Competencias y funcionamiento

cations

In order to develop the optimisation model, it is necessary to use a propagation model to predict the loss or strength of signal at various locations from a given AP. Several types of propagation models have been developed. These models are divided into large-scale prop- agation models and small-scale propagation models [60, 63, 69]. A large-scale model finds the average signal power at a receiver, given various characteristics at both the transmitter and the receiver, and along the path. These models are suitable for estimating the coverage distance of a given AP and, therefore, they are used for coverage planning purposes [63]. On the other hand, a small-scale propagation model characterises rapid fluctuations of the received signal strength over very short distances.

Large-scale models can be categorised into empirical models and ray-tracing based models. Empirical models are developed based on statistical results that are obtained by conducting experiments in different environments. It is proven that these models are easy, accurate, and efficient [29, 60, 63, 69] to use. Many researchers [1, 2, 6, 24, 28, 41, 42, 64, 65, 67, 75, 79, 81, 82, 90] have used empirical models to calculate the loss of signal at the receiver.

Ray-tracing, which is a global illumination based rendering methods, traces rays of light from the source to the eye, and back through the image plane into the scene. Then, these rays are tested against all objects in the scene to determine if they intersect any objects [44, 59,91]. Ray-tracing yields very accurate results, but suffers from computational complexity [10, 23, 42, 79, 90]. Several researchers [23, 30, 42, 79, 90] used ray tracing to find the propagation data.

1.1.1

Free Space Propagation Models

Background 1.1. Preliminaries: Propagation Models in Telecommunications

lengths at the receiver. The path loss model is the core of the signal coverage calculation for any environment [29, 60, 63, 69]. By using path losses, it is possible to calculate the coverage area of the wireless base station and APs, as well as maximum distance between the transmitter and the receiver.

The free space propagation model is used to predict the strength of the signal for en- vironments where there is a clear line of sight (LOS) path between the transmitter and the receiver. The power Pr(aj, ri) received by a receiver ri, which is separated from the AP aj by a distance d(aj, ri), is given by the Friis free space equation [69]:

Pr(aj, ri) =

PtGtGrλ2 (4π)2_d(a

j, ri)2L

(1.1) where Ptis the transmit power; Gtand Grare the transmit and receive antenna gain respec- tively; λ is the wavelength, and is related to carrier frequency f : λ = c/f ; L is the system loss which is related to the transmission line attenuation, filter loss and antenna loss. We assume that there is no loss in the system, therefore, the value of L = 1.

Assuming antennas with unity gain on both sides, the path loss pl(aj, ri) in decibels (dB), which is the loss of signal due to the distance in the free space environments can be defined as [63, 69, 76]: pl(aj, ri)[dB] = 10 log Pt Pr(aj, ri) = 10 log 4πd(aj, ri) λ 2 · (1.2)

This representation is not applicable when the points aj and ri are very close to each other. For this reason, large-scale propagation models use a close-in distance, d0, which is known as the received power reference point. If d(aj, ri) is less than the reference distance, d0, then it is usually assumed that path loss, pl(aj, ri), is constant and equals to the path loss at the reference distance, which can be calculated from [60, 63, 64]:

pl(d0)[dB] = 10 log  4π

λ 2

· (1.3)

Using this equation, (1.2) can be rewritten as

pl(aj, ri)[dB] = pl(d0)[dB] + 20 log

d(aj, ri) d0

· (1.4)

Note that the representation (1.4) is true only in the free space environments. In general, the path loss exponent, n, should be taken into account [22, 60, 69, 74]:

The value of n depends on the specific propagation environment. This value is larger when obstructions are present in the path of the radio waves. In free space, one can take n = 2 and (1.5) reduces to (1.4).

In a small propagation environment, the value of d0 in (1.4) and (1.5) can be taken as equal to 1 [60, 63, 69, 74].

The path loss model given by equation (1.5) can be used to calculate the coverage range for wireless devices such as AP. In this case it is necessary first to evaluate the maximum al- lowable path loss (plmax) using transmit power and receive threshold, and then to determine the corresponding distance.

1.1.2

Obstructed Line of Sight Propagation Models

Tests have shown that the number of floors and obstacles between the transmitter and the receiver have influence on the propagation of radio waves. Different models have been developed that take into account the number of floors and the construction materials.

Path Loss Models Using Building Materials

A building can have a wide variety of partitions and obstacles that affect radio waves. Parti- tions vary in their physical characteristics. Each type of partition results in a particular loss. In one type of partition-dependent model, the path loss exponent n is taken to be equal to 2 (as for free space) and an additional loss is introduced for each type of partition. In this case the path loss formula can be written as [63, 64, 74]:

pl(aj, ri)[dB] = pl(d0)[dB] + 20 log d(aj, ri) d0 + T X t=1 ntlt (1.6)

where ntrepresents the number of obstacles of type t (for example wall, window); ltrepre- sents the loss in dB attributed to this type of obstacle, and T is the number of different types of obstacles. Some authors [1, 2, 75, 81, 82] have used this path loss formula to construct their optimisation models.

Note that function (1.6) is discontinuous because of the presence of obstacles. For ex- ample, a move around the corner of a building or a wall can cause the received signal to drop suddenly. These sudden changes cause the optimisation model to become discontinu-