)


$\displaystyle \min_{\underline{h}}\left\vert \left\vert \left[ \begin{array}{c} {\cal{R}}(\mathbf{A}) \\ {\cal{I}}(\mathbf{A}) \end{array} \right] {\underline{h}} - \left[ \begin{array}{c} {\cal{R}}({\underline{d}}) \\ {\cal{I}}({\underline{d}}) \end{array}\right] \right\vert \right\vert _2^2$ (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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