I would like to know what kind of wireless signal would be able to:

1. Be transmitted from coast to coast and back without any repeater.
2. Would not be detected by a frequency analyzer sweeping in the range
of 30Mhz to 15 GHz.

3. would not interfere with cell phone, tv, radio signals.

4. Would not be blocked by building walls, underground tunnels, etc.

5. Would not be blocked by a magnetically shielded roome like the one
in this system, or be detected by a the equipment used in this system.
http://web.mit.edu/kitmitmeg/sysfull.html#The%20Room%20itself

6. Can transmit an Audio and Video signal.

Check out HAARP: http://en.wikipedia.org/wiki/HAARP

That's a virtually impossible set of contraints.  You could probably
manage at most any two of those six at the same time.

1.  The only way to get reliable coast-to-coast range (I'm assuming
the US coast) is with HF or lower frequencies.  VHF or higher will
require line of sight.  The other alternative is satellite broadcast -
this is probably your best bet!

2.  The power and bandwidth necessary for video would make the signal
very noticable, since you'd be using a large chunck of the spectrum. 
You could use spread sprectrum techniques to hide your signal in the
background noise, but the high power and data rate you'd need for
coast-coast video would make it unlikely that you'd stay hidden for
long.

3.  This is fairly easy if you avoid the frequencies they use, which
you'd have to anyone to get country-wide range (except perhaps for AM
radio)

4.  Normal building walls wouldn't be any worse than the are for AM
radio.  Underground tunnels are much harder - it is virtually
impossible to get RF video through even a few meters of dirt.  There's
especially no way you could manage it with the transmitter located
halfway across the country, without a repeater underground.

5.  It looks like that room is pretty well shielded with mu-metal and
copper to block magnetic and electric fields.  It would be extremely
difficult to get through that from any distance.

6.  This is easy, except when you through in the other contraints!

formica34-ga pretty much beat me to the punch. However:

* If is very hard to transmit video on HF, unless you will use
multiple carriers, and perhaps U/L SSB.

* Why should it not be detectable? If we are speaking about HF, the
transmitter will require multi-kW power. If you want to do it the
right way, you will probably need a license.

* DX'ers can speak throught the world with a big antenna. HF can bend
around the terrain. Research ham radio.

* While you are at MIT, why don't you also develop sub-space
communications? That would certainly solve the problem...

* If you want something undetectable, you can use underground
communications, when the antenna is buried. However, the size of the
antenna, the power, diffictuly of detection, very low frequency and
bandwidth would make it only suitable for military voice and message
communications.

* What the heck are you doing?

* Telepathy should be able to travel coast to coast. Are you trying to
read the UC Berkeley research right from the professors' brains?

* You can always use the HF way of multiple ionoshpere--ground bounces
to transmit. This might require good hardware and software to be
reliable.

* Looks like you will need help from the Stanford Egg experiment and
guests that appear on coasttocoastam.com

* Any more questions?

Correction: Princeton Eggs experiment