Copyright ©2018 Thomas Schwengler.

Chapter 3
Radio Propagation Modeling

This chapter introduces propagation characteristics and models for cellular systems. It summarizes important notions, and expands on aspects of fixed-versus-mobile and indoor-versus-outdoor propagation modeling. 1

Before studying details of propagation we need to define a few notation conventions, which will require the reader to be familiar with the following general concepts of electromagnetic field and wave theory.

Some details of E-field propagation will be studied later with ray tracing; but most of the remainder of the section deals with very simple expressions of power levels for paths loss modeling.

3.1 Propagation Characteristics

Between transmitter and receiver, the wireless channel is modeled by several key parameters. These parameters vary significantly with the environment, they are often separated in three types [1] [3]:

Distance Dependence
of path loss (measured in dB) is approximated by L(d) = L0 +10n×log(d∕d0), where n is the path loss exponent, which varies with terrain and environment, and L0 and d0 are parameters described further in §3.3 and §3.4.
Large-scale Shadowing
causes variations over larger areas, and is caused by terrain, building, and foliage obstructions; its impact on link budgets is detailed further in §4.1.

The large-scale fading due to various obstacles is commonly accepted to follow a log-normal distribution ([17], [18], [19] ch. 7). This means that its attenuation x measured in dB is normally distributed N(m,σ), with mean m, standard deviation σ, and which probability density function is given by the usual Gaussian formula:

       --1---      - (x---m-)2
p(x) = σ√2-π-× exp    2σ2

Small-scale fading
causes great variation within a half wavelength. It is caused by multipath and moving scatterers, and is detailed further in §??. Resulting fades are usually approximated by Rayleigh, Ricean, or similar fading statistics – measurements also show good fit to Nakagami-m and Weibull distributions.

Radio systems rely on diversity, equalizing, channel coding, and interleaving schemes to mitigate its impact.

Copyright ©2018 Thomas Schwengler.

Different spectrum bands have very different propagation characteristics and require different prediction models. Some propagation models are well suited for computer simulation in presence of detailed terrain and building data; others aim at providing simpler general path loss estimates [20].

3.2 Free-Space Propagation

A simple approach to propagation modeling is to estimate the power ratio between transmitter and receiver as a function of the separation distance d, that ratio is referred to as path loss. A physical argument of conservation of energy leads to the Friis’ power transmission formula in free space. A transmitted power source Pt radiates spherically, with an antenna gain Gt; the portion of that power impinging an effective area Ae at a distance d is Pr = PtGtAe(4πd2). The effective area of an antenna is related to antenna gain by Ae∕λ2 = Gr4π, which is used here for the receiving antenna (of gain Gr), and thus yields:

Pr-= GtGr-λ-
Pt    (4πd)2

(Pt and Pr are the transmitted and received power, Gt and Gr are the transmitter and receiver antenna gain, λ is the wavelength of the signal, and d is the separation distance).


Figure 3.1: Spherical free-space propagation: power transmitted over the air radiates spherically, with different gain in different directions. A portion of that power is received on an effective aperture area Ae of the receiving antenna.

This equation shows a free-space dependence in 1∕d2, and is often expressed in decibels (dB): L(dB) = 10 × log (Pt∕Pr ).

In many cases, antenna gains are considered separately, and one choses to focus on the path loss between the two antennas. The path loss reflects how much power is dissipated between transceiver and receiver antennas (without counting any antenna gain). The path loss variation with distance is d2, or 20 log(d) in dB, which is characteristic of a free space model. The exponent (here n = 2) is called the path loss exponent, and its value varies with models. Path loss is often expressed as a function of frequency (f), distance (d), and a scaling constant that contains all other factors of the formula. For instance:

L(dB ) = 32.45 + 20× log(f∕f0)+ 20 × log(d ∕d0)

where f0 = 1 MHz, and d0 = 1 km. These reference values are arbitrary and chosen to be convenient to use values in MHz and km in the formula.

Note that the constant 32.45 changes if the reference frequency f0 and the reference distance d0 are chosen to be different. Many presentations will refer to the formula as L(dB) = 32.45 + 20 × log(f) + 20 × log(d), adding a statement such as “f is in MHz and d is in km”. That shorter notation is equivalent to equation (3.3), but should be treated carefully: with the loss of f0 and d0 in the equation, mistakes can be introduced by further manipulating the expression.

3.3 Ray Tracing

Ray tracing is a method that uses a geometric approach, and examines what paths the wireless radio signal takes from transmitter to receiver as if each path was a ray of light (reflecting off surfaces). Ray-tracing predictions are good when detailed information of the area is available. But the predicted results may not be applicable to other locations, thus making these models site-specific.

Nevertheless, fairly general models may be devised from ray tracing concepts. The well-known two-ray model uses the fact that for most wireless propagation cases, two paths exist from transmitter to receiver: a direct path and a bounce off the ground. That model alone shows some important variations of the received signal with distance. [1] [3] Ray tracing models are extensively used in software propagation prediction packages, which justifies a closer look at them in this section.

3.3.1 Ray Tracing Notations

Rays are an optical approximation of the electromagnetic wave; as seen earlier, in free space it is convenient to focus on the propagation of the electric field. The (complex) transmitted electric field is noted S(t), the received field is R(t), we define the ratio p0(t) = R(t)∕S(t). Then the power link budget between transmitter and receiver can be written Pr∕Pt = |p0|2.

In the next few subsections, we will consider how the electric field propagates, and we will compute various expressions for p0(t) in various conditions; each time the final step will be to take its modulus squared in order to derive the path loss.

3.3.2 Two-Ray Model

With these few notation conventions we can derive a simple but useful model with two rays.


Figure 3.2: 2-ray model geometry: a direct ray links transmitter and receiver, and a second ray bounces off the ground.

Figure 3.2 shows a fixed tower (e.g. in a cellular system) at a height hb, and a client device at a distance d0, and at a height hm (usually lower). The figure shows a direct ray and an indirect ray bouncing off the ground, assumed to be a perfect plane (this assumption is referred to as the flat-earth model). 4

It is easy to see from this figure that the two path lengths are:

    ∘ ---------------
l =   d2 + (h  - h  )2
0      0     b   m

    ∘ ---------------
l′0 =   d20 + (hb + hm )2

The received signal at a distance d0 is therefore:

       ∘ -------  (          -j2πl0∕λ          ′  -j2πl′0∕λ)
R (t) =   (GtGr ) λ- s(t--t0)e-------+  Γ s(t--t0)e′------  e-j2πft
                4π         l0                  l0

where λ = c∕f is the wavelength, t0 = l0∕c (respectively t0 - l0∕c) is the time needed for the wave to propagate over distance l0 (respectively l0), and Γ is the ground reflection coefficient. For now let us simply assume perfect reflection and use Γ = -1; will elaborate further on Γ in §3.3.3. 5

Another important assumption must be made here to simplify the model: we will assume that propagation time differences are small compared to the symbol length of the useful information, that is s(t-t0) = s(t-t0). For a bounce off the ground, that assumption is fairly safe, but in general we will have to recall that such an assumption means that we assume that the delay spread is small compared to transmitted symbol rates.

So we obtain:

        ∘ -------λ ||exp (- j2πl0∕λ)    exp(- j2πl′∕λ)||
|p0(t)| =   (GtGr )--||-------------- + Γ-------′--0---||
                 4π       l0                l0

which simplifies for distances much greater than heights (d0 l0 l0):

        ∘ -------     ||        (      ′     )||
|p0(t)| =  (GtGr )--λ--|1+ Γ exp  j2π(l0 --l0) |
                 4πd0 |               λ      |


Figure 3.3: Simple propagation models: free-space one-slope direct line of sight, and two-ray with direct ray and ground reflected ray. In some places signal add constructively, in others phase differences cause deep fades.

Figure 3.3 represents the path loss attenuation Pr∕Pt = |p0|2 (in dB) as a function of logarithm of distance; it uses hb = 8 m, hm = 2 m, f = 2.4 GHz, Gt = Gr = 0dBi, and Γ as given later in §3.3.3. The direct path, using the first term only of (3.7), leads to the simple one-slope free-space model; the complete expression leads to the two-ray model. The figure shows interesting characteristics:

3.3.3 Reflection and Refraction

Before moving ahead, we need to take a closer look at reflection coefficients used for indirect rays. The details of this analysis come from boundary conditions for electromagnetic waves traveling between two media. In general these boundary conditions vary with the polarization of the wave and the media permittivities. For a wave impinging on the ground with an angle of incidence θ, the ground reflection coefficient depends on the polarization and is given by equation (3.9):

    sinθ - Z        {  1-√ϵr - cos2 θ vert. pol.
Γ = sinθ-+-Z-,  Z =    ϵr√ϵ----cos2-θ   horiz. pol.

Γ is the ground reflection coefficient; Z is the characteristic impedance of the media, as obtained by transmission line theory [76] [19]; θ is the ray angle of incidence (as shown on fig. 3.4); ϵr is the complex relative permittivity of the medium: ϵr = ϵr -jσωϵ-
 0 ϵr -j60σλ where ϵr is the lossless relative permittivity and σ is the conductivity (in Ω-1m-1).

The wave typically has many polarized component to it; even when a transmitter uses vertically polarized antennas, different scatterers in the path may depolarize the wave. Nevertheless, the majority of cellular systems use vertical polarization, which is shown empirically to propagate slightly better in most practical cellular environments. In these cases, the electrical field is near vertical, and the reflection (and refraction) on a surface is shown on figure 3.4.


Figure 3.4: Vertical polarization ground reflection coefficient. The sign of Γ depends on the convention of the direction of incident and reflected electric fields. In our convention a perfect conducting plane has a reflection Γ = +1.


Figure 3.5: Horizontal polarization ground reflection coefficient. With this convention a perfect conducting plane has a reflection Γ = -1.

Similarly, rays bouncing off walls have a reflection coefficient (of course the vertically polarized waves now needs to be considered as impinging the surface with electric field near the surface plane as a horizontally polarized wave does on the ground).

Values for complex permittivities may be used approximately from table 3.1 (from  [19] p. 55, and a few other references); gives ground conductivity maps for the US.

Table 3.1: Relative permittivities for various materials.

Material ϵr σ Comments
(Fm-1) -1m-1)

Vacuum 1 By definition
Air 1.00054 Usually approximated to 1.0
Glass 3.8-8 Varies with glass types
Wood 1.5-2.1
Drywall 2.8
Dry brick 4
Concrete 4.5 Varies 4-6
Limestone 7.5 0.03
Marble 11.6
Fresh water80.2 0.01
Sea water 80.2 5
Snow 1.3
Ice 3.2
Ground 15 (7-30)0.005 (0.001-0.03)Varies with type and humidity

Further refinements may be thought of regarding the thickness of walls: the ground may easily be considered as an infinite semi-plane, but walls are usually thin enough to make that approximation questionable. 6

3.3.4 Multiple Rays


Figure 3.6: Six-ray model (R0,R1,R2) and ten-ray model (R0 to R4) geometry: each line hides two rays: one direct, the other bouncing off the ground.

The above 2-ray approach can easily be extended to add more rays  [3]. We may add rays bouncing off each side of a street in an urban corridor, leading to a 6-ray model (with rays R0,R1,R2 each having a direct and a ground bouncing ray). Adding four more rays (bouncing on both sides: R3,R4 dashed line in figure 3.6) lead to a 10-ray model.

The direct two rays were computed earlier, the additional rays may easily be obtained from geometry of figure 3.6. Let us assume for instance a street corridor of width ws, with a transmitter on a light pole wt from the walls. For simplicity, let us move the receiving point down the street at constant wt from the wall, in that case distance Tx to Rx represented by R1 is d1 = ∘ -----------
  d20 + (2wt)2, so, much like equation (3.7), we have:

         ∘ -------     ||                           ′   ||
|p1(t)| =   (GtGr) λ-Γ 1 ||exp-(- j2πl1∕λ)-+ Γ exp-(--j2′πl1∕λ)||
                  4π          l1                l1

where l1 = ∘d2-+-(2wt)2 +-(hb---hm-)2-
   0, and l1= ∘d2--+-(2wt)2 +-(hb-+-hm)2
    0. Γ1 is the refection coefficient off the nearest wall, and is computed from (3.9), but with angles with respect to the walls.

Additional rays (R2 and more) can be calculated in expressions resembling (3.10), and added to others in order to produce a multiple-ray model.


Figure 3.7: Ray tracing plots of received signal power indicator 20log | i=0i=Npi| as a function of log d0 for N ∈ {0,2,3,5}for a typical suburban case with street width of 20 feet, and average distance from street to home of wt = 10 feet (so ws = 40 feet).

Figure 3.7 shows the increased fading statistic when more rays are taken into account. The figure simply represents the received signal power indicator 20log | i=0i=Npi| as a function of log d0 for N ∈{0,2,3,5}. For that plot a typical suburban case is taken with street width of 20 feet, and average distance from street to home of wt = 10 feet (so ws = 40 feet).

3.3.5 Residential Model

As previously mentioned, that approach is interesting for urban and suburban corridors. We further assume that property lengths and home lengths along the street are approximately identical (say 100 feet and 80 feet respectively). In that case, some rays escape the corridor and never reach the receiver – as illustrated in figure 3.8, R3 rays escape the urban canyon and never reach the receiver. Taking into account these gaps show a slightly modified model (figure 3.9). Alternatively, instead of examining where rays may escape the corridor, a simplified model may be used that takes into account a power loss proportional to the gaps [42].


Figure 3.8: Ray tracing geometry for a street corridor: some rays escape the corridor through gaps between homes.


Figure 3.9: Ray tracing power levels down a street, with gaps between homes.

3.3.6 Indoor Penetration

Most cellular towers are placed outdoors, while eighty percent of phone calls are placed indoors. Therefore the problem of how much of the signal strength propagating down the street might be available indoor is of great interest. Grazing angles of incidence are somewhat concerning in urban and suburban corridors. Figure 3.10 shows a typical case where wireless systems (base stations or access points) may be placed on opposite side of the street to provide coverage to residences.


Figure 3.10: Ray tracing impinging on home walls.

In our previous urban corridor model, the angles of incidences should be restricted to rays illuminating walls (as in figure 3.11). 7 8


Figure 3.11: Angles of incidence illuminating homes in an urban corridor.

tanθ1 = (n - 1)dp + dhh∕2

             ws - wt
tanθ2 = (n---1)d-+-d--∕2-
                p   hh

        (n - 1)dp + dhh∕2
tanθ3 = -----------------
             ws - wt

        (n---1)dp --dhh∕2
tanθ4 =      ws - wt

Angles of incidences between these values should be used to calculate penetration losses such as:

         ∫ θ
L′  = L     4(1 - sin θ)2d θ = L  [3 θ∕2+ 2 cos θ - (sin2θ)∕4]θ4
 ge    ge θ3                  ge                         θ3

For instance in a Lakewood neighborhood a light pole is placed every three homes on opposite street sides (i.e. a pole every 6 homes); we get the values in table 3.2 for the furthest home (n = 3, 100-feet properties, 80-feet long homes, 40-feet wide streets, and wt = 10 feet). And the value Lge 10dB is typical for residential areas. (More details in §3.7).

Table 3.2: Angles of incidence in a suburban area in Lakewood, CO and their resulting estimated penetration losses.

Pole positionθ3 (deg)θ4 (deg)Lge from (3.15)

Across street 19.4 14.6 0.5 Lge
Same side 6.7 5.0 8.0 Lge

3.4 Classic Empirical Models

Empirical models are simple models that provide a first order estimate for a wide range of locations. A handful of empirical models are widely accepted for cellular communications; these models usually simply consist of computing a path loss exponent n from a set of field data, and deriving a model for path loss (in dB) like:

L =  L0 + 10n × log(d∕d0)

(where the intercept L0 is the path loss at an arbitrary reference distance d0). These models are referred to as empirical one-slope models; their applications and domains of validity are well defined and reviewed later. They generally provide a first estimate used by service provider in wireless systems’ design phase.9

A couple of important points should be kept in mind about most propagation models. The first is that large amounts of empirical data are collected usually at cellular or PCS frequencies (800 MHz or 1900 MHz), and extensions to other frequencies are derived as discussed in §3.4.6. The second is that these data points are collected while driving and may not accurately reflect fixed wireless links, which is discussed in more details in §2.10.

3.4.1 COST 231-Hata Model

A one-slope empirical model was derived by Okumura [21] from extensive measurements in urban and suburban areas. It was later put into equations by Hata [22]. This Okumura-Hata model, valid from 150 MHz to 1.5 GHz, was later extended to PCS frequencies, 1.5 GHz to 2 GHz, by the COST project ([23], [24] ch. 4), and is referred to as the COST 231-Hata model; it is still widely used by cellular operators. The model provides good path loss estimates for large urban cells (1 to 20 km), and a wide range of parameters like frequency, base station height (30 to 200 m), and environment (rural, suburban or dense urban).

LHata = c0 + cf log(f∕1MHz )- b(hB ∕1m) - a(hM ∕1m)
        + (44.9- 6.55log(hB∕1m ))log(d∕1km )+ CM

with the following values:

Table 3.3: Values for COST 231 Hata and Modified Hata models.

Frequencyc0 cf b(hB)
(MHz) (dB) (dB) (dB)

150-1500 69.5526.1613.82log(hB1m)
1500-2000 46.3 33.9 13.82log(hB1m)

The parameter a(hM) is strongly impacted by surrounding buildings, and is sometimes refined according to city sizes:

Table 3.4: Values of a(hM) for COST 231-Hata model according to city size.

FrequencyCity size




150-2000 Small-medium

(1.1log(1MfHz-) - 0.7)h1Mm- - 1.56log(1MfHz-) + 0.8

150-300 Large

8.29(log(1.54hM1m))2 - 1.1

300-2000 Large

3.2(log(11.75hM1m))2 - 4.97

And an additional parameter CM is added to take into account city size, and can be summarized for both models as:

Table 3.5: Values of CM for COST 231 Hata model according to city size.

FrequencyCity size




150-1500 Urban


150-1500 Suburban

28MHz))2 - 5.4

150-1500 Open rural

1MHz))2 + 18.33log(--f--
1MHz) - 40.94

1500-2000 Medium city, suburban


1500-2000 Metropolitan center


Empirical values of the model are limited to distances and tower heights that were used to derive the model; consequently the model is usually restricted to:

3.4.2 COST 231-Walfish-Ikegami Model

Another popular model is the Walfisch-Ikegami-Bertoni model [30] [31], also revised the COST project ([23], [24] ch. 4), into a COST 231-Walfisch-Ikegami model. It is based on considerations of reflection and scattering above and between buildings in urban environments. It considers both line of sight (LOS) and non line of sight (NLOS) situations. It is designed for 800 MHz to 2 GHz, base station heights of 4 to 50 m, and cell sizes up to 5 km, and is especially convenient for predictions in urban corridors.

The case of line of sight is approximated by a model using free-space approximation up to 20 m and the following beyond:

LLOS = 42.6+ 26 log(d∕1km ) + 20log(f∕1MHz ) for d ≥ 20m

The model for non line of sight takes into account various scattering and diffraction properties of the surrounding buildings:

LNLOS  = L0 + max {0,Lrts + Lmsd}

where L0 represents free space loss, L


Table 3.6: Values for COST 231-Walfish-Ikegami model.

ParameterValue (dB)

L0 32.4 + 20log(d∕1km) + 20log(f∕1MHz)
Lrts -16.9 - 10log(w∕1m) + 10log(f∕1MHz) + 20log(ΔhM1m) + LOri
w Average street width
ΔhM hRoof - hM
LOri (  - 10+ 0.354ϕ∕1deg         if 0∘ ≤ ϕ < 35∘
{                                ∘         ∘
(  2.5+ 0.0.075 (ϕ ∕1deg- 35)  if 35∘ ≤ ϕ < 55∘
   4.0- 0.114(ϕ∕1deg - 55)   if 55 ≤ ϕ < 90
ϕ Road orientation with respect to direct radio path (see figure(3.12))

Lmsd Lbsh + ka + kd log(d∕1km) + kf log(f∕1MHz) - 9log(b∕1m)
b Average building separation
ΔhB hB - hRoof
Lbsh {  - 18 log(1 + Δh ∕1m )  for h  > h
                 B           B    Roof
   0                     for hB ≤ hRoof
ka (
{  54                          for hB > hRoof
(  54 - 0.8ΔhB  ∕1m             for hB ≤ hRoof and d ≥ 0.5km
   54 - 0.8(ΔhB ∕1m )(2d∕1km )  for hB ≤ hRoof and d < 0.5km
kd {
   18                 for hB > hRoof
   18- 15ΔhB  ∕hRoof  for hB ≤ hRoof
kf {  - 4 + 0.7(f ∕925MHz - 1) medium  cities, suburbs with medium tree density

   - 4 + 1.5(f ∕925MHz - 1) metropolitan centers


Figure 3.12: Definition of street orientation angle ϕ for use in COST-231 Walfish-Ikegami model: in the best case (ϕ = 0) the direction of propagation follows the street; in the worst case (ϕ = 90) the main radio wave is perpendicular to the street.

The model is usually restricted to:

3.4.3 Erceg Model

More recently Erceg et al. [32] proposed a model derived from a vast amount of data at 1.9 GHz, which makes it a preferred model for PCS and higher frequencies. The model was in particular adopted in the 802.16 study group [33] and is popular with WiMAX suppliers for 2.5 GHz products, and even 3.5 GHz fixed WiMAX.

L = L0 + 10γ log(d∕d0)+  s  for d ≥ d0

where free space approximation is used for d < d0. Values for L0, γ, and s are defined in tables 3.7 and 3.8:

Table 3.7: Values for Erceg model.

ParameterValue (dB)

L0 20log(4πd0∕λ) as in free space
d0 100 m
γ (a - bhB + c∕hB) + γ
σ μσ + σ
x,y,z Gaussian random variables N(0,1)

Table 3.8: Values for Erceg model parameters in various terrain categories.


Terrain Category




(Hilly / moderate to heavy tree density)

(Hilly / light tree density or flat / moderate to heavy tree density)

(Flat / light tree density)









c (m)
















The model is usually restricted to:

The model is particularly interesting as it provides more than a median estimate for path loss: it also gives a measure of its variation about that median value in terms of three zero-mean Gaussian random variables of variance 1 (x,y, and z = N(0,1)).

3.4.4 Multiple Slope Models

Further refinements to these models in which multiple path loss exponents (n1,n2) are used at different ranges provide some improvements, especially in heavy multipath indoors environments. For outdoor propagation, two slopes are sometimes used: one near free-space for close points, and another empirically determined. In fact we’ve seen that our 2-ray model could be approximated by a 2-slope model: n1 = 2 and n2 = 4 for distances greater than 4hthr∕λ.

It seems however that variations from site to site generally are such that these multiple slope improvements are fairly small, and simple one-slope models are generaly a good enough first approximation. More detailed site specific models are required for better results; but they require additional efforts and site specific terrain or building data.

3.4.5 In-building

Indoor propagation often has to be estimated by site-specific models with features specific to a particular building: construction material, wall thickness, floor and ceiling material, all have a strong impact on wave guiding within the building. Some models simply approximate the number of walls and floors, with an average loss for each. See in particular the COST 231 approach in §3.7.

A similar model for indoor environment is the Motley-Keenan model ([2],§7.2), which estimates path loss between transmitter and receiver by a free space component (L0) and additive loss in terms of wall attenuation factors (Fwall) and floor attenuation factors (Ffloor).

LdB = L0 + 20 log(d∕d0)+  nwallFwall + nfloorFfloor

Wall attenuation factors vary greatly, typically 10 to 20dB (see table 3.12 in §3.7); and floor attenuation factors are reported to vary between 10 and 40dB depending on buildings. [1]

This model is very site specific, yet sometimes imprecise as it does not take into account proximity of windows external walls, etc; but it can be useful as a guideline to estimate signal strength to different rooms, suites, and floors in buildings.

3.4.6 Frequency Variations

Frequency of operations impacts propagation and path loss estimates. As many models are built on cellular or PCS data measurements, one must be careful about extending them to other frequency ranges.

As seen in equation (3.3) in §3.2, the impact of frequency on free-space propagation is 20logf. Some empirical measurements confirm the trend [37], and the extension is used for instance in the COST-231 Walfish-Ikegami model.

Empirical evidence also shows however that frequency extensions are obtained by adding a frequency dependence in f2.6 (or a 26log f term in dB) as suggested by [40], and used for instance in the Okumura-Hata model [22] and the 802.16 contribution [33].

Finally other important aspects have a significant impact as frequency changes. Spatial diversity gain typically improves with frequency since spatial separation increases when related to wavelength ([41] shows a 2dB diversity gain from cellular 850 MHz to PCS 1.9 GHz). Doppler spread and impact on symbol duration should also be studied separately and may have a significant impact on a change of frequency [43]. Impact on in-building penetration is examined further in §3.7.

3.4.7 Foliage

Foliage attenuates radio waves and may cause additional variations in high wind conditions [44]. Propagation losses and path loss exponents vary strongly with the position of transmitter with respect to the tree canopy; they also vary with the types and density of foliage, and with seasons. [45][46][47][48][49]

We will report in a later chapter on the impact of foliage for fixed wireless links at 3.5 GHz, in a suburban area as foliage grows from the winter months into the spring (see figure 10.9). Studies have been published at different frequencies; some identify empirical attenuation statistic with Raleigh, Ricean, or Gaussian variables, others derive excess path loss, or attenuation per meter of vegetation.

As a rule of thumb, at frequencies around 1 ot 6 GHz single tree causes approximately 10-12 dB attenuation, and typical estimates are 1-2 dB/m attenuation. Deciduous trees in winter cause less attenuation: 0.7-0.9 dB/m. (See table 3.9.)

Table 3.9: Vegetation loss caused by tree foliage, reported for various frequencies: single-tree model loss in dB, and dB/m loss.

Source Frequencysingle tree loss per meter loss Comments
(GHz) (dB) (dB/m)

Benzair [45] 2.0 20.0 1.05 Summer
4.0 27.5 1.40
2.0 9.5 0.70 Winter
4.0 10.7 0.85
Dalley [46] 3.5 11.2 1.9 With leaves
5.8 12.0 2.0
Wang [48] 1.0 10.0 - Single tree
2.0 14.0 -
4.0 18.0 -
Torrico [49] 1.0 - 0.7 With leaves
2.0 - 1.0

Approximation 12.01+7.46 logfGHz0.54+1.40 logfGHz

Another common models for vegetation proposes an empirical exponent both for distance and frequency; thus approximating vegetation loss by L(dB) = Afαdβ, where A,α, and β parameters vary with the type and density of vegetation. See figure 3.13 taken from [50].

Figure 3.13: Summary of tree foliage attenuation models published - table copied from [50].

3.5 Further Modeling Work

The above models are in a sense simplistic as they focus on path loss as a function of distance. Although these models work well in large cellular coverage prediction, they are often deemed insufficient for smaller cells such as wireless LAN’s, especially where multipath is dominant, as in a heavy urban environment or indoor environment.

An interesting and important activity around propagation modeling is the COST project (COperation europénne dans le domaine de la recherche Scientifique et Technique), a European Union Forum for cooperative scientific research that has been useful in focusing efforts and publishing valuable summary reports for wireless communications needs.

COST 207
“Digital Land Mobile Radio Communications”, March 1984 - September 1988, developed channel model used for GSM,
COST 231
“Evolution of Land Mobile Radio (Including Personal) Communications”, April 1989 - April 1996, contributed to the deployment of GSM1800, DECT, HIPERLAN 1 and UMTS, and defined propagation models for IMT-2000 frequency bands [24]
COST 259
“Wireless Flexible Personalised Communications”, December 1996 - April 2000, contributed to wireless LAN modeling, and 3GPP channel model [25]
COST 273
“Towards Mobile Broadband Multimedia Communications”, May 2001 - June 2005, which contributed to standardisation efforts in 3GPP, UMTS networks, provided channel models for MIMO systems [26]. That work was later continued in the COST2100 group[27] in particular to provide further MIMO advances for the wireless industry.

Finally, new interesting activities of research and modeling are taking place at higher frequencies, in millimeter-wave for mobile use with 5G [28][29].

3.6 Dispersive Models

Modern radio systems now make extensive use of multiple paths between transmitter and receiver, even deploying multiple antenna systems (such as MIMO). These systems require more than path loss estimates, as path loss is sensibly the same between transmitting and receiving antenna systems. MIMO channel models are therefore much more complex; several approaches have been used, such as different groups of delayed paths. Of particular interest are the 802.16 model [33], the 802.11n and ac models [34] for various indoor models (from the 802.11n task group on channel modeling), or [38] for mobile cellular models.

IEEE 802.16e and SUI
The work of 802.16 and Wimax provided the first widespread models with interesting multipath considerations and convenient matlab simulations [33]. The work starts with six typical environments modeled by six Stanford University Interim (SUI) channel models. These models are described in terms of terrain types (and propagation from §3.4.3), amount of Doppler, delay spread, and fading statistics.

One way of modeling transmission delay (beyond a simple delay spread value) is to consider a series of successive impluses, each delayed and attenuated. This is referred to as the tapped delay line model. The SUI models define 6 different types of environments, with different tap delays, Doppler effect, and fading statistics: in that manner the model represent different scenarios: pedestrian/vehicular, urban/suburban/rural, indoor/outdoor, etc.

IEEE 802.11n
The IEEE task group TGn produced a series of models [34] for 802.11n (and ac) for LAN applications at 2.4GHz and 5GHz. Consequently these models focus on pedestrian mobility; the group presents six different models (named A to F) aim at describing typical LAN environments, with generally much more multipath than many outdoor cellular environments.

Table 3.10: TGn channel models A to F are used to model MIMO systems in different environment, with different RMS delay spreads (στ in nanoseconds – see table 4.2 in §??).



A 0 Direct


B 15 Residential

In room or room-to-room

C 30 Res. or small office

Conference rooms, classrooms

D 50 Typical office

Cubicles in open office space

E 100 Large office

Large office space, multi-floor

F 150 Large space

Indoor large hangars, outdoor campus / urban

Table 3.11: TGn channel models A to F use two slopes n1 = 2 near transmitter, and different values of n2 beyond a critical distance d0. They also estimate different log-normal shadowing standard deviations σ1 = 3dB near transmitter, and higher values σ2 beyond d0.

Modeld0(m)n1n2 σ1σ2

A 5 2 3.53 4
B 5 2 3.53 4
C 5 2 3.53 5
D 10 2 3.53 5
E 20 2 3.53 6
F 30 2 3.53 6

The 3GPP spatial channel models (SCM) [38] focus on 3G and 4G applications such as UMTS and LTE, in 5MHz channels around 2GHz. There are again different models, based on environment, speed etc, typically modeling N delayed multipaths, each comprised of M subpaths (in typical urban and suburban environments, the model uses N = 6,M = 20). Different parameters are also given for different environments (suburban macrocell, urban macrocell, and urban microcell): pathloss (LOS and NLOS), antenna beamwidth, delay statistics, log-normal shadowing, angles of arrival distribution, etc.

Important work on correlation between these multiple path is also presented, as it is crucial to estimating the MIMO rank important for system capacity (see §9.1.3). The 3GPP spatial channel models (SCM) report [38] is a wonderful source of many other parameters and typical values very useful for any propagation aspects of propagation for mobile communication systems.

3.7 In-Building Penetration

Sending RF signal into buildings means additional building penetration loss in the link budget. Indoor penetration measurements are difficult to perform, and difficult to compare from one experiment to another. Difficulties arises mostly from the fact that indoor and outdoor environments are so different that the method of data collection may cause large variations between the two environments; the following parameters have an influence: antenna beamwidth, angle of incidence, outside multipath, indoor multipath, distance from the walls, etc.

Measurement campaigns show that the distribution of building penetration loss is close to log-normal [17], a Gaussian function is a good approximation of the cumulative distribution function (CDF) of indoor measurements. The mean μi and standard deviation σi of indoor penetration loss vary with frequency, types of homes, and environment around the homes. Variations also depend on the location within the building (near an outside wall, a window, or further inside). Finally the angle of incidence with the outside wall also has a significant impact.

With that in mind, we can consider that wireless systems with in-building penetration have a shadowing statistic with a log-normal random variate which combines two independent log-normal variates: the outdoor shadowing (detailed further in §4.1 with standard deviation σo) and the in-building loss (with standard deviation σi). 10 And the aggregate random variate is also log-normal distributed, and has a standard deviation σ = ∘ -------
  σ2o + σ2

3.7.1 In-Building Models

The COST project proposes models for indoor penetration ([24] §4.6) with variations of angle of incidence. The COST 231 indoor model simply uses a line-of-sight path loss with an indoor component:

L = 32.4+  20logfGHz  + 20log(S + d)+ Lindoor

where S is the outdoor path, d is the indoor path, and

Lindoor = Le + Lge(1- sinθ) + max (Γ 1,Γ 2)

where Le is the normal incidence first wall penetration; the next term represents the added loss due to angle of incidence θ and is sometimes measured over an average of empirical values of incidence, in which case it may be noted Lge = Lge(1 - sinθ)2; and the last term max(Γ1,Γ2) aims at estimating loss within the building, whether going through walls or in a corridor.

Since angles of incidence are not always known the estimate Lge = Lge(1-sinθ)2 is sometimes more convenient. As a rough estimate for angles of incidence between -π∕2 and π∕2 lead to the following:

      ∫ π∕2
L′ge =      Lge(1 - sin θ)2d θ ≈ 0.55Lge

Empirical values of Lge are reported to be 5.7 - 6.4 [54] for residential areas, therefore we may use Lge 10dB. For urban environments, COST-231 reports Lge 20dB.

As for further interior loss, the COST model distinguishes between propagation through walls and propagation down coridors. Through ni interior walls of loss Li each: Γ1 = niLi. In a corridor: Γ2 = α(d′- 2)(1 - sinθ)2, with an empirical propagation loss α = 0.6dB/m.


Figure 3.14: COST-231 indoor penetration loss model.

Typical values for the model reported in [24] and [54] are summarized in table 3.12.

Table 3.12: Penetration loss into buildings, from COST-231 model.

Material FrequencyLe Lge LgeLi

Wood, plaster 900MHz 4 4 4
Concrete w/windows1.8GHz 7 206 10
Residential 2.5GHz 6.2106.1 3


Figure 3.15: Penetration loss into residential buildings, cumulative density distribution for 700 MHz, 900 MHz, 1.9 GHz, and 5.8 GHz.

3.7.2 Residential Homes

In most residential and suburban environments, surfaces involved are mostly made of glass, bricks, wood, and drywall. Penetration is often dominated by paths through windows and roofs, loss are relatively low and go up with frequency.

Precise characterization of in-building penetration is difficult, a rough approximation of an average penetration loss μi around 10 to 15 dB and a standard deviation σi around 6 dB seems to be the norm in published studies. Table 3.13 and figure 3.7.2 summarize some published results for residential homes.

Table 3.13: Penetration Loss into residential buildings: median loss (μi) and standard deviation (σi) from experimental results reported at various frequencies.

Source Frequencyμi σi Comments
(GHz) (dB)(dB)

Aguirre [51][55]0.9 6.4 6.8 7 Boulder residences
1.9 11.6 7.0
5.9 16.1 9.0
Wells [52] 0.86 6.3 6 Sat. meas. into 5 homes
1.55 6.7 6
2.57 6.7 6
Durgin [72] 5.8 14.9 5.6 [72]Table 5 average
Martijn [53] 1.8 12.0 4.0 [53]Table 1
Oestges [54] 2.5 12.3 [54]Table 6 (avg. Le + Lge)
Schwengler 1.9 12.0 6.0 Personal measurements
Schwengler [75]5.8 14.7 5.5 [75]Table 2

Average 0.9 6.4 6.4
2 10.3 6.3
5.8 13.8 6.7


Figure 3.16: In-building loss for residential buildings: measurements campaigns published for different frequencies, in different residential areas.


Figure 3.17: In-building loss for urban office buildings and high-rises: measurements campaigns published for different frequencies, in different urban areas.

3.7.3 Urban Environments

In dense urban areas experiments show different trends as illustrated in figure 3.7.2: some papers show penetration loss increasing with frequency [51][55]; some claim loss are independent of frequency [64][57]; others show a decrease with frequency [61][60][59].

Furthermore the variations between buildings and types of environments nearly always exceed the frequency variations. These environments are dominated by reflections off metal reinforced concrete and heavily reflective glass. In case of high-rises, penetration also depends on the floor and height of neighboring buildings or clutter.

Table 3.14: Penetration loss into vehicles: median loss (μv) and standard deviation (σv) from experimental results reported at various frequencies. (Some references have additional measurements or simulated values.)

Source Frequencyμv σv Comments
(GHz) (dB)(dB)

Hill [66] 0.15 5.3 Head level
0.45 6.9
0.8 3.8
0.9 3.9
Kostanic [67]0.8 8.8 3.0 In minivan
0.8 8.4 3.1 In full size car
0.8 12.0 2.9 In sports car
Tanghe [68] 0.6 16.8 3.2 V pol. Tx rear of van
0.9 7.5 3.0
1.8 9.5 3.4
2.4 13.8 4.1
0.6 4.9 4.7 V pol. Tx front of van
0.9 3.2 3.3
1.8 3.8 4.4
2.4 5.3 4.1

Average 0.6 10.9 4.0
0.8-0.9 6.5 3.1
1.8 6.7 3.9
2.4 9.6 4.1

3.7.4 In-Vehicle Loss

Given the mobile nature of wireless communications, penetration loss into vehicle is important as well. Precise characterization of in-vehicle penetration is difficult as well, and varies with type of vehicles, frequency, polarization, antenna placement in the vehicle, and direction of incidence. [66][67][68] An average penetration loss μv around 8 dB and a standard deviation σv around 3 dB seems to be the norm in published studies – see table 3.14.

Just like in-building penetration, in-vehicle penetration is a log-normally distributed random variate (with standard deviation σv), it is independent of the outdoor large scale shadowing statistic (σo), 11 and the aggregate loss is log-normal distributed with standard deviation σ = ∘ --2---2
  σ o + σv.

3.8 Homework

  1. At the beginning of section 3.2, we start to derive a free-space model from Friis’ equation. (a) Rederive in details the Friis’ formula (3.2). (b) Assume in the above that Gt = Gr = 1 (=0 dBi), derive (3.3).
  2. Find the paper [32] by V. Erceg & al. “An Empirically Based Path Loss Model for Wireless Channels in Suburban Environments”, in IEEE Journal on Selected Areas in Communications, Vol. 17, No 7, July 1999. This popular paper for PCS propagation modelling and design deserves some attention. Read it and answer the following questions:
    1. Summarize data collection campaign methods and size.
    2. Summarize key findings.
    3. A key finding is that path loss exponent variations are Gaussian, how is that proven in the paper?
  3. Plot path loss prediction versus distance and log distance for a cellular system you are designing with the following assumptions: PCS frequency (1900 MHz), base height 20 m, mobile 2 m, suburban area, flat terrain with moderate tree density.
    1. Use and compare the 3 following models: Free space, COST 231-Hata, & Erceg (use a median path loss: i.e. x=y=z=0)
    2. Using typical 140 dB maximum allowable path loss for a CDMA voice system, what is the range (cell radius) according to these models?
  4. Repeat the above problem with unlicensed frequency 5.8 GHz and a link budget of 120 dB. Compare. (Use the same models, including COST 231-Hata and Erceg models even though the frequency exceeds their domain of validity.)
  5. Compare the received power level of free-space (n=2), and 2-ray models for a PCS signal (1900 MHz; use hb = 8 m, hm = 2 m). Write a program (in any language of your choice) to plot a graph of power level versus log of distance (from 10m to 10km). Submit code with comments and explanations, and a resulting figure.
    1. First assume a simple perfect reflection Γ = -1
    2. Then use the actual Γ given for a vertical polarized wave. Is the difference significant?
    3. What cell site radius would be ideal for a system design? Why?
  6. Plot and compare on a same graph the propagation estimate for a radio system at 2.4 GHz and another at 5.8 GHz (all other parameters being equal); use hb = 8 m, hm = 2 m; use a) the two-ray model from §3.3.2, b) the 6-ray model from §3.3.4. Point out the main differences.
  7. Find and read the three papers referenced for in-vehicle penetration [66][67][68], as well as any other such reports or paper if you can.
    1. Produce a table like 3.14 with clear references, but with more details.
    2. Define average values (for mean and standard deviation) that you would recommend using for 700MHz, 800MHz, 1.9GHz, 2.4GHz propagation into vehicles. Justify.
  8. Find and read reference [33].
    1. Implement the software emulations from appendix B (e.g. using Matlab) and produce a simulation of the fading for SUI-3 with omnidirectional antenna at in figure 1 of that appendix B.
    2. Play with some parameters of the simulation, and submit one more, and analyze some noticeable differences.
  9. This problem studies the main differences in link budget to be expected between cellular and PCS systems. This problem is inspired from a 1999 paper by Chu and Greenstein [40] — feel free to refer to it for further details, but some analyses have been simplified, and some values were changed, so do not use any numerical values from the paper or this problem.

    Cellular and PCS systems are usually FDD, thus operating at different frequencies for uplink and downlink. Still in this problem, we neglect that small difference, and we assume cellular frequency to be 800MHz, and PCS frequency to be 1900MHz.

    1. Typical path loss models like one slope models do take into account the variation due to frequency. For simplicity we use here the frequency variation seen in the free space model. Evaluate that variation (in dB).
    2. Calculate the link budget difference between cellular and PCS. Give your answer as LB(cellular)-LB(PCS) in dB.

      We’ll try in the following questions to make up for that difference.

    3. Antenna gain can make up some of that difference. Let’s assume that we use antennas of the same size, meaning that they have the same effective area Ae = Gaλ2(4π). What antenna gain difference ΔGa is there between cellular and PCS?
    4. Another improvement is that of diversity. We consider receive diversity at the base station only; handset receive diversity will be considered negligible. Diversity gain depends (inversely) on the correlation ρ between the signals:
      G  =  10log∘ (1---ρ)

      ρ = exp(- 2π σD ∕λ)

      where λ is the wavelength, D is the antenna separation, and σ is the standard deviation of the angle of arrival, which is difficult to estimate, and we’ll use an empirical estimate of 1 degree, which means σ = 0.0175 rad in the formula above.

      Calculate ρ at both frequencies for 6-feet antenna separation. Calculate ΔGd, the diversity gain difference between cellular and PCS.

    5. Another difference may be the impact of vegetation in the cell: a simple vegetation attenuation formula is sometimes: Lv(dB∕km) = 1.33f0.284 with f in MHz. We’ll consider that a cell is on average 1km radius, therefore we’ll use Lv for our link budget. Calculate the difference in vegetation loss between cellular and PCS.
    6. Noise at the receiver is another important consideration. Handset noise is considered identical, but base station noise parameters vary. The main difference seems to be that ambient noises (mostly man-made) is higher at 800MHz, that impact on the receiver can be approximated (in dB) by Pmm = 2.4 - 2.3log(f∕100) with f in MHz. Calculate that difference between cellular and PCS.
    7. In-building penetration is also likely to vary: from the relevant figure in the class notes, estimate the difference in building penetration loss for the 90th percentile of the CDF. (Meaning the additional loss we should consider to account for 90% of residential buildings.)
    8. Summarize and calculate the link budget difference seen in all questions above. That is the link budget difference between cellular and PCS for the uplink (reverse link).
    9. Conclude and comment about the value of cellular vs. PCS spectrum — e.g. quantify coverage.
  10. Repeat the previous problem to compare 700MHz and AWS (1700MHz) spectrum.

Copyright ©2018 Thomas Schwengler.