# How to include Rayleigh fading into path loss model?

I am doing a student project to localize the physical position of a network's nodes in outdoor environments. It will be done by RSSI values.

Background: To do this I need to get a good model of how the signal strength propagates/attenuates in the environment of choice. I have studied some theories and the most used (as I understand) is the log distance model: PL(d) = 10n*log(d/d0) + PL(d0), where d is the distance, d0 the reference distance and n is the environment path loss coefficient.

This can be expanded to take shadowing into account with a stochastic variable X ∈ N(0,σ) PL = PL(d) = 10n*log(d/d0) + PL(d0) + X

Question: I have read a lot about Rayleigh distributions but I do not see how this shall be implemented in this equation. Can someone help me to understand how it can be included?

• Are you sure that it is even important in your application, rather than the vast inaccuracy of the receivers power measurement? – PlasmaHH Jun 19 '17 at 13:44
• Perhaps multipath seriously dominates Rayleigh in affecting received signal strength. – glen_geek Jun 19 '17 at 13:48
• Rayleigh fading can add or subtract about 20 db to the signal received but it is a statistical thing and will, on occasions create zero received signal. Ricean fading may be more appropriate (one dominant path) – Andy aka Jun 19 '17 at 14:01
• @PlasmaHH No idea, I do not know the magnitude of the fading effects. – tompak Jun 19 '17 at 14:07
• @glen_geek Rayleigh fading is due to multipath. – tompak Jun 19 '17 at 14:08

The instantaneous effect of path loss and channel fading can be modelled as

$$\text{PL}(d) \approx 10\gamma\times\log\left(\frac{d}{d_0}\right) + \text{PL}(d_0) + X_1 + X_2$$

where $X_1$ is zero-mean log-normal distributed and is due to slow fading (i.e shadowing) and $X_2$ is zero-mean Rayleigh distributed if in mobile, NLOS environment. If there is a LOS component then $X_2$ will be Ricean distributed. Note the formula above is not the same one used in communication budget link loss calculation, budget link loss is a similar but slightly different matter.

The log-normal path loss distribution simplifies this model by condensing $X_1$ and $X_2$ into one log-normally distributed variable $X_g$. It is particularly accurate if $X_2$ is negligible because of relatively low mobility (i.e in slow fading channels).

$X_1$ and $X_2$ are stochastic variables drawn from probability distribution functions, not deterministic quantities. If you have a random number source that produces uniformly distributed numbers, $U$, in the interval [0,1]. Then you can easily formulate a log-normal or Rayleigh number source from values of U by using inverse transform sampling. See source [1] and [2].

$$X_1 = \sigma \sqrt {-2\ln U_1} \cos (2\pi U_2)$$ $$X_2 = \sigma \sqrt {-2\log U}$$

or assume just log-normal and use just one variable

$$X_g = \sigma \sqrt {-2\ln U_1} \cos (2\pi U_2)$$

Where $U_i$ is a random variable drawn from the uniform distribution.The standard deviation ($\sigma$) is obtained empirically, you can see sample values of $sigma$ used to evaluate $X_g$ in different environments below [source]:

Note that because $X_1$ and $X_2$ are zero mean their effect can be largely reduced by taking many RSS values and then passing them through an optimal averaging filter. The biggest problem you have is that in a pure NLOS environment the path loss model we have being discussing here will be fairly inaccurate in a lot of settings as it assumes broadly similar environmental conditions existing between the average path distances [$0$,$d_0$] and [$d_0$, $d$], which is not the case in many NLOS cases. Again in NLOS there will be the issue of the average path distance being very different from the actual distance between communicating nodes. You might want to read up on that and maybe try searching for "Wireless Sensor Network RSSI Localisation algorithms" to see common ways the problem you have is commonly tackled in literature.