## Signal strengths of a wireless network

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.

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 puts an emphasis on studying the point process formed from inverse signal strengths, we call the projection process.

## The Standard Model of wireless networks

In the previous post I discussed the signal-to-interference-plus ratio or SIR in wireless networks. If noise is included, then then signal-to-interference-plus-noise ratio or just SINR. But I will just write about SIR, as most results that hold for SIR, will also hold for SINR without any great mathematical difficulty.

The SIR is an important quantity due to reasons coming from information theory.  If you’re unfamiliar  with it, I suggest reading the previous post.

In this post, I will describe a very popular mathematical model of the SIR, which I like to call the standard model. (This is not a term used in the literature as I have borrowed it from physics.)

## Definition of SIR

To define the SIR, we consider a wireless network of $$n$$ transmitters with positions located at $$X_1,\dots,X_n$$ in some region of space. At some location $$x$$, we write $$P_i(x)$$ to denote the power value of a signal received at $$x$$ from transmitter  $$X_i$$. Then at location $$x$$, the SIR with respect to transmitter $$X_i$$ is
$$\text{SIR}(x,X_i) := \frac{P_i(x)}{\sum\limits_{j\neq i} P_j(x)} .$$

Researchers usually attempt to represent the received power of the signal $$P_i(x)$$ with a propagation model. This mathematical model  consists of a random and a deterministic component given by
$$P_i(x)=F_i\ell(|X_i-x|) ,$$
where $$\ell(r)$$ is a non-negative function in $$r\geq 0$$ and $$F_i$$ is a non-negative random variable. The function $$\ell(r)$$  is often called the path loss function. The random variables represent random fading or shadowing.

## Standard model

Based on the three model components of fading, path loss, and transmitter locations, there are many combinations possible. That said, researchers generally (I would guess, say, 90 percent or more) use a single combination, which I call the standard model.

The three standard model assumptions are:

1. Singular power law path loss $$\ell(r)=(\kappa r)^{-\beta}$$.
2. Exponential distribution for fading variables, which are independent and identically distributed (iid).
3. Poisson point process for transmitter locations.

Why these three? Well, in short, because they work very well together. Incredibly, it’s sometimes possible to get relatively a simple  mathematical expression for, say, the coverage probability $$\mathbb{P}[\text{SIR}(x,X_i)>\tau ]$$, where $$\tau>0$$.

I’ll now detail the reasons more specifically.

### Path loss

The $$\ell(r)=(\kappa r)^{-\beta}$$ is very simple, despite having a singularity at $$r=0$$. This allows simple algebraic manipulation of equations.

Some, such as myself, are initially skeptical of this function as it gives an infinitely strong signal at the transmitter due to the singularity in the function $$\ell(r)=(\kappa r)^{-\beta}$$. More specifically, the path loss of the signal from transmitter $$X_i$$ approaches infinity as $$x$$ approaches $$X_i$$ .

But apparently, overall, the singularity does not have a significant impact on most mathematical results, at least qualitatively. That said, one still observe consequences of this somewhat physically unrealistic model assumption. And I strongly doubt enough care is taken by researchers to observe and note this.

Interestingly, the original reason why exponential variables were used is because it allowed the SIR problem to be reformulated into a problem of a Laplace transform of a random variable, which for a random variable $$Y$$ is defined as

$$\mathcal{L}_Y(t)=\mathbb{E}(e^{- Y t}) \, .$$

where $$t\geq 0$$. (This is essentially the moment-generating function with $$-t$$ instead of $$t$$.)

The reason for this connection is that the tail distribution of an exponential variable $$F$$ with mean $$\mu$$  is simply $$\mathbb{P}(F>t)= e^{-t/\mu}$$.  In short, with the exponential assumption, various conditioning arguments eventually lead to Laplace transforms of random variables.

### Transmitters locations

No prizes for guessing that researcher overwhelmingly use a (homogeneous) Poisson point process for the transmitter (or receiver) locations. When developing mathematical models with point processes, if you can’t get any results with the Poisson point process, then abandon all hope.

It’s the easier to work with this point process due to its independence property, which leads to another useful property. For Poisson point process, the Palm distribution is known, which is the distribution of a point process conditioned on a point (or collection of points) existing in a specific location of the underlying space on which the point process is defined.  In general, the Palm distribution is not known for many point processes.

## Random propagation effects can lead to Poisson

A lesser known reason why researchers would use the Poisson point process is that, from the perspective of a single observer in the network, it can be used to capture the randomness in the signal strengths.  Poisson approximation results in probability imply that randomly perturbing the signal strengths can make signals appear more Poisson, by which I mean  the signal strengths behave stochastically or statistically as though they were created by a Poisson network of transmitters.

The end result is that a non-Poisson network can appear more Poisson, even if the transmitters do not resemble (the realization of) a Poisson point process. The source of randomness that makes a non-Poisson network appear more Poisson is the random propagation effects of fading, shadowing, randomly varying antenna gains, and so on, or some combination of these.

A good starting point on this topic is the Wikipedia article Stochastic geometry models of wireless networks. This paper is also good:

• 2009, Haenggi, Andrews, Baccelli, Dousse, Franceschetti, Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks.

This paper by my co-authors and I has some details on standard model and why a general network model behaving Poisson in terms of the signal strengths:

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

Early books on the subject include the two-volume textbooks Stochastic Geometry and Wireless Networks by François Baccelli and Bartek Błaszczyszyn, where the first volume is on theory and the second volume is on applications.  Martin Haenggi wrote a very readable introductory book called Stochastic Geometry for Wireless networks.

Finally, I co-wrote with Bartek Błaszczyszyn, Sayan Mukherjee, and Martin Haenggi a short monograph on SINR models called Stochastic Geometry Analysis of Cellular Networks, which is written at a slightly more advanced level. This book has a section on why signal strengths appear Poisson.

## Signal-to-interference ratio in wireless networks

The fundamentals of information theory say that to successfully communicate across any potential communication link the signal strength of the communication must be stronger than that of the back ground noise, which leads to the fundamental quantity known as signal-to-noise ratio. Information theory holds in very general (or, in mathematical speak, abstract) settings. The communication could be, for example, a phone call on an old wired landline, two people talking in a bar, or a hand-written letter, for which the respective signals in these examples are the electrical current, speaker’s voice, and the writing. (Respective examples of noise could be, for example, thermal noise in the wires, loud music, or coffee stains on the letter.)

In wireless networks, it’s possible for a receiver to simultaneously detect signals from multiple transmitters, but the receiver typically only wants to receive one signal. The other unwanted or interfering signals form a type of noise, which is usually called interference, and the other (interfering) transmitters are called interferers. Consequently, researchers working on wireless networks study the signal-to-interference ratio, which is usually abbreviated as SIR. Another name for the SIR is carrier-to-interference ratio.

If we also include background noise, which is coming not from the interferers, then the quantity becomes the signal-to-interference-plus-noise ratio or just SINR. But I will just write about SIR, though jumping from SIR to SINR is usually not difficult mathematically.

The concept of SIR makes successful communication more difficult to model and predict, as it just doesn’t depend on the distance of the communication link. Putting the concept in everyday terms, for a listener to hear a certain speaker in a room full of people all speaking to the listener, it is not simply the distance to the speaker, but rather 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.

In wireless network research, much work has been done to examine and understand communication success in terms of interference and SIR, which has led to a popular mathematical model that incorporates how signals propagate and the locations of transmitters and receivers.

## Definition

To define the SIR, we consider a wireless network of transmitters with positions located at $$X_1,\dots,X_n$$ in some region of space. At some location $$x$$, we write $$P_i(x)$$ to denote the power value of a signal received at $$x$$ from transmitter $$X_i$$. Then at location $$x$$, the SIR with respect to transmitter $$X_i$$ is
$$\text{SIR}(x,X_i) :=\frac{P_i(x)}{\sum\limits_{j\neq i} P_j(x)} =\frac{P_i(x)}{\sum\limits_{j=1}^{n} P_j(x)-P_i(x)} .$$

The numerator is the signal and the denominator is the interference.  This ratio tells us that increasing the number of transmitters $$n$$ decreases the original SIR values. But then, in exchange, there is a greater number of transmitters for the receiver to connect to, some of which may have larger $$P_i(x)$$ values and, subsequently, SIR values. This delicate trade-off makes it challenging and interesting to mathematically analyze and design networks that deliver high SIR values.

Researchers usually assume that the SIR is random. A quantity of interest is the tail distribution of the SIR, namely

$$\mathbb{P}[\text{SIR}(x,X_i)>\tau ] := \frac{P_i(x)}{\sum\limits_{j\neq i} P_j(x)} \,,$$

where $$\tau>0$$ is some parameter, sometimes called the SIR threshold. For a given value of $$\tau$$, the probability $$\mathbb{P}[\text{SIR}(x,X_i)>\tau]$$ is sometimes called the coverage probability, which is simply the probability that a signal coming from $$X_i$$ can be received successfully at location $$x$$.

## Mathematical models

### Propagation

Researchers usually attempt to represent the received power of the signal $$P_i(x)$$ with a propagation model. 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 $$F_i$$ is a non-negative random variable and $$\ell(r)$$ is a non-negative function in $$r \geq 0$$.

##### Path loss

The function $$\ell(r)$$ is called the path loss 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, which need to be fitted to (or estimated with) real world data.

Researchers generally assume that the so-called path loss function $$\ell(r)$$ is decreasing in $$r$$, but actual path loss (that is, the change in signal strength over a path travelled) typically increases with distance $$r$$. Researchers originally assumed path loss functions to be increasing, not decreasing, giving the alternative (but equivalent) propagation model
$$P_i(x)= F_i/\ell(|X_i-x|).$$

But nowadays researchers assume that the function $$\ell(r)$$ is decreasing in $$r$$. (Although, based on personal experience, there is still some disagreement on the convention.)

With the random variable $$F_i$$, researchers seek to 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. These variables are often called fading or shadowing variables, depending on what physical phenomena they are representing.

Typical distributions for fading variables include the exponential and gamma distributions, while the log-normal distribution is usually used for shadowing. The entire collection of fading or shadowing variables is nearly always assumed to be independent and identically distributed (iid), but very occasionally random fields are used to include a degree of statistical dependence between variables.

### Transmitters locations

In general, we assume the transmitters locations $$X_1,\dots,X_n$$ are on the plane $$\mathbb{R}^2$$. To model interference, researchers initially proposed non-random models, but they were considered inaccurate and intractable. Now researchers typically use random point processes or, more precisely, the realizations of random point processes for the transmitter locations.

Not surprisingly, the first natural choice is the Poisson point process. Other point processes have been used such as Matérn and Thomas cluster point processes, and Matérn hard-core point processes, as well as determinantal point processes, which I’ll discuss in another post.

## Some history

Early random models of wireless networks go back to the 60s and 70s, but these were based simply on geometry: meaning a transmitter could communicate successfully to a receiver if they were closer than some fixed distance. Edgar Gilbert created the field of continuum percolation with this significant paper:

• 1961, Gilbert, Random plane networks.

Interest in random geometrical models of wireless networks continued into the 70s and 80s. But there was no SIR in these models.

Motivated by understanding SIR, researchers in the late 1990s and early 2000s started tackling SIR problems by using a random model based on techniques from stochastic geometry and point processes. Early papers include:

• 1997, Baccelli, Klein, Lebourges ,and Zuyev, Stochastic geometry and architecture of communication networks;
• 2003, Baccelli and Błaszczyszyn , On a coverage process ranging from the Boolean model to the Poisson Voronoi tessellation, with applications to wireless communications;
• 2006, Baccelli, Mühlethaler, and Błaszczyszyn, An Aloha protocol for multihop mobile wireless networks.

But they didn’t know that some of their results had already been discovered independently by researchers working on wireless networks in the early 1990s. These papers include:

• 1994, Pupolin and Zorzi, Outage probability in multiple access packet radio networks in the presence of fading;
• 1990, Sousa and Silvester, Optimum transmission ranges in a direct-sequence spread-spectrum multihop packet radio network.

The early work focused more on small-scale networks like wireless ad hoc networks. Then the focus shifted dramatically to mobile or cellular phone networks with the publication of the paper:

• 2011, Andrews, Baccelli, Ganti, A tractable approach to coverage and rate in cellular networks.

It’s can be said with confidence that this paper inspired much of the interest in using point processes to develop models of wireless networks. The work generally considers the SINR in the downlink channel for which the incoming signals originate from the phone base stations.