Outer Product Expansion

This method performs a principal-components analysis of the multidimensional frequency response, and implements the variable filter as a weighted sum of static filters

With a single control parameter $\Psi_1$ , we have the expansion

Since

$$H_{i,\omega} = \sum_{k=1}^{n} \sigma_k {G_k}_i {\hbox{F}_k}(\omega),$$

we see that any point in control-frequency space $H_{i,\omega}$ is a weighted sum of samples from frequency-space functions $F_k$ (i.e. filters) and control-space functions $G_k$

We can add more control parameters:

$$H_{i_1,\dots,i_m,\omega} = \sum_{k=1}^{n}\left[ \sigma_k \, {{G_1}_k}(i_1) \dots {{G_m}_k}(i_m) {\hbox{F}_k}(\omega)\right]$$