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Given a number of mobile clients, is it possible to find a spot to place a wifi ap to optimize the reception for said clients?

I use FSPL equation to calculate the signal from the clients' position:

$$ \text{FSPL(dB)} = 20\log_{10}(d) + 20\log_{10}(f) + k $$

For d,f in meters and kilohertz, respectively, k becomes -87.55. For d,f in meters and megahertz, respectively, k becomes -27.55. For d,f in kilometers and megahertz, respectively, k becomes 32.45.

https://en.wikipedia.org/wiki/Free-space_path_loss

Any help appreciated

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If we assume that each client has a similar antenna, and is working on the same frequency, then we can simplify the math by simply stating that the signal strength, in decibels, falls off linearly with distance.

There are then various ways you might decide what the "best" compromise in placement is, but one is the centroid, which minimizes the sum of squared Euclidean distances to each client.

Of course given pathological cases it won't work so well. Consider if you have 10 clients in your home, and one on the moon, the centroid will be somewhere in cislunar space, and you'll end up with a WiFi AP that doesn't work for any of the clients.

In practice, real world conditions make this mathematical solution purely academic. Practical applications must consider:

  • reflection off objects
  • attenuation from walls
  • interference from other APs
  • noise from other electronic devices
  • antenna directionality and polarization

These effects make the idealized free-space path loss significantly different from the real loss. And in practice, and especially on the crowded 2.4 GHz band, avoiding interference is at least as critical as getting a good signal.

In practice, AP placement is done by picking a location which seems pretty close to most of the clients, but which is also clear of walls and other obstructions, and which is physically accessible for the installation. Then measurements of signal quality are made in situ, and adjustments are made as necessary.

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  • $\begingroup$ Thank you, your answer is quite enlightening. The AP in fact is mobile and let's assume that as a proof of concept there are no interferences or any other signal loss other than the distance, and also that every client is within the same room with the mobile AP. As a purely academic application, could it be done with centroids? $\endgroup$ – jrsall92 Nov 17 '15 at 21:53
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    $\begingroup$ @jrsall92 I suppose if you want the purely academic, mathematically rigorous answer, you'll need to define what "optimal" means with a similar rigor and precision. To illustrate, you might define "optimal" as any of these: "maximizing the minimum signal strength of all the clients", "maximizing the maximum signal strength of any of the clients", "maximizing the mean of the signal strength of all the clients", or "having the maximum number of clients with signal strength >= -60dBm". $\endgroup$ – Phil Frost - W8II Nov 18 '15 at 3:04

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