Erik,
Full disclosure: We design and manufacture speaker systems (in the USA) that use full DSP, specifically digital crossovers and room correction.
First, DSP (Digital Signal Processing) is a broad term. Basically, anywhere there is digital signal there is DSP. Digital crossovers and room correction are a subset of DSP.
DSP/room correction (DSP/RC) can be much more than amplitude correction, often refereed to EQ. DSP/RC users only applying EQ are using only part of the technology's capabilities. There are simple units that are basically digital equalizers.
Comprehensive DSP/RC systems compensate for amplitude domain and time domain anomalies. Don't underestimate the importance of time domain corrections.
During the development of our Apollo series speaker systems we studied analog passive crossovers, analog active crossovers and digital crossovers for them. We developed the Apollos using all three technologies. Exploring digital crossovers allowed us to experiment with DSP/RC. We found that there are two basic categories: Manual, that requires much experience to sort out and semi-automatic, which are more user friendly. Our supplied DSP unit is user friendly.
We found that applying DSP/RC to a poorly designed speaker doesn’t automatically turn it into a gem. What we did prove to ourselves is that applying DSP/RC to a good speaker allows that speaker to get much closer to its full potential and do so in a much greater variety of listening rooms. And in many cases, improve the measured performance. Time domain issues inherent in the design are corrected.
IMHO, I have found that properly applied high quality DSP/RC makes the system sound more balanced from top to bottom, improves resolution, improves dynamic contrast, eliminates stress and just sounds more complete.
BTW, ASP (Analog Signal Processing) is a term not often used by audiophiles. Basically, anywhere there is an analog signal there is ASP. Technically, ASP/RC can be developed and applied to a system.