Interpolated Outer Product Expansion Filters

where

$$\begin{eqnarray*} \hbox{F}_i &=& \mbox{\lq principal-component' filters}\\ \hbox{g}_{ij} &=& \mbox{interpolated control functions} \end{eqnarray*}$$

• The $\hbox{F}_i$ are designed to the frequency-axis components of the outer-product expansion of the desired response

• The $g_{ij}$ are interpolating functions derived from the control-axes components of the outer-product expansion

• This method overcomes the stability problem of the previous method, since all the filters are static, and can be designed to be stable

• When implemented as shown, we must worry about the filters' phase responses:
  • The design usually ignores the phase of the desired response, to keep $G_i$ real in the expansion), and to keep them from cancelling each other

  • The paper introducing this idea restricted the phases to be equal, and used linear-phase FIR filters