[sdiy] 'State-Variable' vs. 'Multi-Mode' filters - differences?

Donald Tillman don at till.com
Wed Jul 4 19:42:53 CEST 2012

On Jul 4, 2012, at 5:17 AM, Justin Owen wrote:

> With regards to filter types - are the terms 'State-Variable' and 'Multi'Mode' essentially interchangeable or do they refer to two different filter types?
> If they refer to two different filter types - what are the differences?

"State Variable", "Multi Mode" and "BiQuad" are almost always used interchangeably referring to a second order filter built with two integrators.  But they are different.

"State Variable" comes from the analysis of mechanical systems.  That is, where you model a mechanical system as a black box with one or more inputs, one or more values that hold the state inside the box, and make a mix of those for the result.  For the classic mass/spring/friction model you might have the velocity and position of the mass as your state variables, and the rest of the behavior can be calculated from there.

The "BiQuad" filter comes from analog computing, and is an arrangement of opamps that model the above mass/spring/friction system.   One integrator provides the velocity state variable (by integrating the acceleration) and a second integrator provides the position state variable (by integrating the velocity).  The difference between the position state and the input value is equivalent to stretching the spring, which supplies, you guessed it, the acceleration.  

The BiQuad is a wonderful filter architecture for a number of reasons:

   * it's elegant, being an analog of a mechanical system

   * it's easy to implement and robust

   * the Q and resonant frequency can be adjusted independently

   * LP/BP/HP outputs are available simultaneously (your "Multi-Mode" feature)

   * Notch and phase shift outputs can be derived from those

   * lots of variations are possible, both in architecture and implementation details

   * in one variation of the filter, LP/BP/HP inputs are simultaneously available

So, though there are exceptional cases, the terms "State Variable", "Multi-mode" and "BiQuad" usually refer to the same thing.

  -- Don

Don Tillman
Palo Alto, California
don at till.com

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