# Geometry of wireless signal strength

How does wireless signal strength correspond to distance? RSSI lies between -100 and 0 (at least, on my computer). Let's say I walk a distance x towards the router, and my RSSI goes from -60 to -50. Now, lets say instead I walk a distance 2x towards the router. Would this imply that RSSI would go from -60 to -40? I'm curious what the relationship of the metrics is, is RSSI linear/logarithmic/etc with respect to distance? I'm a math guy with little physics/engineering background so some help would be very appreciated. Thanks.

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Your question hugely depends on the geometry of the antenna.

If your antenna is radiating in all directions uniformly, then by taking simple geometry into consideration, you can show that intensity (power/area) falls as $1/r^2$ Energy must be conserved, but the area in which the wave has spread to, increases as $r^2$.

You can look at the problem as follows too: You can ask yourself how much of the original power have I obtained? My receiver has finite dimensions and it can pick signal only at a finite area $A_0$. As you are moving away from the antenna you are getting smaller chunk of the whole power, which equals $\eta = \frac{A_0}{A} = \frac{A_0}{4 \pi r^2}$.

The complication comes from the fact that common antennas, like dipole antenna do not radiate energy evenly in the whole space. In the plane perpendicular to dipole antenna most energy is radiated and energy radiated in direction that is parallel to antenna is negligible. Then calculation gets much tougher and intensity in some directions might fall slower than $1/r^2$

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I'm just talking about standard wifi routers, if that helps at all. – Andrew Apr 23 '12 at 6:55
Standard wifi routers have, as far as I know, dipole antennas. I don't think I could give you the right answer right away - it would require some calculation, but first guess would be that intensity falls approximately between $1/r$ and $1/r^2$, closer to the second (in the plane perpendicular to antenna). The problem with RSSI is that it is unclear if the displayed intensity is logarithmic. Also there are some thresholds intensities - e.g. above certain intensity RSSI shows 100% and below certain intensity it shows 0%. – Pygmalion Apr 23 '12 at 7:08
(Almost) all antennas are dipole antennas when viewed from sufficiently far away. However, intensity cannot fall off slower than $1/r^2$ independently of direction, since that would violate energy conservation. (This is independent of multipolarity, since all the appropriate Bessel functions are asymptotically $\tilde{} e^{ikr}/r$.) Nearer the source (on scales comparable with the source antenna!) the intensity might go up faster than $1/r^2$ due to near-field effects that do depend on the exact shape of the antenna, in which case the dipole approximation probably fails relatively soon. – Emilio Pisanty Apr 24 '12 at 1:05
@episanty Imagine for the sake of argument that antenna's length is infinity. Of course, you would need infinite power supply for it, but intensity would fall as $1/r$. Thus, for geometrical reasons intensity near antenna falls $1/r$ near antenna and $1/r^2$ far away from it. In mid-space it should have mixing of two terms. This is my reasoning. – Pygmalion Apr 24 '12 at 5:50
@Pygmalion: you're right, my argument holds only for bounded sources - we would all love to have infinite antennas but they're hard to come by. Saying an antenna is very large is about the same as saying you're very close to it, and indeed in the near-field regime there are components that go up faster than $1/r^2$, as you point out. – Emilio Pisanty Apr 24 '12 at 18:31

There is no standardized relationship of any particular physical parameter to the RSSI reading. The 802.11 standard does not define any relationship between RSSI value and power level in mW or dBm. Vendors provide their own accuracy, granularity, and range for the actual power (measured as mW or dBm) and their range of RSSI values (from 0 to RSSI_Max).

So whether the implementation is linear or logarithmic in the power received will vary between vendors.

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Typically, a received power would be measured in logarithmic units (for example dBm i.e dB with respect to a milliwatt, or as a field strength dB$\mu$V/m).

A typical received power vs distance law would be of the form

$$RxPower(dBm) = A+Blogr$$

where $A$ and $B$ are constants and $r$ is a the distance from transmitter to receiver. The power law would just be an empirical fit to data, so the units of distance would be specified when giving the law. So the difference in received power when you move by $\Delta r$ can be determined if you know the constant B. B is, in linear terms, an exponent in a power law

$$power \propto r^B$$

For a signal received from via a route which includes absorbers within the Fresnel zone (as it will typically in an indoor environment), the power law will deviate from the free space value B=-2.0 (it will be a faster fall-off). Note also, that these power laws for received signal relate to average values. Locally, there will be a distribution of received values about the predicted average.

The Received Signal Strength Indicator may only display a crude relationship with the Rx Power. (You might at least hope it will be monotonic!)

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## protected by Qmechanic♦Jul 10 '13 at 21:41

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