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Someone tried to explain the process of beam forming to me, and how beam forming is far more efficient than an omnidirectional signal of similar amplitude.

This seemed obvious at the time: the power of the signal is proportional to the integral over time, so since the magnitude is only large in the desired direction, you only need to supply the power for that part.

But from what I understood, beam forming is using interference patterns to make sure that the signal strength in one direction (the 'beam') is much larger than others. I assume that the maximum amplitude of the signal cannot become larger than the sum of the amplitudes of the point sources.

So let's say that I have 2 point sources to form my (admittedly pretty terrible) beam. I model them as both sending out sine waves with amplitude A omnidirectionally. The maximum amplitude of the interference is now 2*A. And the power consumed scales with twice the integral of the sine over time.

Now compare this to a single omnidirectional point source of equivalent amplitude (2*A). This also scales with twice the integral of the sine over time.

So where do the power savings come in? Is it because I ignored the reduction of the signal strength with distance? But that should scale inversely with distance^2 either way, right? Are my cows too spherical?

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    $\begingroup$ Consider how a Yagi antenna works - it isn't quite how you describe it. $\endgroup$ – Jon Custer Jul 23 '18 at 15:59
  • $\begingroup$ Ah, you're right. I was only thinking of active components, but the reflector of a yagi antenna, or the dish on a satellite dish don't require any power input to provide useful interference. If you could write that up as an answer, I'll mark it. $\endgroup$ – securityN00b Jul 23 '18 at 17:08
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    $\begingroup$ Nitpick: signal intensity is inversely proportional to distance squared, not distance cubed. $\endgroup$ – Michael Seifert Jul 23 '18 at 17:08
  • $\begingroup$ You're right, edited $\endgroup$ – securityN00b Jul 23 '18 at 18:21
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Re. "...how beam forming is far more efficient than an omnidirectional signal of similar amplitude." This statement implies that the goal is to transmit the signal in a particular direction. If instead the goal was to transmit in all directions (like a TV station) then obviously beam forming would not work.

In your example (a two element interferometer) even though the input power may be the same in both systems, in the two element system the signal amplitude would be doubled in the directions where the signals from the two elements are in phase. So if these were the desired directions the two element system would be four times as efficient with respect to power. In the directions where the signals are out of phase (zero output) there would be no input power required.

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  • $\begingroup$ But that's not true, is it? That's my point, it takes the same input power (2 * (A * sin)) vs (2A * sin) to generate either omnidirectional 2A amplitude wave, or one interference pattern with peaks of 2A. $\endgroup$ – securityN00b Jul 23 '18 at 18:27
  • $\begingroup$ It will take less power in the two element system to get the same amplitude as the omnidirectional system because its radiating through half the angular area (the two element system has an interferometer pattern--many 'beams') $\endgroup$ – user45664 Jul 23 '18 at 19:09
  • $\begingroup$ But that's the part I don't understand. I want to know where my mental model is wrong. If I take a single point source, and want to double its amplitude, it will take double the power, right? And interference between point sources can't raise the amplitude more than the sum of the amplitude of those point sources, right? $\endgroup$ – securityN00b Jul 24 '18 at 6:56
  • $\begingroup$ Re. first question: Remember...power is proportional to amplitude squared. And yes to your second question. $\endgroup$ – user45664 Jul 24 '18 at 17:42
  • $\begingroup$ Antenna people sometimes use the term 'antenna gain' to refer to main lobe output compared to omnidirectional (or isotropic) output. You can Google that for more info. $\endgroup$ – user45664 Jul 25 '18 at 16:11

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