Amir:
What I'm trying to get at is - in theory is room treatment not a better solution to this resonance & post-ringing as it will suppress both the initial resonance & also each reflection that causes the post-ringing whereas a digital filter just suppresses the resonance?
The whole point of this article is that suppressing the resonance does both in the case of those caused by minimum phase conditions. The preference for electronic correction is that it is very easy to deploy and takes up no space or mess with your decor. Getting 20 to 30 db correction out of acoustic products alone can be very challenging. Our reference theater has a ton of treatment including the entire front wall that is a proprietary bass trap and you still need EQ and lots of it to get a smooth response.
Back to why fixing the resonance fixes both frequency and time domain, here is a single resonance and its response both in Frequecy+Phase domain and Time:
We see the predictable effect we have talked about. The existence of a peak in frequency (top left), results in ringing of the impulse as time goes by on the right. If this is a minimum phase phenomena, then we can apply an inverse minimum phase correction which applies to both amplitude *and* phase at the same time:
The fact that we are correcting frequency response is obvious from that inverse filter. Less obvious is that the phase correction comes along with it and it also gets flatlined per right hand side with its inverse. Fourier theorem says that signals are fully described using their frequency response and phase so the time domain behavior cannot be any different as it corresponds to this view one for one. But we can confirm that. Here is what is going on in time domain:
We see that inverse ringing is created which helps cancel out the ringing from the original resonance. Let's put both views together now:
We see that we have corrected the frequency response (top left), the phase (bottom left) and time domain (right). It is not easy to escape the nature of these .
While I was trying to get permission for these slides, Allan wanted me to note that the way to be sure to find minimum phase response is to perform spatial averaging. That is, measure at multiple points and average. In doing so, the non-minimum phase contributions get reduced. So you would want to do this even if you are optimizing for one seat.
I also have a general issue with the idea that an FFT tells all that is going on so maybe it's not the best tool for using in analysing sound where music reproduction is concerned. Freq & Time might be orthogonal in terms of FFTs but we have to look at the underlying premise of FFTs & check whether it is applicable to music or just to repeating signals?
The implementation of correction filters can and most often is in time domain. There is no FFT done and then the bins manipulated although that could be done too. But rather, a FIR, IIR, or deconvolution is performed. The data does need to be digital however and as I explained, I see no issues with that for bass frequencies which is the topic of this thread even if one has a stance against them .
Amir