In two previous posts, here and here, I discussed the importance of the quantity called the *signal-to-interference ratio*, which is usually abbreviated as *SIR*, for studying communication in wireless networks. In everyday terms, for a listener to hear a certain speaker in a room full of people speaking, the ratio of the speaker’s volume to the sum of the volumes of everyone else heard by the listener. The SIR is the communication bottleneck for any receiver and transmitter pair in a wireless network.

But the strengths (or power values) of the signals are of course also important. In this post I will detail how we can model them using a a simple network model with a single observer.

### Propagation model

For a transmitter located at \(X_i\in \mathbb{R}^2\), researchers usually attempt to represent the received power of the signal \(P_i\) with a *propagation model.* Assuming the power is received at \(x\in \mathbb{R}^2\), this mathematical model consists of a random and a deterministic component taking the general form

$$

P_i(x)=F_i\,\ell(|X_i-x|) ,

$$

where \(\ell(r)\) is a non-negative function in \(r>0\) and \(F_i\) is a non-negative random variable.

The function \(\ell(r)\) is called the *pathloss function*, and common choices include \(\ell(r)=(\kappa r)^{-\beta}\) and \(\ell(r)=\kappa e^{-\beta r}\), where \(\beta>0\) and \(\kappa>0\) are model constants.

The random variables \(F_i\) represent signal phenomena such as *multi-path fading* and *shadowing* (also called *shadow fading*), caused by the signal interacting with the physical environment such as buildings. It is often called fading or shadowing variables.

We assume the transmitters locations \(X_1,\dots,X_n\) are on the plane \(\mathbb{R}^2\). Researchers typically assume they form a random point process or, more precisely, the realization of a random point process.

## From two dimensions to one dimension

For studying wireless networks, a popular technique is to consider a wireless network from the perspective of a single observer or user. Researchers then consider the incoming or received signals from the entire network at the location of this observer or user. They do this by considering the inverses of the signal strengths, namely

$$

L_i(x): = \frac{1}{P_i}=\frac{1}{F_i \,\ell(|X_i-x|) }.

$$

Mathematically, this random function is simply a mapping from the two-dimensional plane \(\mathbb{R}^2\) to the one-dimensional non-negative real line \(\mathbb{R}_0^+=[0,\infty)\).

If the transmitters are located according to a non-random point pattern or a random point process, this random mapping generates a random point process on the non-negative real line. The resulting one-dimensional point process of the values \(L_1,L_2,\dots, \) has been called (independently) *propagation (loss) process* or *path loss (with fading) process*. More recently, my co-authors and I decided to call it a *projection process*, but of course the precise name doesn’t mattter

## Intensity measure of signal strengths

Assuming a continuous monotonic path loss function \(\ell\) and the fading variables \(F_1, F_2\dots\) are iid, if the transmitters form a stationary random point process with intensity \(\lambda\), then the inverse signal strengths \(L_1,L_2,\dots \) form a random point process on the non-negative real line with the intensity measure \(M\).

$$

M(t) =\lambda \pi \mathbb{E}( [\ell(t F)^{-1} ]^2)\,,

$$

where \(\ell^{-1}\) is the generalized inverse of the function \(\ell\). This expression can be generalized for a non-stationary point process with general intensity measure \(\Lambda\).

The inverses \(1/L_1,1/L_2,\dots \), which are the signal strengths, forprocess with intensity measure

$$

\bar{M}(s) =\lambda \pi \mathbb{E}( [\ell( F/s)^{-1} ]^2).

$$

## Poisson transmitters gives Poisson signal strengths

Assuming a continuous monotonic path loss function \(\ell\) and the fading variables \(F_1, F_2\dots\) are iid, if the transmitters form a Poisson point process with intensity \(\lambda\), then the inverse signal strengths \(L_1,L_2,\dots \) form a Poisson point process on the non-negative real line with the intensity measure \(M\).

If \(L_1,L_2,\dots \) form a homogeneous Poisson point process, then the inverses \(1/L_1,1/L_2,\dots \) will also form a Poisson point process with intensity measure \(\bar{M}(s) =\lambda \pi \mathbb{E}( [\ell( F/s)^{-1} ]^2). \)

## Propagation invariance

For \(\ell(r)=(\kappa r)^{-\beta}\) , the expression for the intensity measure \(M\) reduces to

$$

M(t) = \lambda \pi t^{-2/\beta} \mathbb{E}( F^{-2/\beta})/\kappa^2.

$$

What’s striking here is that information of the fading variable \(F\) is captured simply by one moment \(\mathbb{E}( F^{-2/\beta}) \). This means that two different distributions will give the same results as long as this moment is matching. My co-authors and I have been called this observation *propagation invariance*.

## Some history

To study just the (inverse) signal strengths as a point process on the non-negative real line was a very useful insight. It was made independently in these two papers:

- 2008, Haenggi,
*A geometric interpretation of fading in wireless*

*networks: Theory and applications*; - 2010, Błaszczyszyn, Karray, and Klepper,
*Impact of the geometry, path-loss exponent and random shadowing on the mean interference factor in wireless cellular networks*.

My co-authors and I presented a general expression for the intensity measure \(M\) in the paper:

- 2018, Keeler, Ross and Xia,
*When do wireless network signals appear Poisson?.*

This paper is also contains examples of various network models.

## Further reading

A good starting point on this topic is the Wikipedia article Stochastic geometry models of wireless networks. The paper that my co-authors and I wrote has details on the projection process.

With Bartek Błaszczyszyn, Sayan Mukherjee, and Martin Haenggi, I co-wrote a short monograph on SINR models called Stochastic Geometry Analysis of Cellular Networks, which is written at a slightly more advanced level. The book put an emphasis on studying the point process formed from inverse signal strengths, we call the projection process.