Originally Posted by SoundOnSound
Up the Pole
If you've been in the synth game for a while, you'll have heard that 12dB/octave filters are sometimes called '2-pole' filters, and 24dB/octave filters are called '4-pole' filters. You might think it safe to assume, therefore, that each of the 6dB/octave sections in Figure 9 is a 'pole'. Unfortunately, you would be wrong (although not a million miles from the truth).
The name (in this context) is a consequence of a powerful mathematical operation called a 'Laplace Transform'. This transform, while difficult to describe in words, is a convenient operation that allows mathematicians to analyse the responses of linear systems when they are presented with audio signals (as for 'linear systems' and the maths involved... no, don't even dream of asking!) Anyway, the term 'pole' comes about because, when you represent an RC filter using a graph in the 'Laplace Transform domain', it looks like a flat sheet of rubber with a tent-pole pushing it sharply upwards at some point.
A single 6dB/octave RC filter has one such 'tent-pole', and is therefore called a '1-pole' filter, a 12dB/octave filter has two 'poles'... and so on.
Therefore, if you want to create a passive 24dB/octave filter with a single cutoff frequency for each of its four elements, it would seem safe to assume that would you want all the poles in the same place in the graph. And, for once, intuition is correct. Unfortunately, as I've already explained, achieving this using passive components is all but impossible because, when we cascade the sections, they interact and no longer function as they would in isolation. So, instead of the perfect 24dB/octave response of figure 10, the cutoff frequency for each section is different, and the amplitude response of our transfer function has four 'knees', as shown in Figure 11.
This then, leads us to an important conclusion: while a passive 4-pole filter will tend to a 24dB/octave rolloff at high frequencies, it will, to a greater or lesser extent, exhibit regions within which the rolloff is 6dB/octave, 12dB/octave and 18dB/octave. Moreover, if you look closely, you'll see that the transfer functions within these intermediate regions are not quite straight lines, meaning that the relationship between the frequency and the input and output powers are not as straightforward as before.