Tag: far field

Far-field retarded potentials

Key focus: Far-field region is dominated by radiating terms of antenna fields, hence, knowing the far field retarded potentials is of interest.

Introduction

The fundamental premise of understanding antenna radiation is to understand how a radiation source influences the propagation of travelling electromagnetic waves. Propagation of travelling waves is best described by electric and magnetic potentials along the propagation path.

The concept of retarded potentials was introduced in this post.

The electromagnetic field travels at certain velocity and hence the potentials at the observation point (due to the changing charge at source) are experienced after a certain time delay. Such potentials are called retarded potentials.

The retarded potentials at a radial distance r from an antenna source fed with a single frequency sinusoidal waves, is shown to be

\[\begin{aligned} \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(z')e^{-j k R }}{R} d^3 z' \\ A(r) &= \frac{\mu}{4 \pi} \int_V \frac{J(z')e^{-j k R }}{R} d^3 z' \end{aligned} \quad \quad (1)\]

where, the quantity k = ω/c = 2 π/λ is called the free-space wavenumber. Also, ρ is the charge density, J is the current density, Φ is the electric potential and A is the magnetic potential that are functions of both radial distance.

Far-field region

Figure 1 illustrates the two scenarios: (a) the receiver is ‘nearer’ to the antenna source (b) the receiver is ‘far away’ from the antenna source.

Figure 1: Radiation fields when antenna and receiver are (a) near and (b) far away
Figure 1: Radiation fields when antenna and receiver are (a) near and (b) far away

Since the far-field region is dominated by radiating terms of the antenna fields, we are interested in knowing the retarded potentials in the far-field region. The far field region is shown to be

\[\frac{2 l^2}{ \lambda} < r < \infty \quad \quad (2)\]

where l is the length of the antenna element and λ is the wavelength of the signal from the antenna.

In the process of deriving the boundary between far-field and near-field, we used the following first order approximation for the radial distance R.

\[R = r – z' \; cos \theta= r – \hat{r} \cdot z' \quad \quad (3)\]

Far field retarded potential

Substituting this approximation in the numerator of equation (1) and replacing R by r in the denominator

\[\begin{aligned} \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(z')e^{-j k \left( r – \hat{r} \cdot z' \right) }}{r} d^3 z' \\ A(r) &= \frac{\mu}{4 \pi} \int_V \frac{J(z')e^{-j k \left( r – \hat{r} \cdot z'\right) }}{r} d^3 z' \end{aligned} \quad \quad (4)\]

The equation can be written as

\[\begin{aligned} \Phi(r) &= \frac{e^{-jkr}}{4 \pi \epsilon} \int_V \frac{\rho(z')e^{ j k \hat{r} \cdot z' }}{r} d^3 z' \\ A(r) &= \frac{\mu e^{-jkr}}{4 \pi} \int_V \frac{J(z')e^{j k \hat{r} \cdot z' }}{r} d^3 z' \end{aligned} \quad \quad (5)\]
Figure 2: Spherical coordinate system on a cartesian coordinate system

Antenna radiation patterns are generally visualized in a spherical coordinate system (Figure (2)). In a coordinate system, each unit vector can be expressed as the cross product of other two unit vectors. Hence,

\[\begin{aligned}\hat{r} &= \hat{\theta} \times \hat{\phi} \\ \hat{\theta} &= \hat{\phi} \times \hat{r} \\ \hat{\phi} &= \hat{r} \times \hat{\theta} \end{aligned} \quad \quad (6) \]

Therefore, the far-field retarded potentials in equation (5) can be written in terms of polar angle (θ) and azimuthal angle (ɸ)

\[\boxed{\begin{aligned} \Phi(r) &= \frac{e^{-jkr}}{4 \pi \epsilon r} \int_V \rho(z')e^{ j k \left( \hat{\theta} \times \hat{\phi} \right) \cdot z' } d^3 z' \\ A(r) &= \frac{\mu e^{-jkr}}{4 \pi r} \int_V J(z')e^{j k \left( \hat{\theta} \times \hat{\phi} \right) \cdot z' } d^3 z' \end{aligned}} \quad \quad (7)\]

We note that the term inside the integral is dependent on polar angle (θ) and azimuthal angle (ɸ). It determines the directional properties of the radiation. The term outside the integral is dependent on radial distance r. These terms can be expressed separately

\[\begin{aligned} \Phi(r) &= \frac{e^{-jkr}}{4 \pi \epsilon r} \mathbf{Q} \left(\theta, \phi \right) \\ A(r) &= \frac{\mu e^{-jkr}}{4 \pi r} \mathbf{F} \left(\theta, \phi \right) \end{aligned} \quad \quad (8)\]

The terms that determine the directional properties: Q(θ,ɸ) & F(θ,ɸ) are called charge form-factor and radiation vector respectively. The charge form-factor Q(θ,ɸ) and the radiation vector F(θ,ɸ) are three dimensional spatial Fourier transforms of charge density ρ(z’) and current density J(z) respectively.

\[\boxed{\begin{aligned} \mathbf{Q} \left(\theta, \phi \right) & = \int_V \rho(z')e^{ j k \left( \hat{\theta} \times \hat{\phi} \right) \cdot z' }d^3 z' \quad \quad \text{(charge form-factor)}\\ \mathbf{F} \left(\theta, \phi \right) &=\int_V J(z')e^{j k \left( \hat{\theta} \times \hat{\phi} \right) \cdot z' } d^3 z' \quad \quad \text{(radiation vector)} \end{aligned}}\quad \quad (9) \]

The charge-form factor and radiation vector can also be written in terms of direction of the unit vector of radial distance.

\[\boxed{\begin{aligned} \mathbf{Q} \left(\mathbf{k} \right) & = \int_V \rho(z')e^{ j \mathbf{k}\cdot z' }d^3 z' \quad \quad \text{(charge form-factor)} \\ \mathbf{F} \left(\mathbf{k}\right) &=\int_V J(z')e^{j \mathbf{k}\cdot z' } d^3 z' \quad \quad \text{(radiation vector)} \end{aligned}} \quad \quad \boxed{\mathbf{k} = k\hat{r}} \quad \quad (10) \]

Recap

We are in the process of building antenna models. In that journey, we started with the fundamental Maxwell’s equations in electromagnetism, then looked at retarded potentials that are solutions for Maxwell’s equations. Propagation of travelling waves is best described by retarded potentials along the propagation path. Then, the boundary between near-field and far-field regions was defined. Since most of the antenna radiation analysis are focused in the far-field regions, we looked at retarded potentials in the far-field region.

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
(173 votes, average: 3.66 out of 5)

Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
(127 votes, average: 3.58 out of 5)

Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
(134 votes, average: 3.63 out of 5)

Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Near-field – far-field : Where is the boundary ?

Antennas are radiation sources of finite physical dimension. To a distant observer, the radiation waves from the antenna source appears more like a spherical wave and the antenna appears to be a point source regardless of its true shape. The terms far-field and near-field are associated with such observations/antenna measurement. The terms imply that there must exist a boundary between the near field and far field.

Essentially, the near field and far field are regions around an antenna source. Though the boundary between these two regions are not fixed in space, the antenna measurements made in these regions differ significantly. One method of establishing the boundary between the near-field and far-field regions is to look at the acceptable level of phase error in the antenna measurements.

An antenna designer is interested in studying how the phase of the radiation waves launched from the antenna source is affected by the distance between the antenna source and the receiver (observation point). As the distance between the antenna and the receiver increases, there exists a phase difference between the measurements taken along the two lines shown. This phase difference contribute to antenna measurement errors, it also affects retarded potentials and radiation fields.

Near-field and far-field approximations

Figure 1 illustrates the two scenarios: (a) the receiver is ‘nearer’ to the antenna source (b) the receiver is ‘far away’ from the antenna source. The antenna is of standard dimension of length l. The figures show two rays – one from the origin to the observation point P (on the yz plane) and the other from the mid-point of distance z’=l/2 from the origin towards the observation point P.

In Figure(2)(b), the observation point P is at a distance that is very far from the antenna source element. The term ‘far’ implies that the distance r is much greater than the spatial extent of the current distribution of the antenna element, that is, r >> z’. Also, the two rays appear parallel to each other.

Figure 1: Radiation fields when antenna and receiver are (a) near and (b) far away

The essence of the following exercise is to determine the boundary between the ‘near’ and the ‘far’ field regions of the antenna. Once that boundary is established, we can determine whether far field approximation can be used on the antenna measurements or for the calculation of retarded potentials/fields produced by the antenna.

Let’s take a quick look at the retarded potentials derived for a single frequency wave emanating from the antenna source.

\[\begin{aligned} \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(z')e^{-j k R }}{R} d^3 z' \quad\quad (1) \\ A(r) &= \frac{\mu}{4 \pi} \int_V \frac{J(z')e^{-j k R }}{R} d^3 z' \quad\quad (2) \end{aligned}\]

We note that we cannot arbitrarily set R=r, because any small relative difference between R and r, will result in phase errors in the retarded potentials such that e-j k R ≠ e-j k r . Solving for the relationship between R and r is the crux of the radiation boundary problem.

From Figure (1)(a), applying law of cosines, the distance R can be written as

\[R = \sqrt{r^2 – 2 r z' cos \theta + z'^2} \quad \quad (3)\]

which can be expanded using the following Binomial series expansion,

\[ \begin{aligned}(x+y)^{n}&=\sum _{k=0}^{\infty }{n \choose k}x^{n-k}y^{k}\\&=x^{n}+nx^{n-1}y+{\frac {n(n-1)}{2!}}x^{n-2}y^{2}+{\frac {n(n-1)(n-2)}{3!}}x^{n-3}y^{3}+\cdots .\end{aligned}\]

Setting x = r2 and y= – 2 r z’ cos θ + z’ 2 , equation (3) can be expanded as

\[\begin{aligned} R & = r^{2(\frac{1}{2})}+\frac{1}{2}r^{2(\frac{1}{2}-1)} \left( -2 r z’ cos \theta + z’^2 \right )+\cdots .\\ & = r+ \frac{1}{2r} \left( -2 r z’ cos \theta + z’^2 \right ) – \frac{1}{8 r^3} \left(2 r z’ cos \theta \right )^2 + \cdots .\\ & = r – z’ cos \theta + \frac{z’^2}{2r} sin^2 \theta + \cdots . \end{aligned}\]

Neglecting the higher order terms,

\[\begin{aligned} R & \simeq r – z’ cos \theta + \frac{z’^2}{2r} sin^2 \theta \quad \quad (4) \end{aligned}\]

Truncation of equation (4) means we are dealing with the following maximum error in the antenna measurements:

\[\frac{z’^2}{2 r} sin^2 \theta = \frac{z’^2}{2 r} , \quad \quad \text{for } \theta=\frac{\pi}{2}\quad (5)\]

On the other hand, from Figure (1)(b), the distance \(R\) is given by

\[R = r – z’ cos \theta \quad \quad (6)\]

As r → ∞, equation (4) approaches exactly the parallel ray approximation given by equation (6). However, for finite values of r (due to the additional term z’ 2/2r sin2 θ and also the additional terms that were neglected) there exists an error between parallel ray approximation and the actual value of R computed using equation (4).

So the question is: What is the minimum distance over which the parallel ray approximation can be invoked ?

According to text books, for the maximum extent of the antenna (z’ = l/2), when the maximum phase difference is π/8, it produces acceptable errors in antenna measurements.

\[k \frac{z’^2}{2 r} \simeq \frac{\pi}{8} \quad \quad (7)\]

which gives

\[\boxed{r = \frac{2 l^2}{ \lambda}} \quad \quad (8)\]

In these equations, k = ω/c = 2 π/ λ is the free-space wavenumber.

Equation (8) defines the minimum distance (a.k.a the boundary between near and far field regions) over which the parallel ray approximation can be invoked. This minimum distance is called far-field distance – the boundary beyond which the far-field region starts. The quantity l is the maximum dimension of the antenna.

The far-field region, also known as Fraunhofer region, is dominated by radiating terms of the antenna fields. The far-field region is

\[\boxed{\frac{2 l^2}{ \lambda} < r < \infty }\quad \quad (9)\]
Figure 2: Far-field distance and far-field region

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Fresnel zones

An important consideration for propagation models are the existence of objects within what is called the first Fresnel zone. Fresnel zones, referenced in Figure 1 are ellipsoids with the foci at the transmitter and the receiver, where the path length between the direct path and the alternative paths are multiples of half-wavelength (). Rays emanating from odd-numbered Fresnel zones cause destructive interference and the rays from the even-numbered Fresnel zones cause constructive interference.

Figure 1: Fresnal zone illustration

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

For large-scale propagation geometry, the path difference between the LOS and the diffracted rays is

The radius of the (n^{th}) Fresnel zone is approximated as

Given the height of the obstruction (refer Figure 1 given in the single knife-edge diffraction model), we can find out which Fresnel zone is obstructed by the obstruction. Setting in equation (2) and solving for (n) by using equation (1).

As general rule of thumb for point-to-point communication, if of the first Fresnel zone is clear of obstructions, the diffraction loss would be negligible. Any further Fresnel zone clearance does not significantly alter the diffraction loss.

Program 1: FresnalZone.m : Compute radius of a Fresnel zone and safe clearance at first Fresnel zone – Refer the book for Matlab code

As an example, we would like to measure the radius of the first Fresnel zone at the midpoint between the transmitter and receiver that are separated by a distance of and operating at the frequency . The script results in the following output. The radius of the first Fresnel zone will be . It will also inform us that if at-least of the first Fresnel zone is clear of any obstruction, then any calculated diffraction loss can be safely ignored.

Program 2: FresnelzoneTest.m: Computing the diffraction loss using single knife-edge model

d=25e3; %total distance between the tx and the Rx
f=12e9; %frequency of transmission
n=1;% Freznel zone number - affects r_n only
d1=25e3/2; d2=25e3/2; %measurement at mid point
%r_n = radius of the given zone number
%r_clear = clearance required at first zone
[r_n,r_clear] = Fresnelzone(d1,d2,f,1)

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Topics in this chapter

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Modeling diffraction loss : Single knife-edge diffraction model

Modeling diffraction loss

Propagation environments may have obstacles that hinder the radio transmission path between the transmitter and the receiver. Idealized models for estimating the signal loss associated with diffraction by such obstacles are available. The shape of the obstacles considered in these model are too idealized for real-life applications, nevertheless, these models can serve as a good reference.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Single knife-edge diffraction model

The model depicted in Figure 1 considers two idealized cases where a sharp obstacle is placed between the transmitter and the receiver. Using all the geometric parameters as indicated in the figure, the diffraction loss can be estimated with the help of a single, dimension-less quantity called Fresnel-Krichhoff diffraction parameter – . Based on the availability of information, any of the following equation can be used to calculate this parameter [1].

Figure 1: Diffracting single knife-edge obstacle having (a) positive height and (b) negative height

After computing the Fresnel-Krichhoff diffraction parameter from the geometry, the signal level due to the single knife-edge diffraction is obtained by integrating the contributions from the unhindered portions of the wavefront. The diffraction gain (or loss) is obtained as

where, and are respectively the real and imaginary part of the the complex Fresnel integral given by

The diffraction gain/loss in the equation (2) can be obtained using numerical methods which are quite involved in computation. However, for the case where , the following approximation can be used [1].

The following function implements the above approximation and can be used to compute the diffraction loss for the given Fresnel-Kirchhoff parameter.

Program : diffractionLoss.m : Function to calculate diffraction loss/gain – Refer the book for Matlab code

The following snippet of code loops through a range of values for the parameter and plots the diffraction gain/loss (Figure 2).

Program : fresnel_Kirchhoff_diffLoss.m: Diffraction loss for a range of Fresnel-Kirchhoff parameter

v=-5:1:20; %Range of Fresnel-Kirchhoff diffraction parameter
Ld= diffractionLoss(v); %diffraction gain/loss (dB)
plot(v,-Ld);
title('Diffraction Gain Vs. Fresnel-Kirchhoff parameter');
xlabel('Fresnel-Kirchhoff parameter (v)');
ylabel('Diffraction gain - G_d(v) dB');
Figure 2: Loss of signal strength as a function of Fresnel-Kirchhoff diffraction parameter

Finally, the single knife-edge diffraction model can be coded into a function as follows. It also incorporates equation 3 (given in this post) that help us find the Fresnel zone obstructed by the given obstacle. The subject of Fresnel zones are explained in the next section.

Program : singleKnifeEdgeModel.m : Single Knife-edge diffraction model – Refer the book for Matlab code

As an example, using the sample script below, we can determine the diffraction loss incurred for , and at frequency . The computed diffraction loss will be .

Program : Computing the diffraction loss using single knife-edge model

h=20; f=10e9;d1=10e3;d2=5e3;
[L_dB,n]=singleKnifeEdgeModel(h,f,d1,d2)

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] Recommendation ITU-R P.526.11, Propagation by diffraction, The international telecommunication union, Oct 2009.↗

Topics in this chapter

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Two ray ground reflection model

Friis propagation model considers the line-of-sight (LOS) path between the transmitter and the receiver. The expression for the received power becomes complicated if the effect of reflections from the earth surface has to be incorporated in the modeling. In addition to the line-of-sight path, a single reflected path is added in the two ray ground reflection model, as illustrated in Figure 1. This model takes into account the phenomenon of reflection from the ground and the antenna heights above the ground. The ground surface is characterized by reflection coefficient – which depends on the material properties of the surface and the type of wave polarization. The transmitter and receiver antennas are of heights and respectively and are separated by the distance of meters.

Figure 1: Two ray ground reflection model

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

The received signal consists of two components: LOS ray that travels the free space from the transmitter and a reflected ray from the ground surface. The distances traveled by the LOS ray and the reflected ray are given by

Depending on the phase difference () between the LOS ray and reflected ray, the received signal may suffer constructive or destructive interference. Hence, this model is also called as two ray interference model.

where, is the wavelength of the radiating wave that can be calculated from the transmission frequency. Under large-scale assumption, the power of the received signal can be expressed as

where is the product of antenna field patterns along the LOS direction and is the product of antenna field patterns along the reflected path.

The following piece of code implements equation 3 and plots the received power () against the separation distance (). The resulting plot for , , , , is shown in the Figure 2. In this plot, the transmitter power is normalized such that the plot starts at . The plot also contains approximations of the received power over three regions.

twoRayModel.m: Two ray ground reflection model simulation (refer book for Matlab code – click here)

Figure 2: Distance vs received power for two ray ground reflection model and approximations**

** the approximations are shifted down in the plot for clarity, otherwise they will ride on top of the two ray model

The distance that is denoted as in the plot, is called the critical distance. It is calculated . For the region beyond the critical distance, the received power falls-off at rate. For the region where , the received power falls-off at rate and it can be approximated by free space loss equation. Table 1 captures the approximate expressions that can be applied for the three distinct regions of propagation as identified in the plot above.

Table 1: Approximate expressions for three different scenarios of separation distance

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Topic in this chapter

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Hata Okumura model for outdoor propagation

Outdoor propagation models involve estimation of propagation loss over irregular terrains such as mountainous regions, simple curved earth profile, etc., with obstacles like trees and buildings. All such models predict the received signal strength at a particular distance or on a small sector. These models vary in approach, accuracy and complexity. Hata Okumura model is one such model.

In 1986, Yoshihisa Okumura traveled around Tokyo city and made measurements for the signal attenuation from base station to mobile station. He came up with a set of curves which gave the median attenuation relative to free space path loss. Okumura came up with three set of data for three scenarios: open area, urban area and sub-urban area. Since this was one of the very first model developed for cellular propagation environment, there exist other difficulties and concerns related to the applicability of the model. Okumura model can be adopted for computer simulations by digitizing those curves provided by Okumura and using them in the form of look-up-tables [1]. Since it is based on empirical studies, the validity of parameters is limited in range. The parameter values outside the range can be obtained by extrapolating the curves. There are also concerns related to the calculation of effective antenna height. Thus every RF modeling tool incorporates its own interpretations and adjustments when it comes to implementing Okumura model.

Hata, in 1980, came up with closed form expressions based on curve fitting of Okumura models. It is the most referred macroscopic propagation model. He extended the Okumura models to include effects due to diffraction, reflection and scattering of transmitted signals by the surrounding structures in a city environment.

Figure 1: Simulated distance vs. path loss using Hata model, for fc = 1500 MHz , hb = 70 m and hm = 1.5 m

The generic closed form expression for path loss (PL) in dB scale, is given by

where, the Tx-Rx separation distance (d) is specified in kilometers (valid range 1 km to 20 Km). The factors A,B,C depend on the frequency of transmission, antenna heights and the type of environment, as given next.

● fc = frequency of transmission in MHz, valid range – 150 MHz to 1500 MHz
● hb= effective height of transmitting base station antenna in meters, valid range 30 m to 200 m
● hm=effective receiving mobile device antenna height in meters, valid range 1m to 10 m
● a(hm) = mobile antenna height correction factor that depends on the environment (refer table below)
● C = a factor used to correct the formulas for open rural and suburban areas (refer table below)

Table of Parameters for Hata okumura model

The function to simulate Hata-Okumura model is given in the book – Wireless Communication Systems using Matlab. The simulated path loss in three types of environments are plotted in Figure 1. The simulated results are obtained over a range of distances for the following parameter values fc=1500 MHz, hb=70 m and hm=1.5 m.

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] Masaharu Hata, Empirical formula for propagation loss in land mobile radio services, IEEE transactions on vehicular technology, vol. VT-29, no. 3, August 1980.↗

Topic in this chapter

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Large scale propagation models – an introduction

Radio propagation models play an important role in designing a communication system for real world applications. Propagation models are instrumental in predicting the behavior of a communication system over different environments. This chapter is aimed at providing the ideas behind the simulation of some of the subtopics in large scale propagation models, such as, free space path loss model, two ray ground reflection model, diffraction loss model and Hata-Okumura model.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Introduction

Communication over a wireless network requires radio transmission and this is usually depicted as a physical layer in network stack diagrams. The physical layer defines how the data bits are transferred to and from the physical medium of the system. In case of a wireless communication system, such as wireless LAN, the radio waves are used as the link between the physical layer of a transmitter and a receiver. In this chapter, the focus is on the simulation models for modeling the physical aspects of the radio wave when they are in transit.

Radio waves are electromagnetic radiations. The branch of physics that describes the fundamental aspects of radiation is called electrodynamics. Designing a wireless equipment for interaction with an environment involves application of electrodynamics. For example, design of an antenna that produces radio waves, involves solid understanding of radiation physics.

Let’s take a simple example. The most fundamental aspect of radio waves is that it travels in all directions. A dipole antenna, the simplest and the most widely used antenna can be designed with two conducting rods. When the conducting rods are driven with the current from the transmitter, it produces radiation that travels in all directions (strength of radiation will not be uniform in all directions). By applying field equations from electrodynamics theory, it can be deduced that the strength of the radiation field decreases by in the far field, where being the distance from the antenna at which the measurement is taken. Using this result, the received power level at a given distance can be calculated and incorporated in the channel model.

Radio propagation models are broadly classified into large scale and small scale models. Large scale effects typically occur in the order of hundreds to thousands of meters in distance. Small scale effects are localized and occur temporally (in the order of a few seconds) or spatially (in the order of a few meters). This chapter is dedicated for simulation of some of the large-scale models. The small-scale simulation models are discussed in the next chapter.

The important questions in large scale modeling are – how the signal from a transmitter reaches the receiver in the first place and what is the relative power of the received signal with respect to the transmitted power level. Lots of scenarios can occur in large-scale. For example, the transmitter and the receiver could be in line-of-sight in an environment surrounded by buildings, trees and other objects. As a result, the receiver may receive – a direct attenuated signal (also called as line-of-sight (LOS) signal) from the transmitter and indirect signals (or non-line-of-sight (NLOS) signal) due to other physical effects like reflection, refraction, diffraction and scattering. The direct and indirect signals could also interfere with each other. Some of the large-scale models are briefly described here.

The Free-space propagation model is the simplest large-scale model, quite useful in satellite and microwave link modeling. It models a single unobstructed path between the transmitter and the receiver. Applying the fact that the strength of a radiation field decreases as in the far field, we arrive at the Friis free space equation that can tell us about the amount of power received relative to the power transmitted. The log distance propagation model is an extension to Friis space propagation model. It incorporates a path-loss exponent that is used to predict the relative received power in a wide range of environments.

In the absence of line-of-sight signal, other physical phenomena like refection, diffraction, etc.., must be relied upon for the modeling. Reflection involves a change in direction of the signal wavefront when it bounces off an object with different optical properties. The plane-earth loss model is another simple propagation model that considers the interaction between the line-of-sight signal and the reflected signal.

Diffraction is another phenomena in radiation physics that makes it possible for a radiated wave bend around the edges of obstacles. In the knife-edge diffraction model, the path between the transmitter and the receiver is blocked by a single sharp ridge. Approximate mathematical expressions for calculating the loss-due-to-diffraction for the case of multiple ridges were also proposed by many researchers [1][2][3][4].

Of the several available large-scale models, five are selected here for simulation:

Figure 1: Friis free space propagation model (large scale propagation model)

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] K. Bullington, Radio propagation at frequencies above 30 megacycles, Proceedings of the IRE, IEEE, vol. 35, issue 10, pp.1122-1136, Oct. 1947.↗

[2] J. Epstein, D. W. Peterson, An experimental study of wave propagation at 850 MC, Proceedings of the IRE, IEEE, vol. 41, issue 5, pp. 595-611, May 1953.↗

[3] J. Deygout, Multiple knife-edge diffraction of microwaves, IEEE Transactions on Antennas Propagation, vol. AP-14, pp. 480-489, July 1966.↗

[4] C.L. Giovaneli, An Analysis of Simplified Solutions for Multiple Knife-Edge Diffraction, IEEE Transactions on Antennas Propagation, Vol. AP-32, No.3, pp. 297-301, March 1984.↗

Topics in this chapter

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Friis Free Space Propagation Model

Friis free space propagation model is used to model the line-of-sight (LOS) path loss incurred in a free space environment, devoid of any objects that create absorption, diffraction, reflections, or any other characteristic-altering phenomenon to a radiated wave. It is valid only in the far field region of the transmitting antenna [1] and is based on the inverse square law of distance which states that the received power at a particular distance from the transmitter decays by a factor of square of the distance.

Figure 1: Received power using Friis model for WiFi transmission at f=2.4 GHz and f=5 GHz

The Friis equation for received power is given by

where, Pr is the received signal power in Watts expressed as a function of separation distance (d meters) between the transmitter and the receiver, Pt is the power of the transmitted signal’s Watts, Gt and Gr are the gains of transmitter and receiver antennas when compared to an isotropic radiator with unit gain, λ is the wavelength of carrier in meters and L represents other losses that is not associated with the propagation loss. The parameter L may include system losses like loss at the antenna, transmission line attenuation, loss at various filters etc. The factor L is usually greater than or equal to 1 with L=1 for no such system losses.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

The Friis equation can be modified to accommodate different environments, on the reason that the received signal decreases as the nth power of distance, where the parameter n is the path-loss exponent (PLE) that takes constant values depending on the environment that is modeled (see Table below} for various empirical values for PLE).

Table 1: Path loss exponent for various environment

The propagation path loss in free space, denoted as PL, is the loss incurred by the transmitted signal during propagation. It is expressed as the signal loss between the feed points of two isotropic antennas in free space.

The propagation of an electromagnetic signal, through free space, is unaffected by its frequency of transmission and hence has no dependency on the wavelength λ. However, the variable λ exists in the path loss equation to account for the effective aperture of the receiving antenna, which is an indicator of the antenna’s ability to collect power. If the link between the transmitting and receiving antenna is something other than the free space, penetration/absorption losses are also considered in path loss calculation. Material penetrations are fairly dependent on frequency. Incorporation of penetration losses require detailed analysis.

Usually, the transmitted power and the receiver power are specified in terms of dBm (power in decibels with respect to 1 mW) and the antenna gains in dBi (gain in decibels with respect to an isotropic antenna). Therefore, it is often convenient to work in log scale instead of linear scale. The alternative form of Friis equation in log scale is given by

Following function, implements a generic Friis equation that includes the path loss exponent, , whose possible values are listed in Table 1.

FriisModel.m: Function implementing Friis propagation model (Refer the book for the Matlab code – click here)

For example, consider a WiFi (IEEE 802.11n standard↗) transmission-reception system operating at f =2.4 GHz or f =5 GHz band with 0 dBm (1 mW) output power from the transmitter. The gain of the transmitter antenna is 1 dBi and that of receiving antenna is 1 dBi. It is assumed that there is no system loss, therefore L = 1. The following Matlab code uses the Friis equation and plots the received power in dBm for a range of distances (Figure 1 shown above). From the plot, the received power decreases by a factor of 6 dB for every doubling of the distance.

Friis_model_test.m: Friis free space propagation model

%Matlab code to simulate Friis Free space equation
%-----------Input section------------------------
Pt_dBm=52; %Input - Transmitted power in dBm
Gt_dBi=25; %Gain of the Transmitted antenna in dBi
Gr_dBi=15; %Gain of the Receiver antenna in dBi
f=110ˆ9; %Transmitted signal frequency in Hertz d =41935000(1:1:200) ; %Array of input distances in meters
L=1; %Other System Losses, No Loss case L=1
n=2; %Path loss exponent for Free space
%----------------------------------------------------
[PL,Pr_dBm] = FriisModel(Pt_dBm,Gt_dBi,Gr_dBi,f,d,L,n);
plot(log10(d),Pr_dBm); title('Friis Path loss model');
xlabel('log10(d)'); ylabel('P_r (dBm)')

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] Allen C. Newell, Near Field Antenna Measurement Theory, Planar, Cylindrical and Spherical, Nearfield Systems Inc.↗

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Topics in this chapter