3.5 Propagation effects in wireless systems

3.5.1 Free-space propagation

To begin with consider an idealized free-space model (see Figure 3.8), where there are no obstacles between the transmitter and receiver antennas and the transmitted wave propagates along the single possible path, called line-of-sight (LOS).

Let D denote the distance between the transmitter and the receiver. If the transmitter antenna were omnidirectional the transmitted power Pt would be uniformly distributed over the inner surface of the sphere of radius D and power Pt/4 D2 would fall on every unit of area of this surface. The receiving antenna with an effective area Ar would then capture a received power Pr ¼ PtAr/4 D2. If the transmitting antenna is directional it emanates in the receiver direction power which is Gt higher than that of the omnidirectional one, and Gt is called a transmit antenna power gain. In this case the received power becomes Gt times higher as well. To represent the received power in a symmetrical form, we make use of the relation between Ar and a receiving antenna power gain Gr ¼ 4 Ar/ 2w, where w is wavelength. We will come to the Friis free-space formula [4]:

showing that the attenuation of the signal power along the LOS is inversely proportional to the squared distance.

The free-space model may be directly applied to communication links whose environment is well described as an open space, e.g. between space vehicles or aircrafts, ground control centre and space vehicle etc. The propagation medium of terrestrial systems is much less favourable and in its influence on the signal intensity two main components are typically categorized: shadowing and multipath fading.

3.5.2 Shadowing

Shadowing is caused by landscape details obstructing LOS: hills, forests, bushes, buildings and so forth. Due to them the signal intensity drops with distance much faster than equation (3.10) predicts. Of course, the irregular nature of terrestrial patterns makes attempts at creating some universal theoretical model of shadowing impossible or worthless. A great deal of field testing has been carried out to collect knowledge about the general character of the dependence between the received power and the length of the propagation path and a number of empirical models have been proposed [17–19]. One of the most popular with mobile communication specialists is the Okumura–Hata model, according to which the behaviour of an average received power Pr obeys the law Pr ¼ kPt/De where the specific value of the exponent e depends on the kind of landscape, typically ranging from 3 (rural area) to 5 (dense urban area) and the coefficient k is determined by the frequency band and

Figure 3.8 Free-space propagation model

the heights of the antennas [2,6,15,19]. The received power predicted by this model gives only a very rough reference point, being the result of averaging over the different positions of the receiver with the same distance D from the transmitter. Fluctuations of Pr along the arc of radius D centred at the transmitter location are significant and approximated by the lognormal PDF, meaning that the distribution of the decibel content of the received power x ¼ 10 lg Pr is Gaussian (normal):

The standard deviation x of 10 lg Pr in the last expression is commonly accepted in the literature to be between 6 and 12 dB.

Attenuation of power caused by shadowing has a static character and, even when the receiver is in motion, usually changes in time comparatively slowly due to the large scale of landscape components (tens or hundreds of metres). For this reason shadowing is also often referred to as large-scale or long-term fading.

3.5.3 Multipath fading

Let us turn now to the second factor affecting the received signal intensity: multipath propagation. As a matter of fact, the transmitted signal can reach the receiving antenna travelling by many paths. The LOS may appear as one of them or be utterly obstructed, all the other paths emerging as a result of the transmitted wave being reflected by various objects. Examples of such reflectors are buildings, towers, cars, aircrafts, the earth’s surface and many more (see Figure 3.9).

D4

D2 D3

D1

Receiver

Transmitter

Figure 3.9 Illustration of multipath propagation

Suppose that travelling by the ith path, the transmitted signal with the reference complex envelope S_(t) assumes amplitude Ai, delay i and initial phase i. Then the received complex envelope will be found as:

X

S_rðtÞ ¼ AiS_ðt iÞ expðj iÞ ð3:11Þ

i

where without loss of generality the real amplitude of the reference signal is set equal to one. When the delay spread ds, i.e. maximal mutual delay between signals of different paths, is within the signal duration all of the multipath signals will overlap and interfere with each other. To better understand the phenomenon, consider first one simple scenario, which may take place in mobile communications, TV broadcasting or elsewhere.

Example 3.5.1. Figure 3.10 describes the situation where the transmitted signal reaches the receiver through the paths created by two reflectors (buildings, vehicles etc.), the LOS path being totally blocked by an obstacle (e.g. a building) and (3.11) including only two summands. Both reflectors are oriented so that they emit the secondary wave towards each other. The receiver placed on the line connecting the reflectors will observe superposition of the two interfering waves whose phase difference is governed by the ratio of propagation difference ¼ D10 þ D100 D20 D200 to the wavelength w : ¼ 2 / w . With amplitudes of

the reflected signals at the receiver location A1, A2, the resulting amplitude Ar may be found q

by the cosine theorem as Ar ¼ A21 þ A22 þ 2A1A2 cos (see phasor diagram in Figure 3.11a).

The periodicity of Ar as a function of means its periodicity in dependence on the propagation difference . When the receiver moves along the line connecting the reflectors its displacement by w /2 in any direction changes by one wavelength w so that changes by 2 and values of Ar at the points separated by w /2 are identical. In other words, interference of two impinging waves creates a stationary wave with period w /2. Moving along the tested line the receiver will alternate observing maximal A1 þ A2 and minimal jA1 A2j amplitudes each w /2 m. If the

Transmitter

Figure 3.10 The two-path case

Figure 3.11 Interpretation of multipath effects with phasor diagrams

receiver passes the stationary wave nodes. This is exactly the phenomenon called multipath fading. Since the space distance between adjacent peaks of Pr is comparable with a wavelength, for the system operating in metre or decimetre band the time cycles of changing Pr at the moving receiver input will be rather short (typically split seconds).

The plot of Pr in dependence on time in Figure 3.12 gives an example for parameter values typical of mobile communications: w 0:3 m and the receiver carrier speed Vr ¼ 60 km/h. As is seen, even with rather small speed of movement, changes of the received power are very rapid. This explains why multipath fading is also called short-term fading or small-scale fading.

Figure 3.12 Time profile of the received power in the case of two-path fading

Clearly, this example is artificially simplified in order to present the phenomenon in the most explicit way. In practice, the number of multipath signals L received simultaneously may be very large and as a result the interference pattern becomes more complicated. The phasor diagram in Figure 3.11b illustrates a situation of this kind. The chaotic character of the distribution of reflectors or scatterers in the receiver environment makes the interference pattern unpredictable and its statistical description most appropriate.

Example 3.5.2. Figure 3.13 shows the time profile of the received power obtained by modelling in Matlab of the propagation environment with five reflectors located equiprobably within the square of side equal to the transmitter–receiver initial distance D ¼ 30 km. The received power is normalized to the average one. The wavelength and user’s speed are 0.3 m and 60 km/h, respectively. The irregular character of the power change is fairly explicit as well as the presence of deep drops of the received signal intensity.

Figure 3.13 Time profile of the received power in the case of five-path fading

Due to the central limit theorem, a superposition of independent and nearly equally contributing random summands tends to become Gaussian whenever their number grows. Therefore, numerous multipath signals obeying these conditions produce a bandpass Gaussian process at the receiver input. If a dominating deterministic component (like the LOS one) is not present among them the resulting Gaussian process will be a zero-mean one. But the envelope of such a process has Rayleigh distribution (see Section 3.2) and thus we come to a model of Rayleigh fading channel. Now, the received amplitude Ar is not deterministic but, instead, random, meeting the Rayleigh PDF:

Since in the product ArS_(t), the ‘genuine’, actually measurable signal amplitude is split between two cofactors, and this may be done arbitrarily, a convenient normalization is assumed in (3.12) setting the mean square of Ar equal to one: A2r ¼ 1. A plot of PDF (3.12) is shown in Figure 3.14.