Filter Reversing

Given , , and , we can begin the calculations for any filter with:

The coefficients from the spec and are normalized (we know this because is not specified) so we know that:

Stage 1 filter

Cookbook definition of a high-shelf filter

We need an extra variable, which is our gain. If we have gain in decibels , it can be converted with .

Reversing the high-shelf coefficients

Here, we can cheat because we know from the diagram in the spec that . Everything becomes much easier knowing versus having to solve for it as well.

To solve for :

We'll need this form of later to calculate . Right now, we can use our earlier definition of the normalized coefficients to substitute:

and

Now, to solve for we first must solve for :

Time for :

And is really easy:

Stage 2 filter

Cookbook definition of a high-pass filter

Reversing the low-pass coefficients

This filter should be a lot easier to reverse due to the relative simplicity of the , , and definitions.

Again, we start by solving for in terms of one of our normalized coefficients.

Substitute:

We solve for reusing our existing definitions:

And then :

And is the same as done previously.