ESPECIFICACIÓN TEXTUAL DE LOS CASOS DE USO

The way signal strength changes as a function of distance from a radiating source is a function of the environment. The simplest and most exact formulation of that function is applicable to free space. Any other environment contains objects that reflect, absorb or obstruct, or scatter the electromagnetic wave, forcing a modifica-tion of the free-wave signal strength versus distance relamodifica-tionship and the introduc-tion of a probabilistic term to account for the fact that the environment cannot be described exactly or changes with time.

6.2.1 Free Space

In free space, the parameters that directly affect the relationship between received power P_rand distance d at wavelength␭are included in the Friis equation:

P_r=P_tG_tG_r␭²

(4␲)²d² (6.1)

G_tand G_rare transmitter and receiver antenna gains. Note that the receiver cannot calculate the distance to the transmitter only from the received power, but it must be informed of the transmitter’s radiated power—P_tG_t—either from previous knowledge or a message from the transmitter. This is analogous to the situation in a TOF system where a receiver must know an epoch time of transmission in order to find the one-way propagation time. Consequently, distance to a rogue transmitter, for example, cannot be determined by a single receiver without some cooperation from the target terminal.

Equation (6.1) can be made more convenient for purposes of comparison as well as simplified by expressing it as the inverse of the numerical path loss, or path gain, PG. Path loss is the attenuation of the signal as it propagates between transmitter and receiver. We use path gain instead of path loss in order to show more directly the effect on received signal strength. Numerical path gain is the ratio of the received power to the transmitter radiated power,

PG= P_r

P_tG_tG_r =冉^4␲␭d冊^{2 (6.2)}

6.2.2 Free-Space dB

It is usually more convenient to work with logarithmic expressions, for which path gain in decibels is

PG_dB= 20 log冉^4␲␭d冊 ^(6.3)

In free space, when transmitted power and antenna gains are known, distance can be determined with high accuracy from the received signal strength using (6.3).

However, in all other communication links, objects, including the ground, in the vicinity of the transmission path change the relationship between received power and distance. The received power will be a vector sum of signals from the transmitter arriving over different path lengths because of reflections from nearby objects and partial blocking by materials in the signal path. The resulting received power may be greater or less than the line-of-sight signal over the transmission path. When the reflecting objects are moving in respect to the link terminals, the received power will change with time. In addition large obstacles such as buildings, walls, or floors that are present on the line-of-sight path attenuate the direct signal and reduce the received power.

6.2.3 Open Field

A plot showing how received signal strength varies with distance in the presence of one reflector, the ground, is shown in Figure 6.1, which also shows free space path gain for comparison. Frequency is 2.4 GHz and both the transmitting and receiving antennas are vertically polarized and 1.5m high. Vertical antennas are most commonly employed on 2.4-GHz short-range devices because they are nondi-rectional and most convenient to attach to small products. Within the distance span shown in the plot, 100m, the path gain, and consequently the received signal strength, varies significantly from the free space value as expressed in (6.3). As a mobile terminal recedes from a fixed terminal, the signal experiences variable fading, and within short distances the received signal strength grows while the range increases. The mean value however follows closely the free-space curve. Over larger distances than are shown in Figure 6.1, the variations over small distance increments decrease and the open field signal strength is consistently below the that of free space.

0 20 40 60 80 100

−90

−80

−70

−60

−50

−40

Open field Free space

Path gain versus range

Range, meters

Path gain dB

Figure 6.1 Open field and free-space propagation path gain at 2.4 GHz. Polarization is vertical and transmitting and receiver antenna heights are 1.5m.

When the range (d) axis is a logarithmic scale, the mean signal strength curve can be approximated by two linear segments that meet at some distance d₀. The path gain curve shown in Figure 6.1 is plotted in Figure 6.2 with a logarithmic range axis and maximum range extended out to 1,000m. To the left of the vertical dashed line marked as d₀, the average value of the log-log plot has a slope of−2, representing a distance exponent of 2, as in free space, and the segment to the right of d₀ has a slope of −4, showing dependence of a distance exponent of 4.

The plot can be expressed approximately by

PG_dB= −20 log冉^4␲␭d0冊 ^{− 20 log}冉^dd0冊 ^{d≤ d0 (6.4)}

PG_dB= −20 log冉^4␲␭d0冊 − n ⭈ 10 ⭈ log冉^dd0冊 ^{d> d0}

where n is the exponent of the inverse of the distance when d> d₀. The path gain parameters are wavelength␭, d₀, and n. In the case of open field propagation, n= 4 and d0can be approximated by

d₀= (12h₁h₂)/␭ ^(6.5)

where h₁and h₂are the heights of the terminal antennas. In Figure 6.2, h₁ and h₂each equal 1.5m and␭equals 0.125m, resulting in a value for d₀of 216m.

6.2.4 Logarithmic Approximation

Curve approximations expressed by (6.4) with plots similar to Figure 6.2 can be made when there are other reflections in addition to ground. d₀ and n can be

1 10 100 1.10³

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−100

−80

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Open field Free space

Path gain versus range

Range, meters

Path gain, dB

d₀

Figure 6.2 Open field and free space propagation path gain at 2.4 GHz with a logarithmic scale on the range axis. Polarization is vertical and transmitter and receiver heights are 1.5m.

estimated empirically by survey measurements. The slopes and intercepts of (6.4) are calculated by least square regression from the empirical data, choosing d₀by estimation from observing the data.

A simplified propagation model for indoor environments over a range of 0.5m up to several hundred meters is shown in (6.6) and plotted in Figure 6.3. The model is for a frequency of 2.45 GHz and has been suggested for use in wireless personal area networks [1]. The path gain estimation is for free space propagation from 0.5m up to 8m. Beyond 8m, the estimated path gain has a slope of −3.3 (n= 3.3).

PG_dB= −40.2 − 20 log冉^1md 冊 ^{0.5m≤ d ≤ 8m (6.6)}

PG_dB= −58.5 − 33 log冉^8md 冊 ^{d> 8m}

6.2.5 Randomizing Term X

A number of models in addition to (6.6) have been suggested for indoor propagation [2]. Due to the wide variation of propagation conditions in indoor environments, no one formulation can adequately predict received signal strength in every installa-tion. The following factors affect propagation and cause deviations from the various propagation relationships that have been suggested:

• Multipath propagation that depends on the position of the transmitter and receiver relative to floor and ceiling, partitions and furnishings;

0.1 1 10 100 1.10³

−120

−110

−100

−90

−80

−70

−60

−50

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Distance (meters)

Path gain (dB)

Figure 6.3 Example of path gain curve for indoor propagation at 2.45 GHz.

• Shadowing effect of building materials and other objects in the propagation paths;

• Antenna heights and relative polarization;

• Transmission frequency;

• Moving objects, specifically people, in the vicinity of the transmission paths.

In order to keep the propagation formula simple and also indicate deviations from what may be considered a mean large scale value, a term indicating ran-domness due to any of the factors listed above is added to (6.4) and shown in (6.7):

PG_dB= 20 log冉^4␲␭d0冊 + 10 ⭈ n ⭈ log冉^dd0冊 ^{+ X␴ d> d0 (6.7)}

X_␴is a random value in decibels having a standard deviation of ␴. Examples of the variation of n and␴with environment and frequency are shown in Table 6.1 [3].

As mentioned, environmental conditions change with time, and different trans-mission paths, even in a similar locality, have different parameters. Thus, the received power is a random variable and in order to attain desired distance or location accuracy, averaging methods are used, based on multiple measurements.

The parameters d₀, n, and ␴can be found for a particular environment and frequency by taking a set of measurements of signal strengths at known ranges at various positions and times and then using the data to make a least squares estimate of those parameters to fit the curve of (6.7). First, measurement data for range greater than a likely value of d₀should be used to find n and ␴, then the short-range data can be used to find a likely value of d₀.

6.2.6 Outdoor Area Networks

The details of range predications for outdoor mobile and fixed wireless networks are different than those of the indoor systems described above, but they generally are based on log-linear approximations in the form of (6.7). Empirical models have been proposed that apply to specific frequency bands and whose parameters are applied in a manner that depends on terrain or the degree of building density, described as large city, medium city, suburban, or open areas [4]. As an example, one of the models, designated as Stanford University Interim (SUI) Model, is described here briefly. It was developed for the IEEE working group 802.16 for fixed wireless access systems in the band from 2.5 GHz to 2.7 GHz. The systems

Table 6.1 Variation of Propagation Parameters with Environment and Frequency Environment Frequency (MHz) Exponent n Variance␴(dB)

Retail store 914 2.2 8.7

Office, hard partition 1,500 3.0 7.0

Office, soft partition 900 2.4 9.6

Factory, line of sight 1,900 2.6 14.1

are deployed with base station terminals (BS) and customer premises equipment (CPE). The model refers to three types of terrain and is applicable to suburban environments. Terrain type A has highest path loss and is characterized as hilly terrain with moderate to heavy foliage. Type C has minimum path loss and is based on flat terrain with light tree density. Type B is for an intermediate terrain with path loss between that of A and C. The pass loss equation for the model is:

PL = 20 log冉^4␲␭d0冊^{+ 10nlogdd0+ Xf+ Xh+ s for d> d0 (6.8)}

d is the distance between two terminals and d₀= 100m. s is a lognormal distributed factor that accounts for shadowing due to trees and other objects, with a value between 8.2 and 10.6 dB. n, the path loss exponent, is calculated from the expression

n= a − bhb+ c/hb (6.9)

where h_bis the base station antenna height above ground, between 10 and 80m, and constants a, b, and c depend on the terrain type as shown in Table 6.2.

X_fis a correction factor for the operating frequency f in megahertz:

X_f = 6.0 log冉^2,000f 冊 ^(6.10)

The CPE antenna height h_rcorrection is applied to the model as X_h, which depends on the terrain type:

X_h= −10.8 log冉^2,000hr 冊 for terrain types A and B (6.11) X_h= −20.0 log冉^2,000hr 冊 for terrain type C

As in the case of indoor propagation models, the outdoor models generally produce a range of path loss estimations in a given environment. Therefore, they should be applied only under the conditions for which they were developed. When possible, empirical measurement sampling should be done to confirm the applicabil-ity of a given model in a particular situation and to assess the range or location accuracy that can be expected from it.

Table 6.2 Numerical Values for the SUI Model Parameters

Parameter Terrain A Terrain B Terrain C

a 4.6 4.0 3.6

b (m⁻¹) 0.0075 0.0065 0.005

c (m) 12.6 17.1 20

Source: [4].

6.2.7 Path Loss and Received Signal Strength

As discussed previously, range is associated with path loss or path gain. In order to relate the measured received signal strength to distance, through the path loss or path gain expressions, radiated power and receiver antenna gain must be known.

Transmitter power into the antenna, and transmitter and receiver antenna gains, are included in (6.1). In logarithmic terms, using decibels, path gain, PG_dB as a function of received signal strength is:

PG_dB= P_r− (P_t+ G_t + G_r) (6.12) where P_ris received signal strength, P_t is transmitter power to the antenna, and G_tand G_rare transmitter and receiver antenna gains, all in decibels. Path gain is the negative of path loss in decibels.