On 03-Feb-15 03:39, Andrew Simper wrote:
I completely agree! I find it mentally easier to think of "energy stored" in each component rather than "state variables" even though they are the same. So for musical applications it is important that a change in the cutoff and resonance doesn't change (until you process the next sample) the energy stored in each capacitor / inductor / other energy storage component in your model. Direct form structures do not have this energy conservation property, they are only equivalent in the LTI case (linear time invariant - ie don't change your cutoff or resonance ever). Any method that tries to jiggle the states to preserve the energy would only be trying to do what already happens automatically with some of state space model, so I feel it is best to leave such forms for static filtering applications.
I'm not sure whether the choice of the energy-based state variables is indeed the best one (would be nice to try to have some kind of formal proof of that), but at least it seems to me that it might be that the SVF has the optimal (in a way) choice of those. My consideration is the following. Imagine a generic 2nd order mechanical system. Something like a mass on a spring. The external excitation force is the input signal. The natural choice of state variables is the position and the velocity of the mass. We can look at it as at a multimode LP/BP/HP filter. Up to some scaling, the position is the lowpass output, the velocity is the bandpass output and the highpass can be obtained as a linear combination of LP, BP and the input. Since there are not that many possibilities to construct a 2nd order linear differential system, our system is equivalent to an SVF in the LTI sense. The time-varying behavior will depend on which specific coefficients of the 2nd order differential equations are modulated and on the choice of state variables.
So, our state variables are the position and the velocity. Imagine our system has a high cutoff and we are feeding in a sufficiently high sinusoidal signal with the frequency below the cutoff. So the lowpass output (the position) is a similar sinusoid. At the zero-crossing time the velocity will be maximal. Suppose we suddenly lower the cutoff at this moment. Intuitively (I admit that this requires a more rigorous check), at a lower cutoff the velocity will not be changing so quickly any more. This means, the output signal of the system will significantly overshoot the previous output amplitude.
In comparison, the state variables of the SVF are using the velocity divided by the cutoff. Which means a sudden reduction of the cutoff will proportionally reduce the velocity and the overshoot will not be that big anymore.
In order to test your conjecture about the energy-based state variables, one would need to explicitly write down the mass-spring equation and compare the respective choices of state variables to those of SVF. Possibly, the energy-based state variables of the mass-spring system will be equivalent to the state variables of the SVF, which would then be a sign that your idea might be correct.
Regards, Vadim -- Vadim Zavalishin Reaktor Application Architect Native Instruments GmbH +49-30-611035-0 www.native-instruments.com -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
