Week Oct. 12, 2015

Date L/E Time Content Notes

Oct. 12 ----

Oct. 13 L 16.15-18.15 Channel model.

Antenna characteristics.

Free space loss.

Power budget link.

The three components of the wireless channel model.

Large and small scale phenomena.

Attached

[1, cap. 1]

[2, chap. 2]

[3, chap. 4]

Oct. 15 L 16.15-18.15 Propagation loss; general formula.

Shadowing: log-normal distribution

Attached

[1, cap. 1 ]

[2, chap. 2, 3]

[3, chap. 4]

Oct. 16 ----

[1] G. Tartara, L. Reggiani, "Sistemi di radiocomunicazione", Polipress 2009, Milano, marzo 2009.

[2] A. Goldsmith, “Wireless Communications”, Cambridge University Press, 2005.

[3] T. S. Rappaport, "Wireless Communications: Principles and Practice", 2nd edition, Prentice Hall.

Chapter 2

The wireless channel model

This chapter is devoted to the development of appropriate models for the wireless channel. These

models are used for analyzing, simulating and designing the wireless systems and they constitute

a fundamental topic in the field of wireless communications.

2.1 The antenna

The antenna parameters used for characterizing a wireless link are the following:

• directivity function f(φ, ϑ), which expresses the antenna capability of radiating and re-

ceiving e.m. waves on a generic direction, usually identified by the azimuth and elevation

angles (Fig. 2.1). Notice that f(φ, ϑ) is normalized to 1. Fig. 2.2 shows an example of an

antenna directivity for cellular systems.

• Gain, given by the product between the antenna efficiency and the antenna directivity:

G = η · D. G represents the gain provided by the antenna in its direction of maximum

directivity w.r.t. isotropic antenna; in other words, G is the measure of how the transmit

antenna concentrates power density in the direction of maximum irradiation.

• Effective area or aperture A [m2], which returns the amount of power flux density captured

by the antenna. Effective area and gain are related by the universal relation

A

G=

λ2

4π. (2.1.1)

7

8 ϕ ϑ y x z

Figure 2.1: Azimuth and elevation angles (0 ≤ φ < 2π, 0 ≤ ϑ ≤ π).

2.2 Free space loss

The simplest propagation model is the free space model and the resulting path loss or path

attenuation. Given a transmitting and a receiving antenna, the signal flux power density in an

arbitrary direction at a distance d is given by

S(d, φT , ϑT ) =PT

4πd2·GT · fT (φT , ϑT ) [W/m2] (2.2.1)

where fT (φ, ϑ) is the antenna directivity and GT is the antenna gain. In its direction of maximum

directivty, the receiving antenna will intercept a power given by a fraction of density (2.2.1) given

by ARS(d, φ, ϑ), where AR is the effective area of the receiver antenna. Using (2.1.1), we obtain

easily, in a generic direction

PR = AR · fR(φR, ϑR)S(d, φT , ϑT ) = PT

(λ

4πd

)2

GTGRfT (φT , ϑT )fR(φR, ϑR) [W ] (2.2.2)

and, in the maximum directivity direction

PR = PT

(λ

4πd

)2

GTGR [W ] (2.2.3)

9

−200 −150 −100 −50 0 50 100 150 200−25

−20

−15

−10

−5

0

5

Azimuth [gradi]

f R [d

B]

Figure 2.2: Example of a directivity function used for covering a 120 degrees cellular sector.

So the received signal power falls off inversely proportional to the square of the distance between

the transmit and receive antennas and the free space attenuation is given by

AFS =

(4πd

λ

)2

(2.2.4)

In practical scenarios, free path loss attenuation can be considered a valid approximation of the

link attenuation when the first Fresnel’s zone between the transmitter and receiver is free; notice

that only the presence of a LoS link is not enough to assume a free space propagation.

2.3 The three components of the channel model

As can be seen from Fig. 2.3, three channel components compose the overall channel model of

wireless systems:

1. path loss, i.e. the large scale, deterministic relation between the link distance d and the

mean attenuation in a given environment (Sect. 2.3.1).

2. Shadowing, i.e. the large scale (”slow”), random attenuation fluctuation around the path

loss due to presence of large obstacles (buildings, hills, ...) which shadow completely or

partially the propagation path (Sect. 2.3.2).

3. Multipath fading, i.e. the small scale (”fast”), random attenuation fluctuations around the

10

0 0.5 1 1.5 2 2.5 3−30

−20

−10

0

10

20

30

40

50

60

70

log10

(d)

PR

[dB

m]

Figure 2.3: Large and small scale channel attenuation components.

large scale attenuation (path loss + shadowing) due to the presence of multiple reflections

at the receiver. Multipath fading models allow also the definition of the channel impulse

response (Sect. 2.3.3).

We remark that the term large scale characterizes a phenomenon whose variations are associ-

ated with large distance modifications w.r.t. wavelength; on the contrary small scale phenomena

show a clear sensitivity on distance modifications that can be also fractions of the wavelength.

2.3.1 General path loss

This is the main large scale component of the channel model. It is described by a deterministic

model that relates distance d, frequency, eventually other parameters and channel attenuation

or path loss. Of course the path loss can be defined in a linear or dB way by using the ratio

or the difference between the transmitted and received powers (expressed in dBm or dBW). A

general formulation of this component derives from the empirical observation that, for the ranges

of interests of a wireless system, the channel path loss decreases almost linearly in dB with log(d),

so allowing the use of a very simple model, whose general formulation is

APL = C

(d

d0

)γ

(2.3.1)

11

Path loss exponent Environment

Home 2.5 - 3.5

Factories 1.6 - 3.5

Office buildings 1.6 - 3.5 (till to 6 for multiplefloors)

Urban cells 2.5 - 4.5 (till to 6 in particularlydense environments)

Suburban cells 2 - 3.5

Table 2.1: Typical ranges of the path loss exponent γ.

where γ is the path loss exponent, which depends on the environment (typically between 1.6

and 5 in difficult urban or indoor areas, as reported in Table 2.1) and C is the attenuation at

the reference distance d = d0 (d0 is usually set to 1 km or 1 m according to the scenario). So,

generalizing 2.2.3, the received power link budget is expressed as

PR = PTGTGR1

C

(d0d

)γ

= P0

(d0d

)γ

(2.3.2)

where P0 is the power received at distance d0. In dB, the attenuation is expressed by the sum of

two components

APL = 10log10C + 10γlog10

(d

d0

)= α+ βlog10

(d

d0

). (2.3.3)

2.3.2 Shadowing

This random model component is due to the presence of large obstacles on the propagation path.

The large scale fluctuations of the attenuation are well described by a log-normal distribution.

This means that the logarithm of the attenuation assumes a Gaussian density as

fA(ASH) =1√

2πσ2SH

exp

(− (ASH − µ)2

2σ2SH

). (2.3.4)

The variable ASH is the additional attenuation in dB w.r.t. the path loss, µ is the mean,

typically equal to 0, and σSH is the standard deviation in dB domain (between 2 and 8 dB

in typical urban and suburban environments). This model can be justified by keeping in mind

that the global attenuation is given, in dB, by the sum of numerous components corresponding

to the attenuations of single fractions of the overall path (Central Limit theorem). Figs. 2.4

12

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−3

A

f A(A

)

σSH

= 1 dB

σSH

= 2 dB

σSH

= 4 dB

σSH

= 8 dB

Figure 2.4: Log-normal density with varying standard deviations and µ = 0 dB.

and 2.5 show the log-normal density in the linear domain, which can be derived from (2.3.4) by

substituting

A = g(ASH) = 10(ASH/10) (2.3.5)

and operating the random variable transform of the probability density function (2.3.4) by ap-

plying

fA(A) =fA(ASH)

|g′(ASH)||ASH=g−1(A) (2.3.6)

so obtaining

fA(A) =1√

2πσ2SH

10

A · ln(10)exp

(− (10 · log10A− µ)2

2σ2SH

). (2.3.7)

The mean value of A is

E[A] = exp

(µ

ξ+

σ2SH

2ξ2

)(2.3.8)

with ξ = 10/ln10.