System

I guess what it means is that subs can be much slower than speakers. I had one that could clearly be heard bouncing along behind the music, it was almost comical. In fact I can always hear a slight blurring when a sub is used in a system. A good quality servo driven sub may not be within your budget.
How about some more info on your system, or add a system link to your profile.

This article IMO provides the best description I have seen of what the confusing and seemingly oxymoronic term "fast bass" means.

Regards,
-- Al

I have Triangle speakers and the sub I use is a Velodyne Optimum 8. If you have a large room you might want to look at the 10 or 12. In my small space, the 8 is more than enough.
It's a sealed design that is very fast. It also has a room correcting eq that works very well.

I second the Velodyne Optimum 8. That's what I have and it is highly musical and the auto room correction is a must for a music sub. Also, I think a smaller woofer will integrate better with the main speakers. With the OPs budget they would need to find a used one.

There are basically two types of subwoofer sound. The under $1000 home theatre boomers and thudders and the over $1000 (probably more like $1500) for real subs like Velodyne, JL and Martin Logans that can actually integrate well into a music system. Physics and your budget relegates you into the former category, so my advice is to skip it altogether until you can do it right. Otherwise, you're throwing your money away and you will not be satisfied. Trust me, I've still got my old sub in a box somewhere collecting dust....