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If you have two antennas \$T_1\$ and \$T_2\$ each broadcasting 10GHz Gaussian White Noise at the same receiver, each with an EIRP of say 20 dBW with a random phase offset between them, then is there an equation that governs the average received power at the receiver?

This question makes it clear that the sum of two independent white noise sources is still white noise, but it says nothing about the amplitude/received power.

My assumption is the result will be on average the same power as a single transmitter, and if this is the case, is there an alternative type of noise that can be additive despite a random phase offset?

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  • \$\begingroup\$ I would think, docking a ship into port, that you'd see more energy from two lighthouses as opposed to one. \$\endgroup\$
    – rdtsc
    Commented Feb 3, 2022 at 13:21
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    \$\begingroup\$ You receive twice the power (assuming the same distance to each TX). So you can expect sqrt(2) the received amplitude. As it's statistical, you may have to observe for a while to measure accurately... \$\endgroup\$
    – user16324
    Commented Feb 3, 2022 at 13:35
  • \$\begingroup\$ As @user_1818839 wrote, you'll get double the power when receiving two independent white noise sources. Note that for independent sources there is no concept of "a random phase offset". That phrase suggests you have two copies of the same white noise source with some phase offset between them. If you want to know the statistics of that I suggest you ask on the Signal Processing stack. \$\endgroup\$
    – Graham Nye
    Commented Feb 9, 2022 at 20:43

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is there an equation that governs the average received power at the receiver?

Two unrelated signals (voltage or current sources for instance) add up as the sum of squares: -

$$\text{Total RMS} = \sqrt{A_{RMS}^2 + B_{RMS}^2}$$

My assumption is the result will be on average the same power as a single transmitter

No, that isn't the case.

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  • \$\begingroup\$ What about the phase difference, frequency difference ? \$\endgroup\$
    – Wireless Learning
    Commented Feb 4, 2022 at 10:14
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    \$\begingroup\$ @LucasVivi no, gaussian white noise cannot be regarded this way and, as I and others have said (and there are many online references that say the same thing), noises add as \$\sqrt{A^2+B^2}\$. Example. Read also these answers. If you downvoted, can you please explain why? \$\endgroup\$
    – Andy aka
    Commented Feb 4, 2022 at 14:21
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    \$\begingroup\$ No references... \$\endgroup\$
    – Wireless Learning
    Commented Feb 7, 2022 at 8:37
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    \$\begingroup\$ @Andyaka As the phrase you've quoted mentions power it would be helpful if you clarified that the equation you've given relates to voltage or current, but not power. The powers of independent noise sources just add together, using simple addition, of course. \$\endgroup\$
    – Graham Nye
    Commented Feb 9, 2022 at 20:48
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    \$\begingroup\$ @Andyaka I'm aware of that. It would be an improvement if you clarified that point in your answer for the benefit of those who aren't. It is normal to define the terms used in an equation. Your equation is sandwiched between two mentions of power so it's easy to misinterpret it as also dealing with power. \$\endgroup\$
    – Graham Nye
    Commented Feb 9, 2022 at 22:19

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