# Introduction

Line of Sight analysis is a commonly used technique in telecommunication industry for A/I (Air Interface) equipment planning and allocation. With the simplest terms LoS is the question whether a point on N-dimensional space is visible by an other observer point. The question can be used to answer where to locate a transceiver on terrain so that it can serve customers on some region A.

Before relatively more complicated problems, let’s start with an easy example focusing on two dimensional terrains. Throughout the post, we will use R for coding which is my favorite option for any mathematical problem (statistics, plotting, linear algebra, optimization, etc.). But you can easily adapt coding material to Mathlab, Python,or your favorite language.

We will start by defining a mathematical function to be used to generate our pseudo terrains. For this purpose trigonometric functions (sin, cos) and polynomial functions are the best ones because of their wavy shapes. Here is an example of trigonometric terrain

Figure 1 Trigonometric Terrain

In order to generate this two dimensional one use the following code piece

x <- seq(from=4,to=10,by=0.01)
y <- sin(x)+cos(2*x)+sin(3*x)+cos(4*x)+3

windows()
plot(x,y,'l',
main="y=sin(x)+cos(2x)+sin(3x)+cos(4x)+3",
ylab="height",col="blue")
You may choose to use a polynomial terrain also

Figure 2 Polynomial Terrain

To obtain this terrain, use the following R script piece

x <- seq(from=0,to=6,by=0.01)
y <- x*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)+100

windows()
plot(x,y,'l',
main="y=x(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)+100",
ylab="height",col="blue")

Combining polynomial terrain functions with trigonometric ones will give you fancier ones.

# What is LoS ?

You can think that we have already answered this question but this was an informal try which is not very useful for solving the problem. In order to solve this problem methodically we need to understand what makes a target visible (within LoS) by the observer.

As you see on Figure 3, green point is within line of sight of observer (blue point). However there is pseudo hill between red point and observer. The difference is that the line connecting observer and green point is always greater than the terrain function whereas this is not valid for the line connecting observer and red point (for x ε [~2.5, ~3.5] red line is under the terrain curve).

Figure 3 LoS vs non-LoS

This was the first point (blocking) we should define. The second one is an easier one related with maximum Euclidean distance between observer and target. The distance between observer and target may cause a phase shift in signal if the distance is sufficiently long or depending on weather conditions and terrain properties you may observer diffraction problems (actually there might be more than those). In return this will cause signal quality issues or call drops.  On Figure 3, although blocking is not an issue between observer and  yellow point, target is out of  visible range (say 8 units) of observer.

You can generate Figure 3 using the following R script

# Terrain Function
height <- function (x) {
x*x/3+sin(x)+cos(2*x)+sin(3*x)+cos(4*x)+sin(5*x)+cos(6*x)+3
}

# Observer location
observer<-c(1.5,8.9)

# Generate terrain points with a tolerance of 0.1
x<-seq(from=-0.1,to=6.1,by=0.1)
terrainHeight<-height(x)

windows()
# Draw terrain
plot(x,terrainHeight,type='b',
xlim=range(x),ylim=range(terrainHeight),
main="Line of Sight (LoS)",
ylab="Height",xlab="")

# Not LoS
points(x=c(observer[1],x[41]),
y=c(observer[2],terrainHeight[41]),
col="red",type='b')

# LoS
points(x=c(observer[1],x[5]),
y=c(observer[2],terrainHeight[5]),
col="green",type='b')

# LoS but far
points(x=c(observer[1],x[length(x)]),
y=c(observer[2],terrainHeight[length(x)]),
col="yellow",type='b')

# Draw Observer
points(x=c(observer[1]),
y=c(observer[2]),
col="blue",pch=10)

# Method to Decide LoS

Finally let’s define a method to find all visible, invisible, and “far” points on any terrain. Since it is not “easy” to decide analytically whether the line connecting observer and target “is above” the terrain for any terrain function, we will use a simple numeric method.

We will define a step size small enough (around Spatial Tolerance) to generate all x values between observer and target. seq function is a good choice for doing this (Code Lines 33-36).  Evaluate these x values for line function connecting observer and target and terrain function. Evaluation is simple for terrain function using height function (Code Lines 4-7). Evaluation of line function is held by function linear  using parametric definition of line function (Code Lines 9-14) . Next step is to search for any x value having a line evaluation less than terrain evaluation (Code Line 44-28). The rest is simple as to evaluate euclidean distance and assigning values to status variable.

##################
# Functions
##################
# Terrain Function
height <- function (x) {
x*x/3+sin(x)+cos(2*x)+sin(3*x)+cos(4*x)+sin(5*x)+cos(6*x)+3
}

# Linear Function
linear <- function (x, observer, target) {
v <- observer - target

((x - observer[1])/v[1])*v[2]+observer[2]
}

# Linear Function
distance <- function (p0,p1) {
sqrt(sum((p0-p1)^2))
}

##################
# Input
##################
# Observer location
observer<-c(1.5,9)

# Target on terrain
target <- c(5, height(5))

# Max visible distance
maxVisibleDistance = 4

# Generate points with a step size of 0.1
x <- seq(from=min(observer[1],target[1]),
to=max(observer[1],target[1]),
by=0.1)

# Terrain Height
h <- height(x)

# y Values
y <- linear(x, observer, target)

# LoS Analysis
aboveTerrain <- round((y-h),2) >= 0.00

# First Rule
visible <- !is.element(FALSE,aboveTerrain)
if (visible){
# Second Rule
d <- distance(observer, target)
if(d <= maxVisibleDistance){
status <- "LoS"
}else{
status <- "non-LoS due to Distance"
}
}else{
status <- "non-LoS due to Blocking"
}