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Suppose that you have a signal source, a set of point-transducers, and a handful of moderately powerful DSPs. Is it possible to construct an arrangement of the transducers such that the original signal propagates within a given region, but is absent or significantly attenuated outside of that region?

The obvious application would be sound systems that don't leak loud sound into neighboring rooms. But this would also be useful for limiting RF propagation to a physical space. In the audio example, you can shield the room with sound-absorbing materials. You could also build a faraday cage for the RF case, but either physical solution is likely to be prohibitively expensive. Now that powerful DSPs are cheap, can this be done entirely in the signal domain?

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+1 I really hope this does not turn into a who has a bigger forcefield contest. –  Argus Jul 10 '12 at 0:34
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For the simple problem where you have a set of point sources and you want them to cancel at infinity in all directions (i.e. there is very little signal except in a bounded region), the answer is yes. You just have to arrange for the sources at the surface to absorb the radiation going outwards.

An estimate of how many signal sources this will take (in the audio case): The highest frequencies are the worst. At 20KHz, the wavelength is about 1/20 feet so that is about how close your sources will have to be. For a 8 foot cube sized room this will be something like 200,000 sources (all 6 walls and a half inch apart) and will be pretty expensive.

On the other hand, you know that it's a lot cheaper to insulate high frequency sound than low frequency. So you hear your neighbors bass but not his treble. So if you are interested in sounds only below 300 Hz, now you need elements separated by about 3 feet and you only need something like 42 sources.

Given that these are audio frequencies, I'm thinking that the cost of the DSPs will be quite small compared to the cost of the speakers.

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can you elaborate on why the spacing of the point sources needs to be at the wavelength that you want to cancel? –  ObscureRobot Jul 10 '12 at 11:19
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It's cause you're emulating a wall that absorbs (or reflects) the radiation. If you leave a region uncovered, you'll be emulating a wall with a hole in it. A hole a lot bigger than a wave length lets the wave out. This is why, for example, the holes in the metal shield around a microwave oven cavity are limited in size to something smaller than the wavelength of the microwave radiation (which has a wavelength of around 12 cm): hyperphysics.phy-astr.gsu.edu/hbase/waves/mwoven.html –  Carl Brannen Jul 11 '12 at 5:09
    
If you treat the center of the space as a point source, and then extend a radius out to the distance you want to cancel sound, you would have a 360-degree view of the exiting wavefront. If you broke that down into sectors, couldn't you use fewer speakers and mount them near the center of the space? Cancellation wouldn't be perfect, but you might be able to get acceptable results with eight or sixteen speakers. –  ObscureRobot Jul 11 '12 at 5:24
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Yes, if you mount the noise cancelling sources close enough to the point source. However, what you're doing is making the room (where sound is allowed) smaller. It takes a lot less material to stop the sound from escaping when you use headphones, for example. –  Carl Brannen Jul 12 '12 at 1:30
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