"Wow. We are talking about 1-3ms latency/response time as bad one. But having 200-400ms latency like in Theremini is beyond evil." - Vadim
Well, there's two kinds of latency. One is "transport delay" or pure delay, like an echo. The other is "intertial delay" which is like phase delay through an RC network. Transport delay is much worse than inertial delay in a feedback (human playing a Theremin) scenario.
"Hmm. When I'm feeding LC tank with 2MHz 5V via 300 ohm resistor, isn't 50cm antenna radiating as a good transmitter?"
I should have an equation for this, but I believe a Theremin antenna acts like a monopole above a ground, so the closer the antenna length is to a multiple of the wavelength at the operating frequency the more efficiently it radiates RF. At 2 MHz the wavelength is 3x10^8 m/s / 2x10^6 = 300 / 2 = 150 m. If it's a direct ratio (I don't really know) then 0.5 m / 150 m = 1 / 300, so very little RF is being radiated. The radiated power puts an upper limit on tank Q, because it acts like a resistor (power sink). You can pretty much tell that not much power is radiated with a Theremin oscillator on the bench, because with a good coil you can get 100's of volts swing at the antenna with one mA @ 3.3V supply.
RF transmission loss is an argument against making the Theremin sensors look like antennas. Plates - or ideally spheres I suppose - should radiate less because the largest dimension is smaller for a given surface area (which sets intrinsic C and sensitivity).
"Is there any method to check if some device exceeds FCC rules?"
I should have a ready answer for this too but I haven't even looked into it. I know that FCC radiated emissions were a huge deal when I was working on telephony equipment, they'd spend weeks sniffing the equipment rack with all kinds of antennas, installing shorting tooth conductors between the doors, ferrite beads on the cables, etc. Everyone got nervous at the end of a design cycle, and they'd often have to play games with the emissions results to get a pass ("assuming the cabinet doors stay closed", etc.).
Vadim, have you seen this?: http://humancond.org/wiki/user/ram/electro/capsense/0main. One way to meet emissions requirements in general is to spread the energy in the peaks out. The techniques in that article are quite interesting, but the main problem with it (other than the super low bandwidth) is the sensing instability. He's driving the shield, which exposes the shield / plate capacitance, which drifts with temperature. I think one could perhaps do like he does and design a low Q tank, drive it with a band-limited pseudo-random sequence (NRZ), and synchronously AM detect it, but without using an explicit shield. Might get around the drive / sense / emissions issues with high Q LC. Something to ponder, anyway.