Filters and Ringing: Part 7

I’m going to start this part by doing something I very, very rarely do: to quote Wikipedia.

“In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.”

However, in my defence, one of the references attached to that statement is Julius O. Smith III, so that makes it okay.

Let’s unwrap that sentence and see if we know enough to know what it’s telling us.

We don’t care about control theory. So let’s ignore that part. We’re only interested in signal processing, where our signal is audio; so we move on.

We already know what a ‘linear, time-invariant” system (like our filters) is, and we now know that we can say that that system is ‘minimum-phase’ if:

  • the system (our peak filter in the previous part, for example)
  • and its inverse (our dip filter in the previous part, for example)
  • are causal
  • and stable

Let’s deal with the ‘stable’ part first. We know that our two filters are stable because we saw that their poles are inside the unit circle in the Z-Plane representation. (We also know it because they both have ringing that decays instead of increases over time.)

We also know that their zeros are also inside the unit circle, since the zeros of each filter are in the same place as the poles of the other filter, which we already said, are inside the unit circle.

So, what does ‘causal’ mean? It’s really just a fancy word that means that the output of our filter is determined by either the past or the present, or some combination of the two. In real life, all filters and systems are causal, since they can’t do something based on what will happen in the future.

However, if you are not working in real time, you can easily create systems and filters that are non-causal and have outputs that are created by events in the future. One simple example of this is to record your voice, reverse the track, add some reverb, and then reverse it back again. Now you have reverb that ramps up to a sound before it starts. This is non-causal.

Do I care?

Not yet. But keep the two conditions in mind:

  • Both the filter and its inverse must be ‘causal’. The output of a minimum phase filter can only be the result of the present or the past, never the future.
  • Both the filter and its inverse must be stable. We like stable…