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## **Meet the editor**

Sandra Costanzo received her Laurea degree (summa cum laude) in Computer Engineering (Electronics and Telecommunication profile) from the University of Calabria in 1996 and her PhD degree in Electronic Engineering from the University of Reggio Calabria in 2000. Currently, she is an associate professor at University of Calabria, Italy, where she teaches the courses of electromagnetic

wave propagation, radar and remote sensing and signal and waves in telecommunications. Her research interests are focused on near-field far-field techniques, antenna measurement techniques, antenna analysis and synthesis, numerical methods in electromagnetics, millimeter-wave antennas, reflectarrays, synthesis methods for microwave structures, electromagnetic characterization of materials and innovative antennas and technologies for radar applications. She serves as a reviewer for various international journals. She is an associate editor of *IEEE Antennas and Wireless Propagation Letters* and *IEEE Access and Radioengineering* journals, and she is a member of the Editorial Board for the *Electrical and Electronic Engineering* journal and Scientific and Academic Publishing. She is the lead editor of three Special Issues titled "Reflectarray Antennas: Analysis and Synthesis Techniques" (*International Journal of Antennas and Propagation*), "Advances in Radar Technologies" (*Journal of Electrical and Computer Engineering*), "Compressed Sensing: Applications in Radar and Communications" (*The Scientific World Journal*). She is the editor of the book *Microwave Materials Characterization* (ISBN 978-953-51-0848-1, InTech, 2012). She has coauthored more than 160 contributions in international journals, books and conferences.

## Contents

## **Preface XI**

## Preface

Chapter 8 **A Rain Estimation System Based on Electromagnetic**

Shulabh Gupta and Christophe Caloz

Chapter 10 **Manipulating Electromagnetic Waves with Zero Index**

Chapter 9 **Real-Time Electromagnetic Signal Processing: Principles and**

Caridi

**VI** Contents

**Illustrations 155**

**Materials 177**

Zhifang Lin

**Propagation Models and DVB-S Opportunistic Sensors 143** Daniele Caviglia, Matteo Pastorino, Andrea Randazzo and Andrea

Shiyang Liu, Jialin Zhou, Ying Han, Xinning Yu, Huajin Chen and

The present book gives a collection of some recent studies on relevant challenges for the design of innovative telecommunication systems, with particular focus on the propagation aspects and radiating system design. The book is divided into two sections, both including original research contributions. Section 1 is devoted to the illustration of advanced results in terms of microwave propagation at high frequencies, while Section 2 is focused on new elec‐ tromagnetic concepts and applications.

Section 1 consists of Chapters 1–5. Chapter 1 by S. Costanzo discusses the basic properties of nondiffracting Bessel beams, providing an overview of some potential applications.

In Chapter 2 the ionospheric propagation effects including higher order terms for micro‐ wave signals up to 100 GHz frequencies are analyzed by M. M. Hoque et al., with the aim to enhance the accuracy of precise positioning, navigation and frequency transfer.

Chapter 3 by L. Luini et al. deals with all aspects of characterizing the propagation of elec‐ tromagnetic waves in the Ka, Q, V and W bands, spanning from the main impairments in‐ duced by the troposphere to how extreme atmospheric conditions can be handled making use of suitable fade mitigation techniques.

A conceptual frame for understanding the propagation of light through a dispersive medi‐ um is presented by P. Robles and F. Pizarro in Chapter 4, considering both the classical ap‐ proach based on electromagnetic waves and a quantum description in terms of photons.

In Chapter 5, the problem of shaped beam scattering by typical particles with regular shapes is faced by Y. P. Han et al, and relevant results for some complex particles are discussed.

Section 2 consists of Chapters 6–10. G. Di Massa in Chapter 6 discusses some relevant appli‐ cations of the interaction between slow electromagnetic waves and dielectric or corrugated media, to be helpful in the design of compact and easily integrable microwave devices and antennas.

In Chapter 7, the scattering response of non-planar frequency selective surfaces (FSS) and curved elements is numerically investigated by G. Leone et al, with the aim to provide a useful design tool for conformal FSS to be adopted in high-performance applications.

In Chapter 8, a real-time and low-cost system for precipitation detection from Satellite Digi‐ tal Video Broadcasting (DVB-S) microwave signals is discussed by D. Caviglia et al.

In Chapter 9, real-time electromagnetic signal processing paradigm is presented by S. Gupta and C. Caloz, with the aim to control electromagnetic signals at low cost directly in the ana‐ log domain.

Chapter 10 by S. Liu et al. presents the design of zero-index materials with magnetic meta‐ materials based on multiple scattering theory and effective-medium theory.

I wish to thank all the book contributors for the original submitted results that came from their research activities. A special acknowledgement is due to the editorial assistants for their professional support during the book preparation.

> **Sandra Costanzo** University of Calabria Italy

**Electromagnetic Propagation in Near-Future Telecommunication Systems**

Chapter 10 by S. Liu et al. presents the design of zero-index materials with magnetic meta‐

I wish to thank all the book contributors for the original submitted results that came from their research activities. A special acknowledgement is due to the editorial assistants for

> **Sandra Costanzo** University of Calabria

> > Italy

materials based on multiple scattering theory and effective-medium theory.

their professional support during the book preparation.

VIII Preface

Provisional chapter

## **Localized Bessel Beams: Basic Properties and Emerging Communication Applications** Localized Bessel Beams: Basic Properties and Emerging Communication Applications

Sandra Costanzo Sandra Costanzo

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68780

## Abstract

Relevant properties of Bessel beams in terms of nondiffracting propagation over ideally infinite range, with unchanged transverse profile and self-healing capability, are revised and discussed in the present chapter. Promising applications in the framework of newgeneration communication systems are also outlined.

Keywords: focused systems, nondiffracting fields, Bessel beam, propagation, slow waves

## 1. Introduction

Diffractive phenomena are strictly related to the wave nature of light: any field of wavelength λ, initially confined to a finite area having radius r, will be subjected to a diffractive spreading beyond a characteristic distance equal to r2 λ, known as the Rayleigh range [1]. In the late 1980s, a class of diffraction-free mode solutions has been introduced [2] to describe well-defined nonspreading beams with extremely narrow central spot. They are called Bessel beams, due to the particular shape in the transverse plane, which is mathematically described by a Bessel function [3]. A qualitative comparison of Gaussian (diffractive) and ideal Bessel (localized) beams is illustrated in Figures 1 and 2: it is evident that Gaussian beam (Figure 1) soon diffracts for very short distance range, while the Bessel beam (Figure 2) ideally maintains a stationary transverse pattern along the propagation distance.

As deeply addressed in Section 2, Bessel beams are solutions to the Helmholtz equation in circular cylindrical coordinates, revealing many appealing features, namely

• they theoretically guarantee an indefinite extension along the axial (propagation) direction;

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Figure 1. Gaussian beam propagating into a homogeneous medium.

Figure 2. Bessel beam propagating into a homogeneous medium.

• they own the property of reconstruction after obstruction ("self-healing"), that is, if an obstacle occurs at the center of the beam, it will not block the rays, as they will interfere each other to reform the beam.

The ideal Bessel beam, having an infinite number of rings and covering an infinite distance, cannot physically exist, as it requires an infinite energy. Thus, physically meaningful beams are the apertured Bessel beams, possessing nearly diffracting properties within a limited axial distance. In this chapter, an overview of the basic properties of localized Bessel beams is provided, and some specific applications in the framework of emerging communication technologies are discussed. In particular, mathematical details on Bessel beams solutions are presented in Section 2, where the possibility to express the Bessel beam in terms of plane waves traveling on a cone with the same phase velocity is demonstrated. In Section 3.1, a practical realization example of microwave Bessel beam source to be applied for focused high-penetration applications is described, while in Sections 3.2 and 3.3, a discussion about potential applications in the framework of high-performance communication systems is outlined.

## 2. Localized Bessel beam solutions

Let us consider the scalar wave equation:

$$
\nabla^2 \psi - \frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2} = 0 \tag{1}
$$

where c is the free-space velocity.

In any system of cylindrical coordinates (u1, u2, z), the wave equation (1) is satisfied by a solution of the form [4]:

$$
\psi(\rho, z, t) = f(\rho) e^{j(k\_z z - \omega t)} \tag{2}
$$

where

$$
\rho = \sqrt{u\_1^2 + u\_2^2} \tag{3}
$$

is the transverse distance, and the radial function shape f (ρ) is assumed to be preserved along the propagation axis z.

When replacing the solution (2) into Eq. (1), the following equation is obtained:

$$
\rho^2 \frac{d^2 f(\rho)}{d\rho^2} + \rho \frac{df(\rho)}{d\rho} + \rho^2 (k^2 - k\_z^2) f(\rho) = 0 \tag{4}
$$

where k 2 ¼ ω2=c2.

Eq. (4) satisfied by the radial function f (ρ) can be easily recognized as Bessel's equation, namely

$$\mathbf{x}^2 \frac{d^2 I\_p(\mathbf{x})}{d\mathbf{x}^2} + \mathbf{x}\frac{J\_p(\mathbf{x})}{d\rho} + (\mathbf{x}^2 - p^2)I\_p(\mathbf{x}) = \mathbf{0} \tag{5}$$

where Jp (…) is the cylindrical Bessel function of order p.

Thus, a solution of the wave equation (1) which maintains unchanged its radial shape can be written as

$$
\psi(\rho, z, t) = f\_o(k\_\rho \rho) e^{j(k\_z z - \omega t)} \tag{6}
$$

where

• they own the property of reconstruction after obstruction ("self-healing"), that is, if an obstacle occurs at the center of the beam, it will not block the rays, as they will interfere

propagation distance [Km]

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2

propagation distance [Km]

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2

The ideal Bessel beam, having an infinite number of rings and covering an infinite distance, cannot physically exist, as it requires an infinite energy. Thus, physically meaningful beams are the apertured Bessel beams, possessing nearly diffracting properties within a limited axial distance. In this chapter, an overview of the basic properties of localized Bessel beams is provided, and some specific applications in the framework of emerging communication technologies are discussed. In particular, mathematical details on Bessel beams solutions are presented in Section 2, where the possibility to express the Bessel beam in terms of plane waves traveling on a

each other to reform the beam.

Figure 2. Bessel beam propagating into a homogeneous medium.

Figure 1. Gaussian beam propagating into a homogeneous medium.

4 Wave Propagation Concepts for Near-Future Telecommunication Systems

transverse distance [Km]

−1.5

transverse distance [Km]

−1.5 −1 −0.5 0 0.5 1 1.5 2

−1

−0.5

0

0.5

1

1.5

2

#### 6 Wave Propagation Concepts for Near-Future Telecommunication Systems

$$k\_{\rho} = k^2 - k\_z^2$$

$$k^2 = \frac{\omega^2}{c^2} = k\_1^2 + k\_2^2 + k\_z^2 = k\_{\rho}^2 + k\_z^2\tag{7}$$

Recalling that

$$k\_1 = k \sin \theta \cos \phi, \ k\_2 = k \sin \theta \sin \phi, \ k\_z = k \cos \theta \tag{8}$$

we have

$$k\_{\rho}^2 = k\_1^2 + k\_2^2 = k^2 \sin^2 \theta \tag{9}$$

Replacing expression (9) into the solution (6), we can write

$$
\psi(\rho, z, t) = f\_o(\rho k \sin \theta) e^{j(kz \cos \theta - \omega t)} \tag{10}
$$

In particular, for θ = 0�, the solution (10) reduces to a plane wave propagating along the zdirection.

Now, let us recall the Bessel function can be written as

$$J\_o(\rho) = \frac{1}{2\pi} \int\_0^{2\pi} e^{j\rho \cos \alpha} d\alpha \tag{11}$$

When imposing

$$
\mu = \phi - \phi', \ u\_1 = \rho \cos \phi', \ u\_2 = \rho \sin \phi' \tag{12}
$$

Eq. (11) can be written as

$$J\_o(\rho) = \frac{1}{2\pi} \int\_0^{2\pi} e^{j(u\_1 \cos \phi + u\_2 \sin \phi)} d\phi \tag{13}$$

Then, if replacing expression (13) into solution (10), we have

$$\Psi(\rho, z, t) = \frac{1}{2\pi} \int\_0^{2\pi} e^{j\mathbf{k}(u\_1 \sin\theta \cos\phi + u\_2 \sin\theta \sin\phi + z \cos\theta)} e^{-j\omega t} d\phi = \frac{1}{2\pi} \int\_0^{2\pi} e^{j(\underline{k}\cdot\underline{y} - \omega t)} d\phi \tag{14}$$

The latter equation demonstrates that a nondiffracting Bessel beam can be expressed in terms of plane waves, as it describes a cone of plane waves having the same inclination angle θ with respect to the propagation axis z.

Let us consider the solution expressed by Eq. (10). For 0 < kρ ≤ ω c , it represents a nondiffracting beam whose intensity profile decays at a rate proportional to the product kρρ. The maximum value of the radial wavenumber kρ to have a nonevanescent field is given by kρ ¼ ω c . From this condition, the spot width 2ρ of the Bessel beam can be easily derived to be equal to 3λ 4 (Figure 3). Localized Bessel Beams: Basic Properties and Emerging Communication Applications http://dx.doi.org/10.5772/intechopen.68780 7

Figure 3. Spot width of Bessel beam.

kρ ¼ k

k 2 ¼ ω2 c2 ¼ k 2 1 þ k 2 2 þ k 2 z ¼ k 2 ρ þ k 2

6 Wave Propagation Concepts for Near-Future Telecommunication Systems

Replacing expression (9) into the solution (6), we can write

Now, let us recall the Bessel function can be written as

k 2 ρ ¼ k 2 1 þ k 2 2 ¼ k2

Recalling that

we have

direction.

When imposing

Eq. (11) can be written as

Ψðρ, z, tÞ ¼ 1

respect to the propagation axis z.

2π ð2π 0

2 � k 2 z

k1 ¼ ksinθcosφ, k2 ¼ ksinθsinφ, kz ¼ kcosθ ð8Þ

ψðρ, z, tÞ ¼ JoðρksinθÞejðkzcosθ�ωtÞ ð10Þ

sin2

In particular, for θ = 0�, the solution (10) reduces to a plane wave propagating along the z-

, u1 ¼ ρcosφ0

ejkðu1sinθcosφþu2 sin θsinφþzcosθÞ

The latter equation demonstrates that a nondiffracting Bessel beam can be expressed in terms of plane waves, as it describes a cone of plane waves having the same inclination angle θ with

beam whose intensity profile decays at a rate proportional to the product kρρ. The maximum

value of the radial wavenumber kρ to have a nonevanescent field is given by kρ ¼ ω

condition, the spot width 2ρ of the Bessel beam can be easily derived to be equal to 3λ

ejðu1cosφþu2sinφÞ

e �jωt dφ ¼ 1 2π ð2π 0 e jð ¯ k� ¯ u�ωtÞ

JoðρÞ ¼ 1 2π ð2π 0 e

α ¼ φ � φ0

Then, if replacing expression (13) into solution (10), we have

Let us consider the solution expressed by Eq. (10). For 0 < kρ ≤ ω

JoðρÞ ¼ 1 2π ð2π 0

z ð7Þ

θ ð9Þ

jρcosαdα ð11Þ

, u2 ¼ ρsinφ0 ð12Þ

dφ ð13Þ

c , it represents a nondiffracting

c

. From this

4 (Figure 3).

dφ ð14Þ

As a matter of fact, the first null of the Bessel beam is equal to

$$k\_{\rho}\rho = 2.325\tag{15}$$

Thus, taking into account that kρ ¼ 2π λ , it results

$$
\rho = 0.37\lambda \tag{16}
$$

As yet outlined in Section 1, the solution (10) describes an ideal Bessel beam, with extremely narrow intensity profile and infinite propagation distance. However, from simple energy argumentations, it can be demonstrated that such a Bessel beam is not physically realizable. The intensity field distribution given by Eq. (10) is described by the Bessel function Jo (…), which is not square integrable (it decays as 1 ρ). This means that each ring contains the same amount of energy as the central spot. An infinite energy would be required to create the Bessel beam over the entire plane, due to the presence of an infinite number of rings. Thus, we should conclude that only an approximated Bessel beam can be practically realized over a finite area. It however maintains all appealing features of the ideal Bessel beam in terms of nondiffracting propagation over a finite distance zmax, as illustrated in Figure 4.

To derive the expression of the finite range zmax, let us consider the Bessel beam as given by the superimposition of plane waves, all having the same amplitude and traveling at the same angle θ relative to the propagation z-axis.

From Eq. (9), we have

$$\theta = \sin^{-1}\left(\frac{k\_{\rho}\lambda}{2\pi}\right) \tag{17}$$

Then, applying geometrical optics considerations, the following expression can be derived:

$$z\_{\text{max}} = \frac{r}{tg\theta} = r\sqrt{\frac{1}{\sin^2\theta} - 1} = r\sqrt{(\frac{2\pi}{k\_\rho\lambda})^2 - 1} \tag{18}$$

r being the aperture radius (Figure 4).

From Eq. (18) we can observe that, in order to produce a Bessel beam propagating on a long distance, either the radius r should be taken large, or the radial wavenumber kρ should be small, or alternately we should increase the operating frequency.

Let us consider the power Pr contained in a Bessel beam up to a radius r:

$$P\_r = \int\_0^{2\pi} \int\_0^r |\psi(\rho)|^2 \rho d\rho d\phi = P\_r = \int\_0^{2\pi} \int\_0^r |I\_o(k\_\rho \rho)|^2 \rho d\rho d\phi = r^2 [I\_o^2(k\_\rho r) + I\_1^2(k\_\rho r)] \tag{19}$$

When imposing relation (15), we have

$$P\_r = \pi r^2 f\_1^2(2.325) \tag{20}$$

If comparing the above result with that relative to a Gaussian beam produced by the same aperture, an increased power delivery can be demonstrated.

Figure 4. Finite propagation range of Bessel beam from finite aperture of radius r.

## 3. Potential applications of localized Bessel beams

zmax ¼ r

8 Wave Propagation Concepts for Near-Future Telecommunication Systems

small, or alternately we should increase the operating frequency.

Let us consider the power Pr contained in a Bessel beam up to a radius r:

ρdρdφ ¼ Pr ¼

aperture, an increased power delivery can be demonstrated.

r

r being the aperture radius (Figure 4).

Pr ¼

ð2π 0

ðr 0 jψðρÞj2

When imposing relation (15), we have

tgθ ¼ r

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sin2θ � 1

From Eq. (18) we can observe that, in order to produce a Bessel beam propagating on a long distance, either the radius r should be taken large, or the radial wavenumber kρ should be

> ð2π 0

Pr ¼ πr 2 J 2

ðr 0

If comparing the above result with that relative to a Gaussian beam produced by the same

q

zmax

Figure 4. Finite propagation range of Bessel beam from finite aperture of radius r.

jJoðkρρÞj2

¼ r

ð 2π kρλ Þ 2 � 1

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρdρdφ ¼ r

2 ½J 2 o ðkρrÞ þ J

1ð2:325Þ ð20Þ

2

1ðkρrÞ� ð19Þ

ð18Þ

The unique properties of Bessel beams in terms of self-reconstruction and profile stability over large distances make them ideal candidates in a variety of applications, requiring highly localized energy and/or diffraction mitigation. In this section, a few specific application examples to be adopted in the framework of new-generation telecommunication systems are reviewed. They include a microwave realization of Bessel beam launcher and the potential use of Bessel beams as an efficient transmission medium to increase data rate and overcome diffraction limits in long distance communications.

## 3.1. Waveguide-based microwave Bessel beam launcher

Most of existing works in the literature are mainly focused on the generation of Bessel beams in the optical regime, through the adoption of annular slit, axicons, or lasing devices [5–9], while much less results exist in the microwave regime.

In a recent ESA (European Space Agency) research study on "Microwave Drilling" [10], a practical X-band realization of Bessel beam launcher to be adopted as a focused near-field source has been performed. Following the theoretical approach outlined in [11], the microwave Bessel beam is generated as the aperture field at the open end of a metallic circular waveguide. In particular, a transverse electric (TE) representation is adopted for the zeroorder Bessel beam, with an expansion in terms of a finite number of propagating TE0n modes of the circular waveguide. These modes are produced by a set of elementary loop antennas, whose excitation coefficients are properly chosen to obtain the prescribed beam intensity and spot size. By imposing the Bessel beam to have n distinct annular sections, and choosing the radial wavenumber kρ to match the relative cutoff wavenumber χcn of the modes inside the circular waveguide [12], the Bessel beam can be fully represented by the single TE0n mode.

The schematic configuration of X-band Bessel beam launcher developed in [10] is illustrated in Figure 5. A single-loop antenna is considered for the proper field generation inside the circular waveguide of radius r = a. The excitation antenna is located at a distance equal to λo 2 (at the central operating frequency fo) from the short-circuited side of the circular waveguide, in order to match with the maximum value of the current, so to realize the maximum coupling with the magnetic field.

A Bessel beam with n = 3 distinct annular rings is chosen, so deriving the value of radius a from the relation [13]:

$$k\_{\rho}a = \chi\_{03} = 10.1735\tag{21}$$

which gives a ffi 1:5λo.

The realized prototype, for a design frequency f o ¼ 8:74GHz, is illustrated in Figure 6, where the microstrip loop antenna, designed on a standard substrate Arlon Diclad 880 (Er ¼ 2:2, thickness t = 0.762 mm) is also reported.

Figure 5. Schematic configuration of microwave Bessel beam launcher (taken from [10]).

Figure 6. Photograph of realized microwave Bessel beam launcher (taken from [10]): (a) side view and (b) front view.

In order to verify the nondiffracting feature along the propagation range, near-field measurements are performed on various planes placed at different distances from the Bessel beam launcher. All near-field acquisitions are realized on a square grid of 43 � 43 points, with uniform spacing equal to λo 2 , to satisfy the Shannon-Nyquist criterion. Near-field tests are executed into the anechoic chamber at the Microwave Laboratory of the University of Calabria, by adopting a standard X-band rectangular waveguide as a measuring probe. Contours profiles reported in Figure 7, referring to the acquisition planes at distances equal to 8 and 10 cm, respectively, reveal that Bessel beam produced by the designed microwave launcher properly maintains its shape on a long propagation distance.

#### 3.2. Bessel beam as information carrier in telecommunication systems

2a=12 cm 20 cm

(max of current) l/2

10 Wave Propagation Concepts for Near-Future Telecommunication Systems

Figure 5. Schematic configuration of microwave Bessel beam launcher (taken from [10]).

(a) (b)

Figure 6. Photograph of realized microwave Bessel beam launcher (taken from [10]): (a) side view and (b) front view.

The impressive property of unchanged shape over extended propagation distances makes Bessel beams appealing also in communication systems as information carriers. High-order Bessel beams expressed by

$$J\_p(k\_\rho \rho)e^{j\nu\phi} \tag{22}$$

have azimuthal index p which gives an additional degree of freedom to create custom fields [14–16] for carrying encoded information, by adopting the expansion:

$$\sum\_{p} a\_{p}(\rho)e^{jp\phi} \tag{23}$$

whose harmonics are independent of spatial scale and orthonormal over the azimuthal plane.

Researchers are currently looking at this interesting application of Bessel beam as a mean of transferring data, with a special focus on the development of efficient techniques to perform modal decomposition, but avoiding false detections due to cross-talk effects between neighboring modes.

Figure 7. Intensity profile of measured Bessel beam at distances equal to (a) 8 cm and (b) 10 cm (taken from [13]): (a) side view and (b) front view.

## 3.3. Bessel beam application in free-space optical communication

Free-space optical (FSO) communication is a robust method to transmit information with high capacity, high speed, and security [17]. Gaussian beams are typically adopted to realize the propagation; however, they suffer from limitations caused by diffraction, leading to the spread of the beam's energy, thus lowering the signal-to-noise ratio (SNR) at the receiver and increasing the bit error rate (BER).

In order to investigate the above effects, nondiffracting Bessel beams can be successfully adopted as alternative to Gaussian beams. An efficient FSO communication system should have a transmission beam as small as possible, with high peak intensity and high power. As deeply discussed in [18], these criteria are fully satisfied by Bessel beams. First of all, thanks to relation (15), the aperture radius required for a Bessel beam to transmit the half power is smaller than that required by a Gaussian beam, with a reduction of about 25% [18]. Furthermore, when comparing Bessel and Gaussian beams generated with the same aperture, the peak intensity of Bessel beam results to be about 1.2 times greater than that of the Gaussian beam.

As a validation example, the intensity cross-sections for long range propagation at a distance of 22 km through atmosphere are simulated in Figure 8 for Bessel beams (left hand) and Gaussian beams (right hand) and three different aperture radii. The presented results are taken from [18], vertical axis giving the radial distance in [m] and horizontal axis representing the propagation distance in [km]. The outperformances of Bessel beams are clearly visible.

The actual challenge to really achieve FSO communication systems with improved power delivery features for long propagation distances still remains the realizations of launchers less complex than standard axicons but able to produce Bessel beams efficiently.

Figure 8. Intensity cross-sections for long-range propagation (taken from [18]): (a) Bessel beams and (b) Gaussian beams.

## 4. Conclusions

3.3. Bessel beam application in free-space optical communication

12 Wave Propagation Concepts for Near-Future Telecommunication Systems

ing the bit error rate (BER).

beam.

clearly visible.

Free-space optical (FSO) communication is a robust method to transmit information with high capacity, high speed, and security [17]. Gaussian beams are typically adopted to realize the propagation; however, they suffer from limitations caused by diffraction, leading to the spread of the beam's energy, thus lowering the signal-to-noise ratio (SNR) at the receiver and increas-

In order to investigate the above effects, nondiffracting Bessel beams can be successfully adopted as alternative to Gaussian beams. An efficient FSO communication system should have a transmission beam as small as possible, with high peak intensity and high power. As deeply discussed in [18], these criteria are fully satisfied by Bessel beams. First of all, thanks to relation (15), the aperture radius required for a Bessel beam to transmit the half power is smaller than that required by a Gaussian beam, with a reduction of about 25% [18]. Furthermore, when comparing Bessel and Gaussian beams generated with the same aperture, the peak intensity of Bessel beam results to be about 1.2 times greater than that of the Gaussian

As a validation example, the intensity cross-sections for long range propagation at a distance of 22 km through atmosphere are simulated in Figure 8 for Bessel beams (left hand) and Gaussian beams (right hand) and three different aperture radii. The presented results are taken from [18], vertical axis giving the radial distance in [m] and horizontal axis representing the propagation distance in [km]. The outperformances of Bessel beams are

The actual challenge to really achieve FSO communication systems with improved power delivery features for long propagation distances still remains the realizations of launchers

(a) (b)

Figure 8. Intensity cross-sections for long-range propagation (taken from [18]): (a) Bessel beams and (b) Gaussian beams.

less complex than standard axicons but able to produce Bessel beams efficiently.

The basic features of nondiffracting Bessel beams have been reviewed in the present chapter, and mathematical discussions have been outlined to derive the relevant properties, such as the spot size, the maximum nondiffracting propagation range, and the delivered power. Assuming a waveguide-based structure as a beam launcher, it has been shown that three degrees of freedom exist to maximize the propagation distance of a Bessel beam without spreading, namely the radius of the aperture from which the beam is generated, the radial wavenumber (in turn depending on the launcher geometry and the dielectric medium properties), and the operating frequency (higher propagation ranges can be achieved when increasing frequency). Finally, potential applications of Bessel beam as an efficient information carrier for long range communication systems have been outlined.

## Author details

Sandra Costanzo

Address all correspondence to<EMAIL_ADDRESS>
University of Calabria, Italy

## References

Provisional chapter

## **Transionospheric Microwave Propagation: Higher-Order Effects up to 100 GHz** Transionospheric Microwave Propagation: Higher-Order Effects up to 100 GHz

Mohammed Mainul Hoque, Norbert Jakowski and

#### Jens Berdermann Mohammed Mainul Hoque, Norbert Jakowski and Jens Berdermann

[9] McGloin D, Garces-Chavez V, Dholakia K. Interfering Bessel beams for optical microma-

[10] Costanzo S, Di Massa G, Borgia A, Raffo A, Versloot T. Ariadna study 'Microwave

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[13] Costanzo S, Di Massa G, Borgia A, Raffo A, Versloot T, Summerer L. Microwave Bessel beam launcher for high penetration planetary drilling operations. In: 2016 10th European Conference on Antennas and Propagation, EuCAP 2016, Davos (Switzerland); April 11–

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14 Wave Propagation Concepts for Near-Future Telecommunication Systems

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Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/66659

## Abstract

Ionospheric refraction is considered as one of the major accuracy limiting factors in microwave space-based geodetic techniques such as the Global Positioning System (GPS), Satellite Laser Ranging (SLR), very-long-baseline interferometry (VLBI), Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), and satellite altimetry. Similarly, a high-performance ground-to-space and space-to-ground microwave link is considered to be very important for synchronizing clocks in global networks. Moreover, precise time and frequency transfer may lead to new applications in navigation, Earth observation, solar system science, and telecommunications. However, all transionospheric microwave signals are subject to ionospheric refraction and subsequent delays in the travel time. Since the ionosphere is a dispersive medium for radio signals, the first-order propagation effect can be removed by combining signals at two or more frequencies. Anyway, higher-order ionospheric effects remain uncorrected in such combinations. The residuals can significantly affect the accuracy of precise positioning, navigation, as well as the performance of time and frequency transfer. Here, we studied ionospheric propagation effects including higher-order terms for microwave signals up to 100 GHz frequencies. The possible combination between the L, S, C, X, Ku, and Ka band frequencies is studied for the first-order ionosphere-free solutions. We estimated the higher-order propagation effects such as the second- and third-order terms and raypath bending effects in the dual-frequency group delay and phase advance computation. Moreover, the correction formulas originally developed for global navigation satellite systems (GNSS) L-band frequencies are tested for mitigating residual errors at higher frequencies up to 100 GHz.

Keywords: ionospheric group delay, phase advance, ray-path bending, correction model

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

## 1. Introduction

The propagation of a radio wave through the ionospheric plasma can be described by the refractive index of the ionosphere given by the Appleton-Hartree formula [1]. At very high frequencies (>100 MHz), the refractive index depends mainly on the electron density and on the strength and the direction of the geomagnetic field in relation to the ray path. It becomes evident that the spatial distribution of the electron density along the ray path and the corresponding geomagnetic field relationships determine the ionospheric impact on the electromagnetic wave.

Since the ionosphere is a dispersive medium, radio wave propagation is frequency dependent. Therefore, by combining two signals, more than 99% of the ionospheric propagation delay can be corrected. However, higher-order propagation effects such as the second- and third-order terms in the refractive index and ray-path bending errors remain uncorrected in dual-frequency ionosphere-free combination. The range (or travel time) computation using transionospheric signal is affected up to several centimeters (or 1.e−9 s) due to higher-order ionospheric terms. Therefore, higher-order effects should not be neglected in precise time and positioning applications, especially during times of enhanced total electron content (TEC).

Early work was done by Brunner and Gu [2] in computing higher-order ionospheric effects and developing correction formulas for them. Since then higher-order ionospheric effects have been studied by different authors during the last two decades, e.g., Bassiri and Hajj [3], Jakowski et al. [4], Strangeways and Ioannides [5], Kedar et al. [6], Fritsche et al. [7], Hawarey et al. [8], Hoque and Jakowski [10–16], Hernandez-Pajares et al. [17], Datta-Barua et al. [18], Morton et al. [19], etc.

Brunner and Gu [2] considered the second-order term and the curvature correction term for the dual-frequency ionospheric correction of the Global Positioning System (GPS) observations. Similarly, Bassiri and Hajj [3] studied the second-order ionospheric term assuming an Earth-centered tilted dipole approximation for the geomagnetic field. Jakowski et al. [4] studied ionosphere induced ray-path bending effects in precise satellite positioning systems such as GPS (20,000 km altitude) and Navy Navigation Satellite System (NNSS) signals (1000 km altitude). Strangeways and Ioannides [5] considered the ratio of the curved path lengths and the geometric path length and the ratio of the TECs along GPS L1 and L2 ray paths for determining the geometric distance of the Earth-GPS path.

Hawarey et al. [8] investigated the second-order ionospheric term for very-long-baseline interferometry (VLBI) and found that further improvements can be achieved using a more realistic magnetic field model such as the International Geomagnetic Reference Field (IGRF) model. Fritsche et al. [7] found that Global Navigation Satellite Systems (GNSS) satellite positions can also be improved to the centimeter level by applying higher-order ionospheric corrections.

Datta-Barua et al. [18] estimated the higher-order ionospheric errors using data from the federal aviation administration's Wide Area Augmentation System (WAAS). They found that during ionospheric storms when slant range delays at GPS L1 can be as high as 100 m, the higher-order group errors in the GPS L1–L2 or L1–L5 dual-frequency combination can be tens of centimeters. Morton et al. [19] studied the second-order error analysis based on an extensive collection of electron density profiles measured by the Arecibo incoherent scatter radar and geomagnetic field vectors generated using the IGFR model.

Hoque and Jakowski [10–12] derived different approximation formulas to correct the secondand third-order terms, errors due to excess path length in addition to the free space path length, and TEC difference at two GNSS frequencies for L-band signals. Petrie et al. [21] investigated the potential effects of the bending terms on global GPS network.

Recently, Hernandez-Pajares et al. [17, 20] made a comprehensive summary of the secondorder effect in receiver position and clock, tropospheric delay, geocenter offset, and GNSS satellite position and clock. They considered all the relevant higher-order contributions such as the second- and third-order terms, geometric bending, and slant total electron content bending (i.e., the difference between the slant total electron content (STEC) for straight and bent paths).

In the present paper, we have investigated the ionospheric propagation effects including higher-order terms for microwave signals up to 100 GHz frequencies. The possible combination between the L, S, C, X, Ku, and Ka (1–2, 2–4, 4–8, 8–12, 12–18, 27–40 GHz, respectively) band frequencies is studied for the first-order ionosphere-free solutions. We estimated the remaining higher-order propagation effects such as the second- and third-order terms and ray-path bending effects in the dual-frequency group delay and phase advance computation.

In our previous work, we developed correction formulas for mitigating the second-order ionospheric term [10, 11], the third-order term, and errors due to ray-path bending [4, 12, 15, 16] in the dual-frequency ionosphere-free linear combination. The correction formulas are developed and validated mainly for precise (e.g., centimeter and millimeter level) GNSS positioning at L-band frequencies. In the present paper, we investigated the performance of these correction formulas for mitigating remaining errors at higher frequencies up to 100 GHz.

## 2. First-order ionospheric term

1. Introduction

16 Wave Propagation Concepts for Near-Future Telecommunication Systems

tromagnetic wave.

Morton et al. [19], etc.

The propagation of a radio wave through the ionospheric plasma can be described by the refractive index of the ionosphere given by the Appleton-Hartree formula [1]. At very high frequencies (>100 MHz), the refractive index depends mainly on the electron density and on the strength and the direction of the geomagnetic field in relation to the ray path. It becomes evident that the spatial distribution of the electron density along the ray path and the corresponding geomagnetic field relationships determine the ionospheric impact on the elec-

Since the ionosphere is a dispersive medium, radio wave propagation is frequency dependent. Therefore, by combining two signals, more than 99% of the ionospheric propagation delay can be corrected. However, higher-order propagation effects such as the second- and third-order terms in the refractive index and ray-path bending errors remain uncorrected in dual-frequency ionosphere-free combination. The range (or travel time) computation using transionospheric signal is affected up to several centimeters (or 1.e−9 s) due to higher-order ionospheric terms. Therefore, higher-order effects should not be neglected in precise time and positioning applications, especially during times of enhanced total electron content (TEC).

Early work was done by Brunner and Gu [2] in computing higher-order ionospheric effects and developing correction formulas for them. Since then higher-order ionospheric effects have been studied by different authors during the last two decades, e.g., Bassiri and Hajj [3], Jakowski et al. [4], Strangeways and Ioannides [5], Kedar et al. [6], Fritsche et al. [7], Hawarey et al. [8], Hoque and Jakowski [10–16], Hernandez-Pajares et al. [17], Datta-Barua et al. [18],

Brunner and Gu [2] considered the second-order term and the curvature correction term for the dual-frequency ionospheric correction of the Global Positioning System (GPS) observations. Similarly, Bassiri and Hajj [3] studied the second-order ionospheric term assuming an Earth-centered tilted dipole approximation for the geomagnetic field. Jakowski et al. [4] studied ionosphere induced ray-path bending effects in precise satellite positioning systems such as GPS (20,000 km altitude) and Navy Navigation Satellite System (NNSS) signals (1000 km altitude). Strangeways and Ioannides [5] considered the ratio of the curved path lengths and the geometric path length and the ratio of the TECs along GPS L1 and L2 ray paths for

Hawarey et al. [8] investigated the second-order ionospheric term for very-long-baseline interferometry (VLBI) and found that further improvements can be achieved using a more realistic magnetic field model such as the International Geomagnetic Reference Field (IGRF) model. Fritsche et al. [7] found that Global Navigation Satellite Systems (GNSS) satellite positions can also be improved to the centimeter level by applying higher-order ionospheric corrections.

Datta-Barua et al. [18] estimated the higher-order ionospheric errors using data from the federal aviation administration's Wide Area Augmentation System (WAAS). They found that during ionospheric storms when slant range delays at GPS L1 can be as high as 100 m, the higher-order group errors in the GPS L1–L2 or L1–L5 dual-frequency combination can be tens

determining the geometric distance of the Earth-GPS path.

The travel time τ of the signal for traveling the geometric distance between the transmitting satellite S and the receiver R can be written in units of seconds as (by dividing the true range expression given in [15] by the speed of light):

$$\tau = \frac{L}{c} + \frac{1}{c} \int\_{S}^{R} (1 - n) ds - t\_I^{len} \tag{1}$$

where the optical distance L ¼ ðR S nds is the line integral of the refractive index between the transmitting satellite and the receiver along the ray path, c is the velocity of light in a vacuum, 1 c ðR S ð Þ 1−n ds is the ionospheric time delay or advance, and t len I is the excess time delay due to ionospheric ray-path bending in which ds is the ray-path element and n is either group or phase refractive index of the ionosphere. The Appleton-Hartree formula of the refractive index and its expansion for phase advance and group delay can be found in [3, 9, 15] and references therein. The ionospheric group delay tIgr and phase advance tI can be written in units of seconds as (by dividing ionospheric delay expressions given in [15] by the speed of light):

$$t\_{l\mathcal{g}\tau} = \frac{1}{c} \int\_{S}^{R} (n\_{\mathcal{g}\tau} - 1) ds = \frac{p}{cf^2} \mp \frac{q}{cf^3} + \frac{u}{cf^4} \tag{2}$$

$$t\_l = \frac{1}{c} \int\_S^R (1 - n) ds = \frac{p}{c\mathfrak{f}^2} \mp \frac{q}{2c\mathfrak{f}^3} + \frac{u}{3c\mathfrak{f}^4} \tag{3}$$

where

$$p = 40.3 \int n\_e ds = 40.3 \cdot TEC \tag{4}$$

$$q = 2.2566 \times 10^{12} \int n\_e B \cos \Theta ds \tag{5}$$

$$
\mu = 2437 \int n\_e^2 ds + 4.74 \times 10^{22} \int n\_e B^2 (1 + \cos^2 \Theta) ds\tag{6}
$$

The upper (−) and lower (+) signs in the expressions (2) and (3) are related to the ordinary and extraordinary waves, respectively. The group refractive index ngr is greater than unity causing travel time greater than the speed of light, whereas the phase refractive index n is less than unity causing travel time less than the speed of light.

The quantity ne is the electron density, B is magnetic induction, and Θ is the angle between the Earth's magnetic field vector and the propagation vector. The presence of the Earth's magnetic field makes the ionosphere anisotropic that means the refractive index as well as the signal propagation depends on the propagation direction. The integral ð

neds along a signal path (i.e., curved path) is defined as the total electron content (TEC) and often measured in TEC units

Table 1. Estimates of the first-order term at different frequencies for TEC = 1 TECU.

(1 TECU = 1016 electrons/m2 ). The parameters p, q, and u given by Eqs. (4)–(6) together with signal frequency f determine the first-, second-, and third-order ionospheric refraction effects, respectively.

ionospheric ray-path bending in which ds is the ray-path element and n is either group or phase refractive index of the ionosphere. The Appleton-Hartree formula of the refractive index and its expansion for phase advance and group delay can be found in [3, 9, 15] and references therein. The ionospheric group delay tIgr and phase advance tI can be written in units of seconds as (by dividing ionospheric delay expressions given in [15] by the speed of light):

ngr � 1 � �ds ¼ p

ð Þ 1 � n ds ¼ p

ð

q ¼ 2:2566 · 1012

e ds þ 4:74 · 1022

The upper (−) and lower (+) signs in the expressions (2) and (3) are related to the ordinary and extraordinary waves, respectively. The group refractive index ngr is greater than unity causing travel time greater than the speed of light, whereas the phase refractive index n is less than

The quantity ne is the electron density, B is magnetic induction, and Θ is the angle between the Earth's magnetic field vector and the propagation vector. The presence of the Earth's magnetic field makes the ionosphere anisotropic that means the refractive index as well as the signal

curved path) is defined as the total electron content (TEC) and often measured in TEC units

Frequency [GHz] Ionospheric delay [s] Ionospheric delay [m]

 3.3607e−10 0.1007 2.1004e−11 0.0063 5.9745e−12 0.0018 1.4936e−12 4.4778e−04 3.7341e−13 1.1194e−04 1.3443e−13 4.0300e−05

p ¼ 40:3

cf 2 ∓ q cf 3 þ

cf 2 ∓ q 2cf 3 þ

ð

ð

neB2 1 þ cos 2

ð

u

u

neds ¼ 40:3∙TEC (4)

neBcosΘds (5)

Θ � �ds (6)

neds along a signal path (i.e.,

cf 4 (2)

3cf 4 (3)

tIgr ¼ 1 c ðR S

18 Wave Propagation Concepts for Near-Future Telecommunication Systems

tI ¼ 1 c ðR S

u ¼ 2437

unity causing travel time less than the speed of light.

ð n2

propagation depends on the propagation direction. The integral

Table 1. Estimates of the first-order term at different frequencies for TEC = 1 TECU.

where

Figure 1. Frequency dependence of the first-order term at different levels of ionospheric ionization and elevation angles. Left and right scales correspond to the units of seconds and meters, respectively.

The first-order ionospheric effect (group delay or phase advance) contributes above 99% of the total ionospheric effect and can be written as

$$t\_{lgr}^{(1)} = -t\_l^{(1)} = \frac{40.3}{cf^2} \left\{ n\_\epsilon ds = \frac{40.3}{cf^2} TEC \right. \tag{7}$$

in units of seconds where c is in m/s, TEC is in electrons/m2 , and f is in Hz. Eq. (7) indicates that the travel time caused by the first-order term is equal in magnitude but opposite in sign for group delay and phase advance.

The first-order term is directly proportional to the total number of electrons TEC, encountered by the signal during its travel through the ionosphere and inversely proportional to the square of the signal frequency. So, if frequency and link-related TEC are known, the first-order propagation effect can be derived by Eq. (7). The first-order ionospheric delay on different frequencies is determined for 1 TECU and given in Table 1.

The vertical TEC may vary between 1 TECU and 300 TECU depending on a number of factors such as local time, geographic/geomagnetic location, season, solar activity level, etc. The frequency dependence of the first-order ionospheric term up to 100 GHz frequency at different levels of ionospheric ionization characterized by the vertical total electron content (VTEC) = 300, 150, 50, and 5 TECU is plotted for elevations 10°, 30°, and 60° in Figure 1. The vertical total electron content of 300 and 150 TECU corresponds to VTEC during extreme space weather conditions, 50 TECU corresponds to midlatitude day time, and 5 TECU corresponds to midlatitude night time VTEC. The following obliquity factor or mapping function is used to convert the vertical TEC to slant TEC [15]:

Table 2. Estimates of the first-order term at different frequencies for VTEC = 300 and 5 TEC units and elevation = 10°.

$$MF = \frac{\text{STEC}}{\text{VTEC}} = \frac{h\_m + R\_E}{\sqrt{\left(h\_m + R\_E\right)^2 - \left(R\_h + R\_E\right)^2 \cos^2\beta}}\tag{8}$$

where hm is the maximum ionization height (e.g. 350 km), RE is the Earth's mean radius (~6371 km), Rh (~0) is the receiver height from the Earth's surface, and β is the elevation angle.

Table 2 gives estimates of the first-order term at 2, 8, 15, 30, 60, and 100 GHz frequencies for 10° elevation angle and VTEC = 300 and 5 TEC units.

As Figure 1 and Table 2 demonstrate, the first-order ionospheric delay can reach up to the 1.e −09 and 1.e−10 s levels at 30 and 60 GHz frequencies, respectively.

## 3. Higher-order terms

The first-order ionospheric effect (group delay or phase advance) contributes above 99% of the

the travel time caused by the first-order term is equal in magnitude but opposite in sign for

The first-order term is directly proportional to the total number of electrons TEC, encountered by the signal during its travel through the ionosphere and inversely proportional to the square of the signal frequency. So, if frequency and link-related TEC are known, the first-order propagation effect can be derived by Eq. (7). The first-order ionospheric delay on different

The vertical TEC may vary between 1 TECU and 300 TECU depending on a number of factors such as local time, geographic/geomagnetic location, season, solar activity level, etc. The frequency dependence of the first-order ionospheric term up to 100 GHz frequency at different levels of ionospheric ionization characterized by the vertical total electron content (VTEC) = 300, 150, 50, and 5 TECU is plotted for elevations 10°, 30°, and 60° in Figure 1. The vertical total electron content of 300 and 150 TECU corresponds to VTEC during extreme space weather conditions, 50 TECU corresponds to midlatitude day time, and 5 TECU corresponds to midlatitude night time VTEC. The following obliquity factor or mapping function is used to

VTEC [TECU]

First term

 2.81e−07 4.69e−09 1.76e−08 2.93e−10 6.00e−09 8.33e−11 1.25e−09 2.08e−11 3.13e−10 5.21e−12 1.13e−10 1.88e−12

300 5

[s] [s]

Table 2. Estimates of the first-order term at different frequencies for VTEC = 300 and 5 TEC units and elevation = 10°.

neds ¼ 40:3

cf 2 TEC (7)

, and f is in Hz. Eq. (7) indicates that

total ionospheric effect and can be written as

group delay and phase advance.

convert the vertical TEC to slant TEC [15]:

Frequency [GHz]

t ð Þ1 Igr ¼ �t

20 Wave Propagation Concepts for Near-Future Telecommunication Systems

in units of seconds where c is in m/s, TEC is in electrons/m2

frequencies is determined for 1 TECU and given in Table 1.

ð Þ1 I ¼ 40:3 cf 2 ð

> Although higher-order effects are less than 1% of the total ionospheric effects, they cannot be ignored in precise time and position applications. In precise applications dual-frequency measurements are commonly used to eliminate the major part (first-order effect) of the ionospheric propagation effects. The remaining higher-order terms can be up to 30 cm at L-band frequencies (15, 18). However, they will be much less at C, X, Ku, and K band frequencies. When analyzing ionospheric effects on signal propagation, it is a common practice to consider the code pseudo-range and carrier-phase observation equations for group delay and phase advance computation. Considering extraordinary wave of propagation, the code pseudo-range and carrier-phase expressions can be written in terms of ionospheric effects as (by dividing pseudo-range and carrier-phase expressions given in [12] by the speed of light):

$$
\Psi = \mathfrak{r} + t\_{lgr} + t\_l^{len} = \mathfrak{r} + \frac{p}{cf^2} + \frac{q}{cf^3} + \frac{u}{cf^4} + t\_l^{len} \tag{9}
$$

$$\Phi = \pi - t\_I + t\_I^{len} = \pi - \frac{p}{cf^2} - \frac{q}{2cf^3} - \frac{u}{cf^4} + t\_I^{len} \tag{10}$$

where τ is the travel time in vacuum. Although the measured travel times Ψ and Φ are biased by satellite and receiver clock errors, instrumental biases and tropospheric effect, and multipath effects, we have ignored them for simplicity in Eqs. (9) and (10). Due to the dispersive nature of the ionosphere, the propagation effect is frequency dependent, and we can eliminate the first-order term by combining signals at two different frequencies (f1, f2). The remaining terms can be written in units of seconds as (e.g., [12])

$$\frac{f\_1^2}{f\_1^2 - f\_2^2} \Psi\_1 - \frac{f\_2^2}{f\_1^2 - f\_2^2} \Psi\_2 = \tau - \Delta t\_{\rm TEC} - \Delta t\_2 - \Delta t\_3 - \Delta t\_{\rm len} \tag{11}$$

$$\frac{f\_1^2}{f\_1^2 - f\_2^2} \Phi\_1 - \frac{f\_2^2}{f\_1^2 - f\_2^2} \Phi\_2 = \tau + \Delta t\_{\rm TEC} + \frac{\Delta t\_2}{2} + \frac{\Delta t\_3}{3} - \Delta t\_{\rm lin} \tag{12}$$

$$\Delta t\_{\rm TEC} = \frac{40.3 \times (TEC\_2 - TEC\_1)}{c(f\_1^2 - f\_2^2)} = \frac{40.3 \times (\Delta TEC\_{bend2} - \Delta TEC\_{bend1})}{c(f\_1^2 - f\_2^2)}\tag{13}$$

$$TEC\_{1,2} = \int \mathfrak{n}\_{\varepsilon} d\mathfrak{s} = \left( TEC\_{LoS} + \Delta TEC\_{bound1,2} \right) \tag{14}$$

$$
\Delta t\_2 = \frac{2.2566 \times 10^{12} \int n\_e B \cos \Theta ds}{cf\_1 f\_2 (f\_1 + f\_2)} \tag{15}
$$

$$
\Delta t\_3 = \frac{2437}{cf\_1^2 f\_2^2} \left[ n\_e^2 ds + \frac{4.74 \times 10^{22}}{cf\_1^2 f\_2^2} \right] n\_e \mathcal{B}^2 \left( 1 + \cos^2 \Theta \right) ds \tag{16}
$$

$$
\Delta t\_{lcn} = \frac{t\_2^{lcn} f\_2^2 - t\_1^{lcn} f\_1^2}{c \left(f\_1^2 - f\_2^2\right)} \tag{17}
$$

where Ψ1 and Ψ2 are the measured pseudoranges and Φ1 and Φ1 are the measured carrier phases on f1 and f2. The terms Δt2 and Δt3 are the dual-frequency second- and third-order ionospheric terms. Due to the dispersive nature of the ionosphere, the bending effects as well as the total electron content (TEC) along the f1 path will be different from that along the f2 path. Considering this, the additional terms Δtlen and ΔtTEC are introduced in Eqs. (11) and (12) referring the dual-frequency residual errors due to the excess path and TEC difference, respectively. The quantities ΔTECbend1 and ΔTECbend2 are the excess TEC due to bending in addition to the straight line of sight (LoS) TEC, and t len 1 and t len 2 are the excess path length in addition to the LoS path length for f1 and f2 signals, respectively.

In the following sub sections, we estimated various dual-frequency residual terms for signal combination at different frequency bands considering different levels of ionospheric ionization.

#### 3.1. Dual-frequency second-order term

The imposed Earth's magnetic field causes an electron in the plasma to oscillate around the magnetic field line with the gyrofrequency fg = eB/(2πm) which is usually less than 1.4 MHz [22]. The value of B can be derived as ~5 +10−5 Tesla for fg = 1.4 MHz and considered constant throughout the propagation. Therefore, assuming the worst case condition with B = 5 +10−5 Tesla and Θ = 0, the maximum estimates of the dual-frequency second-order term (Eq. (15)) for group delay can be written as

$$
\Delta t\_2 = \frac{11.28 \times 10^7}{cf\_1 f\_2 (f\_1 + f\_2)} \left[ n\_e ds = \frac{11.28 \times 10^7}{cf\_1 f\_2 (f\_1 + f\_2)} \text{TEC} \tag{18}
$$

The elevation angle dependence of the Δt2 at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU is plotted in Figure 2.

Table 3 gives the maximum estimates of the second-order residual term at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU and elevation = 10°, 30°, and 60°.

ΔtTEC ¼ 40:3 · ð Þ TEC2 � TEC1 c f 2 1 � f 2 2

22 Wave Propagation Concepts for Near-Future Telecommunication Systems

TEC1, 2 ¼

ð n2 e ds þ

Δt3 ¼ 2437 cf 2 1f 2 2

to the straight line of sight (LoS) TEC, and t

3.1. Dual-frequency second-order term

[22].  The terms Δt2 and Δt3 are the dual-frequency second- and third-order ionospheric terms. Due to the dispersive nature of the ionosphere, the bending effects as well as the total electron content (TEC) along the f1 path will be different from that along the f2 path. Considering this, the additional terms Δtlen and ΔtTEC are introduced in Eqs. (11) and (12) referring the dual-frequency residual errors due to the excess path and TEC difference, respectively. The quantities ΔTECbend1 and ΔTECbend2 are the excess TEC due to bending in addition

> len 1 and t

In the following sub sections, we estimated various dual-frequency residual terms for signal combination at different frequency bands considering different levels of ionospheric ioniza-

The imposed Earth's magnetic field causes an electron in the plasma to oscillate around the magnetic field line with the gyrofrequency fg = eB/(2πm) which is usually less than 1.4 MHz

Tesla and Θ = 0, the maximum estimates of the dual-frequency second-order term (Eq. (15)) for

The elevation angle dependence of the Δt2 at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz

neds ¼ 11:28 · 107 cf 1f 2 f 1 þ f 2

+

Δt2 ¼ 11:28 · 107 cf 1f 2 f 1 þ f 2 � �

for VTEC = 300 and 5 TECU is plotted in Figure 2.

throughout the propagation. Therefore, assuming the worst case condition with B = 5

ð

len

c f 2 1 � f 2 2

neB2 1 þ cos 2

neds ¼ ð Þ TECLoS þ ΔTECband1, 2 (14)

� � (15)

� � (17)

2 are the excess path length in addition to

10−5 Tesla for fg = 1.4 MHz and considered constant

� � TEC (18)

+10−5

� � (13)

Θ � �ds (16)

Figure 2. Elevation angle dependence of the Δt2 at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU. Left and right scales correspond to the units of seconds and millimeters, respectively.

Table 3. Maximum estimates of the second-order residual term in the dual-frequency group delay computation.

Figure 2 and Table 3 show that the dual-frequency second-order residual term is bigger than the 1.e−14 and 1.e−15 s levels for frequency combinations (2–15) and (8–30) GHz, respectively, even at VTEC = 5 TECU. Therefore, the second-order residual term cannot be ignored if 1.e−16/ 1.e−17 s level accuracy is required in the time transfer.

#### 3.2. Dual-frequency third-order term

Eq. (16) indicates that the third-order residual term is proportional to the square of the electron density as well as geomagnetic induction B and angle between the geomagnetic field vector and propagation direction. The dependency on n2 e can be simplified assuming that the ionosphere is composed of Chapman layer function [23]. In this case, the so-called shape parameter η ¼ Ð n2 e ds Nm · TEC where Nm is the maximum ionization can be assumed to be 0.66 [2, 24], and thus the integral Ð n2 e ds in Eq. (16) can be written as 0.66NmTEC. Therefore, assuming the worst case condition with B = 5 +10−5 Tesla and Θ = 0, Eq. (16) can be simplified as

$$\Delta t\_3 = \frac{1608.4}{c f\_1^2 f\_2^2} N\_m T \text{EC} + \frac{23.7 \times 10^{13}}{c f\_1^2 f\_2^2} T \text{EC} = \left(1608.4 N\_m + 23.7 \times 10^{13}\right) \frac{T \text{EC}}{c f\_1^2 f\_2^2} \tag{19}$$

where Δt3 is measured in seconds, c is in meters, TEC is in electrons/ m2 , f in Hz, and the maximum ionization Nm is measured in electrons/m3 . The maximum ionization Nm can be estimated for a Chapman profile using the expression given by [12]

$$VTEC = 4.13HN\_m \tag{20}$$

where VTEC is the TEC in vertical direction and H is the atmospheric scale height. The VTEC will be measured in electrons/m2 when H is in meters and Nm is in electrons/m3 . The scale height H can be assumed as 70 km for a rough estimation of the third-order ionospheric term. The elevation angle dependence of the Δt3 at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU is plotted in Figure 3.

Table 4 gives the maximum estimates of the third-order residual term at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU and elevation = 10°, 30°, and 60°.

Figure 2 and Table 3 show that the dual-frequency second-order residual term is bigger than the 1.e−14 and 1.e−15 s levels for frequency combinations (2–15) and (8–30) GHz, respectively, even at VTEC = 5 TECU. Therefore, the second-order residual term cannot be ignored if 1.e−16/

Table 3. Maximum estimates of the second-order residual term in the dual-frequency group delay computation.

300 TECU 5 TECU

2–15 6.17e−12 3.88e−12 2.51e−12 1.03e−13 6.46e−14 4.19e−14 8–30 3.45e−13 2.17e−13 1.41e−13 5.75e−15 3.61e−15 2.34e−15 8–60 9.65e−14 6.06e−14 3.93e−14 1.61e−15 1.01e−15 6.55e−16

Elevation [deg] Elevation [deg]

10 30 60 10 30 60

Eq. (16) indicates that the third-order residual term is proportional to the square of the electron density as well as geomagnetic induction B and angle between the geomagnetic field vector

sphere is composed of Chapman layer function [23]. In this case, the so-called shape parameter

10−5 Tesla and Θ = 0, Eq. (16) can be simplified as

where VTEC is the TEC in vertical direction and H is the atmospheric scale height. The VTEC

height H can be assumed as 70 km for a rough estimation of the third-order ionospheric term. The elevation angle dependence of the Δt3 at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz

Table 4 gives the maximum estimates of the third-order residual term at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU and elevation = 10°, 30°, and 60°.

will be measured in electrons/m2 when H is in meters and Nm is in electrons/m3

23:7 · 1013 cf 2 1f 2 2

where Δt3 is measured in seconds, c is in meters, TEC is in electrons/ m2

estimated for a Chapman profile using the expression given by [12]

Nm · TEC where Nm is the maximum ionization can be assumed to be 0.66 [2, 24], and thus

e ds in Eq. (16) can be written as 0.66NmTEC. Therefore, assuming the worst case

TEC ¼ 1608:4Nm þ 23:7 · 1013 � � TEC

VTEC ¼ 4:13HNm (20)

e can be simplified assuming that the iono-

cf 2 1f 2 2

. The maximum ionization Nm can be

, f in Hz, and the

. The scale

(19)

1.e−17 s level accuracy is required in the time transfer.

24 Wave Propagation Concepts for Near-Future Telecommunication Systems

and propagation direction. The dependency on n2

NmTEC þ

maximum ionization Nm is measured in electrons/m3

for VTEC = 300 and 5 TECU is plotted in Figure 3.

+

Δt3 ¼ 1608:4 cf 2 1f 2 2

3.2. Dual-frequency third-order term

η ¼ Ð n2 e ds

the integral Ð

condition with B = 5

Freq-comb. (code) [GHz]

n2

Figure 3. Elevation angle dependence of the Δt3 at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU. Left and right scales correspond to the units of seconds and millimeters, respectively.

Figure 3 and Table 4 show that the dual-frequency third-order residual term can be bigger than the 1.e−13 and 1.e−15 s levels for frequency combinations (2–15) and (8–30) GHz, respectively, during times of high TEC such as VTEC = 300 TECU.

Table 4. Estimates of the third-order residual term in the dual-frequency group delay in units of seconds.

Figure 4. Elevation angle dependence of the ΔtTEC at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU. Left and right scales correspond to the units of seconds and millimeters, respectively.

## 3.3. Dual-frequency excess TEC term

Freq-comb. (code) [GHz]

300 TECU 5 TECU

26 Wave Propagation Concepts for Near-Future Telecommunication Systems

2–15 5.25e−13 3.30e−13 2.14e−13 2.66e−16 1.67e−16 1.08e−16 8–30 8.20e−15 5.15e−15 3.34e−15 4.16e−18 2.61e−18 1.69e−18 8–60 2.05e−15 1.29e−15 8.35e−16 1.04e−18 6.53e−19 4.24e−19

Figure 4. Elevation angle dependence of the ΔtTEC at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300

and 5 TECU. Left and right scales correspond to the units of seconds and millimeters, respectively.

Table 4. Estimates of the third-order residual term in the dual-frequency group delay in units of seconds.

Elevation [deg] Elevation [deg]

10 30 60 10 30 60

Due to ray-path bending or curvature effect, the estimate of electron density integrated over a curved path is slightly greater that that along the straight line of sight propagation (TECLoS). The excess TEC along a curved path in addition to the TECLoS can be written as [12]

$$
\Delta TEC\_{bend} = \frac{1.108 \times 10^{-3} \exp\left(-2.1844\beta\right) TEC^2}{f^2 H h\_m^{0.3}} \tag{21}
$$

where ΔTECbend is measured in TECU, atmospheric scale height H is in km, the maximum ionization height hm is in km, the signal frequency f is in GHz, TEC is in TECU, and elevation angle β is in radians. Typical values for the parameters H = 70 km and hm = 350 km are used in the present studies. Thus, knowing ΔTECbend at two frequencies, we can calculate ΔtTEC by Eq. (21). The elevation angle dependence of the ΔtTEC at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300, and 5 TECU is plotted in Figure 4.

Table 5 gives the maximum estimates of the residual excess TEC term at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU and elevation = 10°, 30°, and 60°.

Figure 4 and Table 5 show that the dual-frequency residual excess TEC term can be bigger than the 1.e−12 and 1.e−14 s levels for frequency combinations (2–15) and (8–30) GHz, respectively, at 10° elevation angle during times of high TEC such as VTEC = 300 TECU.

Eq. (21) requires the knowledge of the ionospheric parameters H and hmF2. If actual parameters are not known, the formula may not be used in practical cases. Considering this, Hoque and Jakowski [16] derived the following correction approach considering excess TEC dependency only on TEC and elevation angle.

$$
\Delta TEC\_{bend} = \frac{d\_1}{f^2} \left( \frac{1}{\sqrt{1 - d\_2 \cos^2 \beta}} - 1 \right) TEC^2 \tag{22}
$$

where d1 = 1.4563 and d2 = 0.8260. The coefficients are derived based on a nonlinear fit with ray-tracing results in least square senses. Although the correction given by the original approach Eq. (21) gives the best performance in comparison with the ray-tracing results, the correction results given by the new approach Eq. (22) is comparable.

Table 5. Estimates of the residual excess TEC term in the dual-frequency code combinations in units of seconds.

#### 3.4. Dual-frequency excess path term

Again the estimate of the total path length along a curved path is slightly larger than the LoS path length, and their difference is defined as the excess path length and denoted by t len I in

Figure 5. Elevation angle dependence of the Δtlen at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU. Left and right scales correspond to the units of seconds and millimeters, respectively.

Eq. (17). To estimate the excess path length, we follow the estimates based on numerical raytracing computations given by [4]

3.4. Dual-frequency excess path term

28 Wave Propagation Concepts for Near-Future Telecommunication Systems

Again the estimate of the total path length along a curved path is slightly larger than the LoS path length, and their difference is defined as the excess path length and denoted by t

Figure 5. Elevation angle dependence of the Δtlen at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and

5 TECU. Left and right scales correspond to the units of seconds and millimeters, respectively.

len I in

$$t\_I^{len} = \frac{b\_1}{cf^4} \left(\frac{1}{\sqrt{\left(1 - b\_2 \cos^2 \beta\right)}} - 1\right) TEC^2 \tag{23}$$

where b1 = 2.495 +108 , b2 = 0.859, and β is the elevation angle. The excess path length will be estimated in milliseconds when β is measured in radians, f is in MHz, TEC is in TECU, and c is in m/s. Now the dual-frequency residual excess path term Δtlen can be calculated by Eq. (17) in conjunction with Eq. (23). The elevation angle dependence of the Δtlen at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU is plotted in Figure 5.

Figure 5 and Table 6 show that the dual-frequency residual excess path term can be bigger than the 1.e−13 and 1.e−14 s levels for frequency combinations (2–15) and (8–30) GHz, respectively, at 10° elevation angle during times of high TEC such as VTEC = 300 TECU.

Table 6. Estimates of the residual excess path term in the dual-frequency group delay computation in units of seconds.

Table 6 gives the estimates of the residual excess path term at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU and elevation = 10°, 30°, and 60°.

The estimation of excess path length can be improved if additional ionospheric parameters such as scale height H and peak density heights hmF2 are known. Such a formula is derived by [12].

$$d\_{I}^{len} = \frac{7.5 \times 10^{-5} \exp\left(-2.13 \beta\right) \text{TEC}^2}{f^4 H \text{(hmF}\_2\text{)}^{1/8}} \tag{24}$$

where the excess path is measured in meters, TEC in TEC units, frequency f in GHz, atmospheric scale height H and height of maximum ionization hmF2 in kilometers, and elevation angle β in radians.

#### 3.5. Total dual-frequency residuals in group delay and phase advance

In the previous sections, we estimated different dual-frequency-residual terms separately. It is worthy to estimate their combined effects on group delay and phase advance computation.

Figure 6. Elevation angle dependence of the total residuals (absolute values) in group delay computation at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU. Left and right scales correspond to the units of seconds and millimeters, respectively.

Table 7. Estimates of the total residual error in the dual-frequency group delay in units of seconds.

3.5. Total dual-frequency residuals in group delay and phase advance

30 Wave Propagation Concepts for Near-Future Telecommunication Systems

In the previous sections, we estimated different dual-frequency-residual terms separately. It is worthy to estimate their combined effects on group delay and phase advance computation.

Figure 6. Elevation angle dependence of the total residuals (absolute values) in group delay computation at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU. Left and right scales correspond to the units of

seconds and millimeters, respectively.

Figure 7. Elevation angle dependence of the total residuals in phase advance measurements at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU. Left and right scales correspond to the units of seconds and millimeters, respectively.

Table 8. Estimates of the total residual term in the dual-frequency phase advance in units of seconds.

The elevation angle dependence in group delay (ΔtTEC + Δt2 + Δt3 + Δtlen, see Eq. 11) at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU is plotted in Figure 6.

Table 7 gives the maximum estimates of the total dual-frequency residuals in the group delay at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU and elevation = 10°, 30°, and 60°.

Similarly the elevation angle dependence of the total dual-frequency residuals in the phase advance (ΔtTEC + Δt2/2 + Δt3/3 − Δtlen, see Eq. (12)) at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU is plotted in Figure 7.

Table 8 gives the maximum estimates of the total dual-frequency residuals in the phase advance at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU and elevation = 10°, 30°, and 60°.

Figures 6 and 7 and Tables 7 and 8 show that the residual error in the dual-frequency group delay and phase advance cannot be ignored if 1.e−16/1.e−17 s level accuracy is required in the time transfer.

## 4. Higher-order term correction

In the following sections, we have discussed the possibility of higher-order propagation effects correction at very high frequencies.

## 4.1. Possibilities of second-order term correction

For the second-order term correction, the following correction formula is proposed by [11] for the global navigation satellite systems (GNSS) users in Europe with geographic latitude 30–65° N and longitude 15° W–45° E. Although the formula gives the best performance for the L-band signals, the second-order term correction at S (2–4 GHz), C (4–8 GHz), X (8–12 GHz), Ku (12– 18 GHz), K (18–27 GHz), and Ka (27–40 GHz) band signals is also possible by this formula.

Single frequency (deduced from Eqs. 2 and 5):

Transionospheric Microwave Propagation: Higher-Order Effects up to 100 GHz http://dx.doi.org/10.5772/66659 33

$$\mathbf{t}\_{\rm lgr}^{(2)} \mathbf{q} = \frac{2.2566 \times 10^{12}}{\mathbf{c} \mathbf{f}^3} \left[ \mathbf{n}\_{\rm t} \mathbf{B} \cos \Theta \mathbf{ds} = \frac{2.2566 \times 10^{12}}{\mathbf{c} \mathbf{f}^3} \overline{\mathbf{B} \cos \Theta} \times \text{TEC} \tag{25}$$

Dual frequency (deduced from Eq. 15):

$$
\Delta \mathbf{t}\_2 = \frac{2.2566 \times 10^{12} \times \overline{\mathbf{B} \cos \Theta} \times \text{TEC}}{\text{cf}\_1 \text{f}\_2 (\mathbf{f}\_1 + \mathbf{f}\_2)} \tag{26}
$$

In which

The elevation angle dependence in group delay (ΔtTEC + Δt2 + Δt3 + Δtlen, see Eq. 11) at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU is plotted in Figure 6. Table 7 gives the maximum estimates of the total dual-frequency residuals in the group delay at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU and

Table 8. Estimates of the total residual term in the dual-frequency phase advance in units of seconds.

300 TECU 5 TECU

2–15 4.27e−12 2.23e−12 1.36e−12 5.18e−14 3.24e−14 2.10e−14 8–30 1.91e−13 1.13e−13 7.19e−14 2.88e−15 1.81e−15 1.17e−15 8–60 5.39e−14 3.16e−14 2.01e−14 8.06e−16 5.05e−16 3.28e−16

Elevation [deg] Elevation [deg]

10 30 60 10 30 60

Similarly the elevation angle dependence of the total dual-frequency residuals in the phase advance (ΔtTEC + Δt2/2 + Δt3/3 − Δtlen, see Eq. (12)) at pair of frequencies (2, 15), (8, 30), and (8, 60)

Table 8 gives the maximum estimates of the total dual-frequency residuals in the phase advance at pair of frequencies (2, 15), (8, 30), and (8, 60) GHz for VTEC = 300 and 5 TECU

Figures 6 and 7 and Tables 7 and 8 show that the residual error in the dual-frequency group delay and phase advance cannot be ignored if 1.e−16/1.e−17 s level accuracy is required in the

In the following sections, we have discussed the possibility of higher-order propagation effects

For the second-order term correction, the following correction formula is proposed by [11] for the global navigation satellite systems (GNSS) users in Europe with geographic latitude 30–65° N and longitude 15° W–45° E. Although the formula gives the best performance for the L-band signals, the second-order term correction at S (2–4 GHz), C (4–8 GHz), X (8–12 GHz), Ku (12– 18 GHz), K (18–27 GHz), and Ka (27–40 GHz) band signals is also possible by this formula.

elevation = 10°, 30°, and 60°.

Freq-comb. (code) [GHz]

and elevation = 10°, 30°, and 60°.

4. Higher-order term correction

4.1. Possibilities of second-order term correction

Single frequency (deduced from Eqs. 2 and 5):

correction at very high frequencies.

time transfer.

GHz for VTEC = 300 and 5 TECU is plotted in Figure 7.

32 Wave Propagation Concepts for Near-Future Telecommunication Systems

Figure 8. Estimates of second-order term by ray-tracing simulation and correction formula at 30 GHz for VTEC = 300 and 5 TECU (top and bottom plots, respectively) at elevation = 10°, 30°, and 60°.

$$\begin{aligned} \overline{B\cos\Theta = -y\_1\cos\alpha + \left|\sqrt{r\_1^2 - y\_1^2\sin^2\alpha}\right| - 2r\_2\cos\alpha} \end{aligned} \tag{27}$$

where BcosΘ is the average value of the longitudinal component of the Earth's magnetic field along the ray path. The parameters r1, r2, and y1 are the functions of the receiver-to-satellite elevation angle β, geographic latitude ϕ, and longitude λ at the receiver position. The quantity α is the receiver-to-satellite azimuth angle and α' is the modified azimuth angle. For details of formulation, we refer to [11].

To check the performance of the correction formula at very high frequency, e.g., 30 GHz, we estimated the second-order term by a 2D ray-tracing tool [12] and also by the correction formula Eqs. (25) and (27). The effect of the Earth's magnetic field on the radio wave propagation is taken into account by considering the international geomagnetic field (IGRF) model in the ray-tracing tool. Figure 8 gives their comparisons.

We found that during times of high TEC such as VTEC = 300 TECU, the differences between the ray-tracing and correction results are in the level of 1.e−15 s.

Figure 9. Estimates of the third-order term by ray-tracing simulation and correction formula at 30 GHz for VTEC = 300 and 5 TECU as a function of the elevation.

#### 4.2. Possibilities of third-order term correction

BcosΘ ¼ �y1cosα þ

34 Wave Propagation Concepts for Near-Future Telecommunication Systems

the ray-tracing tool. Figure 8 gives their comparisons.

the ray-tracing and correction results are in the level of 1.e−15 s.

formulation, we refer to [11].

and 5 TECU as a function of the elevation.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 sin 2α

� � �

� � 2r2 cos α (27)

r2 1 � y2

where BcosΘ is the average value of the longitudinal component of the Earth's magnetic field along the ray path. The parameters r1, r2, and y1 are the functions of the receiver-to-satellite elevation angle β, geographic latitude ϕ, and longitude λ at the receiver position. The quantity α is the receiver-to-satellite azimuth angle and α' is the modified azimuth angle. For details of

To check the performance of the correction formula at very high frequency, e.g., 30 GHz, we estimated the second-order term by a 2D ray-tracing tool [12] and also by the correction formula Eqs. (25) and (27). The effect of the Earth's magnetic field on the radio wave propagation is taken into account by considering the international geomagnetic field (IGRF) model in

We found that during times of high TEC such as VTEC = 300 TECU, the differences between

Figure 9. Estimates of the third-order term by ray-tracing simulation and correction formula at 30 GHz for VTEC = 300

� � q � �

To correct the third-order ionospheric term, we proposed a correction formula based on analytical integration of the Chapman layer [12].

Single frequency (deduced from Eqs. 2 and 6):

$$n\_{lgr}^{(3)} = \frac{2487}{cf^4} \int n\_e^2 ds = \frac{1608}{cf^4} N\_m TEC \tag{28}$$

Dual frequency (deduced from Eqs. 15 and 36):

$$
\Delta t\_3 = \frac{1608}{c f\_1^2 f\_2^2} N\_m TEC \tag{29}
$$

The third-order term Δt3 will be measured in meters when f is measured in Hertz and the maximum ionization Nm and TEC in electrons/m3 and electrons/m2 , respectively.

The above correction formula was validated for the GNSS L-band signals in former studies. Therefore, to check the performance of the correction formula at very high frequency, e.g., 30 GHz, we have estimated the third-order term by a 2D ray-tracing tool [12] and also by the correction formula Eq. (28). Figure 9 gives their comparisons.

Figure 10. Estimates of the excess path term by ray-tracing simulation and correction formula at 30 GHz for VTEC = 300 TECU as a function of the elevation.

We see that during times of high TEC such as VTEC = 300 TECU, the differences between the ray-tracing and correction results are in the level of 1.e−18 s.

## 4.3. Possibilities of excess path term correction

To correct the excess path length, Jakowski et al. [4] proposed a correction formula based on simulation studies of the Chapman layers. The formula is already given by Eq. (23) for single frequency applications.

The correction formula was mainly derived for the GNSS L-band signals. Therefore, to check the performance of the correction formula at very high frequency, e.g., 30 GHz, we estimated the excess path length by the 2D ray-tracing tool [12] and also by the correction formula Eq. (23). Figure 10 gives their comparisons.

We found that during times of high TEC such as VTEC = 300 TECU, the differences between the ray-tracing and correction results are in the level of 1.e−17 s.

## 5. Conclusions

From our simulation studies, we have found that the time delay due to the first-order ionospheric term is in the range of 1.e−9–5.e−12 s depending on the ionospheric ionization and elevation angles at 30–60 GHz frequencies. The corresponding second- and third-order terms are in the ranges of 1.e−13–2.e−16 s and, respectively. The corresponding excess path length error due to bending is in the range of 1.e−15–2.e−20 s.

We found that for the second-order ionospheric term, the differences between the ray-tracing and correction results are in the order of 1.e−15 s for a microwave link at 30 GHz when assuming high vertical total electron content (VTEC) such as VTEC = 300 TECU. The corresponding differences are at the level of 1.e−18 and 1.e−17 s for the third-order and raypath bending terms, respectively.

European Space Agency (ESA) Atomic Clock Ensemble in Space (ACES) mission on board International Space Station (2014–2016) is aimed to transfer clock signals from low earth orbiting (LEO) satellites to the Earth using a time and frequency transfer link in the microwave domain supporting the clock performance at the level of 10−16 s [25]. High-performance frequency comparison of optical clocks between space and ground is essential for supporting future missions in the field of fundamental physics. Such a metrology link must provide frequency and time comparison and dissemination with an uncertainty level of 10−18 and beyond. Our investigation shows that for achieving such an accuracy level in the time and frequency transfer using trans-ionospheric microwave links, the higher-order ionospheric propagation effects must be corrected for.

## Author details

We see that during times of high TEC such as VTEC = 300 TECU, the differences between the

To correct the excess path length, Jakowski et al. [4] proposed a correction formula based on simulation studies of the Chapman layers. The formula is already given by Eq. (23) for single

The correction formula was mainly derived for the GNSS L-band signals. Therefore, to check the performance of the correction formula at very high frequency, e.g., 30 GHz, we estimated the excess path length by the 2D ray-tracing tool [12] and also by the correction formula

We found that during times of high TEC such as VTEC = 300 TECU, the differences between

From our simulation studies, we have found that the time delay due to the first-order ionospheric term is in the range of 1.e−9–5.e−12 s depending on the ionospheric ionization and elevation angles at 30–60 GHz frequencies. The corresponding second- and third-order terms are in the ranges of 1.e−13–2.e−16 s and, respectively. The corresponding excess path length

We found that for the second-order ionospheric term, the differences between the ray-tracing and correction results are in the order of 1.e−15 s for a microwave link at 30 GHz when assuming high vertical total electron content (VTEC) such as VTEC = 300 TECU. The corresponding differences are at the level of 1.e−18 and 1.e−17 s for the third-order and ray-

European Space Agency (ESA) Atomic Clock Ensemble in Space (ACES) mission on board International Space Station (2014–2016) is aimed to transfer clock signals from low earth orbiting (LEO) satellites to the Earth using a time and frequency transfer link in the microwave domain supporting the clock performance at the level of 10−16 s [25]. High-performance frequency comparison of optical clocks between space and ground is essential for supporting future missions in the field of fundamental physics. Such a metrology link must provide frequency and time comparison and dissemination with an uncertainty level of 10−18 and beyond. Our investigation shows that for achieving such an accuracy level in the time and frequency transfer using trans-ionospheric microwave links, the higher-order ionospheric

ray-tracing and correction results are in the level of 1.e−18 s.

36 Wave Propagation Concepts for Near-Future Telecommunication Systems

the ray-tracing and correction results are in the level of 1.e−17 s.

error due to bending is in the range of 1.e−15–2.e−20 s.

path bending terms, respectively.

propagation effects must be corrected for.

4.3. Possibilities of excess path term correction

Eq. (23). Figure 10 gives their comparisons.

frequency applications.

5. Conclusions

Mohammed Mainul Hoque\*, Norbert Jakowski and Jens Berdermann

\*Address all correspondence to<EMAIL_ADDRESS>
German Aerospace Center (DLR), Neustrelitz, Germany

## References

**Provisional chapter**

## **Ka-to-W Band EM Wave Propagation: Tropospheric Effects and Countermeasures Effects and Countermeasures**

**Ka-to-W Band EM Wave Propagation: Tropospheric**

Lorenzo Luini, Roberto Nebuloni and Carlo Riva

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/66660

Lorenzo Luini, Roberto Nebuloni

#### **Abstract**

and Carlo Riva

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[15] Hoque MM, Jakowski N. Ionospheric propagation effects on GNSS signals and new correction approaches. In: Shuanggen J, editor. Global Navigation Satellite Systems:

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38 Wave Propagation Concepts for Near-Future Telecommunication Systems

Signal, Theory and Applications. InTech; 2012. DOI: 10.5772/1134

2010, 162–173

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2013JB010568

111434172X

demic Press; 1969.

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DOI: 10.1029/2006JB004707

2009;44:RS1002. DOI: 10.1029/2008RS003888

JUL 2011

Near future satellite and terrestrial telecommunication (TLC) systems are expected to benefit from the use of operational frequencies spanning the Ka, Q, V and W bands, the main advantages being the availability of larger bandwidths and the smaller antenna size for a given gain. Moreover, the possibility of using on‐board antennas with enhanced directivity is attractive for satellite systems whose coverage area is subdivided into spot beams for frequency reallocation or regional services. For example, the W band is attrac‐ tive for fixed satellite services (FSS), especially for geostationary high‐throughput systems (HTSs), in which the use of such frequencies for the feeder link (i.e. large available band‐ width) could reduce significantly the number of gateways with respect to Ka and Q/V bands. As for deep space missions, the main driver for the interest in using frequencies in the Ka to W bands is the possible increase in the on‐board antenna gain with respect to the values at X band considered for current or planned missions. The drawback of using electromagnetic waves at frequencies in Ka, Q, V and W bands is the definite impact of the impairments caused by the troposphere. As a consequence, the design of TLC systems at such frequencies, and in particular satellite‐based ones, cannot rely on the classical approach of simply assigning an extra power margin to counteract atmospheric fades. The extensive use of fade mitigation techniques (FMTs), such as link power control (LPC), site diversity or on‐board adaptive power allocation, from the propagation side, adap‐ tive coding and modulation (ACM) and data rate adaptation (DRA), from the telecom‐ munication side, is mandatory. A reduction of the quality of service (QoS) should also be considered. This chapter deals with all these aspects characterizing the propagation of electromagnetic waves in the Ka, Q, V and W bands, spanning from the main impair‐ ments induced by the troposphere (and how they change as the frequency increases), to how extreme atmospheric conditions can be handled making use of suitable FMTs.

**Keywords:** electromagnetic wave propagation, atmospheric effects, satellite communications, fade mitigation techniques

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Tropospheric effects at Ka band and above**

Satellite and terrestrial telecommunication (TLC) using very high frequency bands (Ka band and above) benefit from the available larger bandwidths and the smaller antenna size for a fixed gain or the larger gain for a fixed antenna size. Unfortunately, radiowaves at Ka band and above are more severely affected by impairments due to tropospheric elements [1–4] and, in particular, gases, clouds, rain and turbulence.

Oxygen and water vapour are the relevant gaseous components for radio wave propaga‐ tion in the 20–300 GHz frequency range. Oxygen attenuation is weakly dependent on tem‐ perature and atmospheric pressure and its space and time variations are quite limited. The dependence on frequency is much larger with a resonant absorption band at 60 GHz (with specific attenuation that can exceed 10 dB/km) and smaller absorption (less than 0.1 dB/km) below 40–50 GHz and above 70–80 GHz. The water vapour absorption strongly depends on temperature, pressure and humidity, exhibiting marked daily, seasonal and geographical variations. Water vapour‐specific attenuation is characterized by a broadened peak around the H2 O absorption line at 22.2 GHz (up to about 0.2 dB/km in reference conditions1 ) and monotonically increases in the 40–100 GHz band (up to about 1 dB/km in reference condi‐ tions). As shown in **Figure 1** [5], the specific gas (oxygen and water vapour) attenuation in reference conditions is around 0.1 dB/km (with a peak of 0.2 dB/km at 22.2 GHz) in Ka band, while it increases with frequency from 0.1 to 0.4 dB/km in Q/V band. Due to the opposite

**Figure 1.** Oxygen (dashed line), water vapour (dotted line) and total gas (solid line) specific attenuation at reference conditions (water vapour density 7.5 g/m3, temperature 15°C, atmospheric pressure 1013 hPa).

1 Temperature 15°C, pressure 1013 hPa, water vapour density 7.5 g/m3 [5].

dependence of oxygen and water vapour attenuation on frequency, the specific gas attenu‐ ation in the W band is included in the interval between 0.36 and 0.5 dB/km, being higher at the bandwidth edges (70 and 95 GHz) and lower in the centre. The total gas attenuation along a slant path depends on the atmospheric profile of temperature, relative humidity and pressure. The models currently recommended by the International Telecommunication Union‐Radiocommunication (ITU‐R) sector for the prediction of the slant path gas attenu‐ ation [5] rely on solid physical bases and are expected to be quite accurate at least up to 100 GHz. Such models introduce simplifications relying on the description of vertical pro‐ files of oxygen and water vapour concentrations by using effective parameters, like the oxy‐ gen effective height (analytical function of frequency and pressure) and the water vapour total content (global maps of this parameter are available in Ref. [6]). The mean yearly dis‐ tribution of oxygen and water vapour attenuation at the zenith is calculated by multiply‐ ing their specific attenuation (in reference conditions) by the effective parameters and the slant path attenuation is then obtained through the cosecant law, reasonably assuming the local homogeneity of the atmosphere. As an example, the predicted zenith gas attenuation exceeding 1% of the average yearly time at 20 (Ka band), 50 (V band) and 80 (W band) GHz ranges almost all over the globe in the intervals 0.1–1, 1‐2.5 and 1–3 dB, respectively.

**1. Tropospheric effects at Ka band and above**

40 Wave Propagation Concepts for Near-Future Telecommunication Systems

in particular, gases, clouds, rain and turbulence.

the H2

1

Satellite and terrestrial telecommunication (TLC) using very high frequency bands (Ka band and above) benefit from the available larger bandwidths and the smaller antenna size for a fixed gain or the larger gain for a fixed antenna size. Unfortunately, radiowaves at Ka band and above are more severely affected by impairments due to tropospheric elements [1–4] and,

Oxygen and water vapour are the relevant gaseous components for radio wave propaga‐ tion in the 20–300 GHz frequency range. Oxygen attenuation is weakly dependent on tem‐ perature and atmospheric pressure and its space and time variations are quite limited. The dependence on frequency is much larger with a resonant absorption band at 60 GHz (with specific attenuation that can exceed 10 dB/km) and smaller absorption (less than 0.1 dB/km) below 40–50 GHz and above 70–80 GHz. The water vapour absorption strongly depends on temperature, pressure and humidity, exhibiting marked daily, seasonal and geographical variations. Water vapour‐specific attenuation is characterized by a broadened peak around

O absorption line at 22.2 GHz (up to about 0.2 dB/km in reference conditions1

monotonically increases in the 40–100 GHz band (up to about 1 dB/km in reference condi‐ tions). As shown in **Figure 1** [5], the specific gas (oxygen and water vapour) attenuation in reference conditions is around 0.1 dB/km (with a peak of 0.2 dB/km at 22.2 GHz) in Ka band, while it increases with frequency from 0.1 to 0.4 dB/km in Q/V band. Due to the opposite

**Figure 1.** Oxygen (dashed line), water vapour (dotted line) and total gas (solid line) specific attenuation at reference

conditions (water vapour density 7.5 g/m3, temperature 15°C, atmospheric pressure 1013 hPa).

Temperature 15°C, pressure 1013 hPa, water vapour density 7.5 g/m3 [5].

) and

In non‐precipitating atmosphere, the attenuation due to clouds must be also considered. Since in the 10–100 GHz band the size of suspended water droplets in clouds is smaller than the wavelength, the extinction cross sections can be calculated by using the Rayleigh approxi‐ mation, according to which the cloud attenuation turns out to depend only on the liquid water content, on the droplets temperature and on frequency. Cloud attenuation changes with climate and reveals a large temporal variability, depending on the presence of clouds along the link and on their liquid water content. It is monotonically increasing with frequency in the 10–100 GHz band. In ITU‐R recommendation P.840‐6 [7], cloud attenuation is calcu‐ lated through the reduced liquid water total content (kg/m2 ) or (mm), an effective parameter that depends on frequency and is available through global maps (presently calculated for the 20–50 GHz band) [7]. Cloud attenuation depends also on the water permittivity, the frequency and the elevation angle. The predicted zenith cloud attenuation exceeding 1% of the average yearly time at 20 GHz does not exceed 2 dB all over the globe, while it can exceed 10 dB in equatorial and tropical regions at 80 GHz, as shown in **Figure 2**. It is then clear that, even in absence of rain, cloud attenuation only limits availability in the W band.

As for precipitating atmosphere, the wavelength above 10 GHz becomes comparable to the size of hydrometeors (few millimetres), causing absorption and scattering, and consequently attenu‐ ation, of the incident wave. The rain attenuation along a slant path can be calculated by integrat‐ ing the specific attenuation, which statistically depends on rain intensity, *R* (mm/h), as follows:

$$\boldsymbol{\gamma} = k \, \mathbf{R}^a \tag{1}$$

In Eq. (1), *k* and *α* are coefficients (provided, for instance, in recommendation ITU‐R P.838‐3 [8]), which depend on frequency, elevation angle, drop temperature, wave polarization (as raindrops are not spherical), drop size distribution (DSD) and shape. DSD and shape can vary during a single event or from one event to the other, leading to significant differences in specific rain attenuation. In the W band, the effect of the DSD is more evident than in the

**Figure 2.** Zenith cloud attenuation (dB) at 80 GHz exceeded for 1%, prediction according to recommendation ITU‐R P.840‐6 [7].

lower bands, with a marked dependence on rain intensity: for example, at 10 mm/h, the peak to peak relative variation of the specific attenuation for various DSD is about 40% (for both polarizations). The difference between specific attenuation in vertical and horizontal polariza‐ tion is well below 5%. As for the prediction of rain attenuation in the 10–100 GHz band, the *k* and *α* coefficients in Ref. [8] can be used, though they originate from a best‐fitting procedure up to V band, and their accuracy in W band is still to be verified.

The rain attenuation exceeded for a certain time percentage in a year is related to the local rain rate exceeded for the same percentage, which may vary significantly across the globe; as an example, **Figure 3** shows the global map [9] of the rain rate (mm/h) exceeded for 0.01% of the average year (which corresponds approximately to 1 h in a year). The probability to have rain, *P*0 , is very close to the probability to have rain attenuation, *P*A; typically, for temperate climate, *P*0 ≈ 4–8% [9], and *P*A ≈ 6–10%, depending on the link elevation angle.

Several models have been developed in the last decades to predict rain attenuation statistics from the local rain rate statistics. Their accuracy in the 10–50 GHz range [10] is generally satis‐ factory. Since experimental statistics from earth‐to‐satellite links are not available for predic‐ tion models testing above 50 GHz, in this frequency range, physically based methods should

Ka-to-W Band EM Wave Propagation: Tropospheric Effects and Countermeasures http://dx.doi.org/10.5772/66660 43

**Figure 3.** Global map of rain rate (mm/h) exceeded for 0.01% of the average year.

lower bands, with a marked dependence on rain intensity: for example, at 10 mm/h, the peak to peak relative variation of the specific attenuation for various DSD is about 40% (for both polarizations). The difference between specific attenuation in vertical and horizontal polariza‐ tion is well below 5%. As for the prediction of rain attenuation in the 10–100 GHz band, the *k* and *α* coefficients in Ref. [8] can be used, though they originate from a best‐fitting procedure

**Figure 2.** Zenith cloud attenuation (dB) at 80 GHz exceeded for 1%, prediction according to recommendation ITU‐R

The rain attenuation exceeded for a certain time percentage in a year is related to the local rain rate exceeded for the same percentage, which may vary significantly across the globe; as an example, **Figure 3** shows the global map [9] of the rain rate (mm/h) exceeded for 0.01% of the average year (which corresponds approximately to 1 h in a year). The probability to have rain,

, is very close to the probability to have rain attenuation, *P*A; typically, for temperate climate,

Several models have been developed in the last decades to predict rain attenuation statistics from the local rain rate statistics. Their accuracy in the 10–50 GHz range [10] is generally satis‐ factory. Since experimental statistics from earth‐to‐satellite links are not available for predic‐ tion models testing above 50 GHz, in this frequency range, physically based methods should

up to V band, and their accuracy in W band is still to be verified.

42 Wave Propagation Concepts for Near-Future Telecommunication Systems

≈ 4–8% [9], and *P*A ≈ 6–10%, depending on the link elevation angle.

*P*0

P.840‐6 [7].

*P*0

be privileged, as their accuracy is not expected to depend significantly on the frequency. The SC EXCELL (Stratiform Convective EXponential CELL) model [11, 12], for example, describes the precipitation through a set of synthetic isolated rain cells with exponential rain rate profile; moreover, SC EXCELL takes separately into account stratiform and convective rain contributions, using different rain heights for the two types of precipitation. Finally, the con‐ tribution due to melting layer (the layer, lying just on the top of rain, where snow and ice par‐ ticles gradually melt during their fall to the ground) is added only to stratiform rain through an equivalent rain slab whose height depends on frequency. **Figure 4** shows an example of rain attenuation statistics (specifically, the complementary cumulative distribution function [CCDF]) predicted by SC EXCELL for a satellite slant path (elevation angle 35°) at 20, 50 and 80 GHz (vertical polarization). It is evident from the figure that, due to rain effects only, a system availability of 99.99% (outage of 0.01%) is realistic only in Ka band (20 GHz), with a propagation power margin of about 20 dB. In V (50 GHz) and W (80 GHz) band, a maximum availability of 99% is probably feasible, if no adequate fade mitigation techniques (FMTs) are implemented.

In case of dual polarization systems, another effect induced by rain must be taken into account: depolarization (specifically, cross polarization discrimination, XPD), which is introduced by non‐spherical drops. It is worthwhile observing that anisotropic ice crystals also produce the

**Figure 4.** CCDF of rain attenuation (predicted by the SC EXCELL model [11]) for a satellite slant path (elevation angle 35°) from a ground station located in Milano (lat. 45.5°N, long. 9.2°E, alt. 120 m a.m.s.l.) at 20 GHz (black solid line), 50 GHz (blue dashed line) and 80 GHz (red dotted line) with vertical polarization.

same effect in presence or absence of rain. The physical parameters that cause XPD are the anisotropy (i.e. the polarization dependent forward scattering of the particles, both for hydro‐ meteors and ice crystals) and the angle of the symmetry axes of the particles with respect to the polarization plane, named canting angle. Like rain attenuation, also XPD increases dra‐ matically with the frequency; by scaling anisotropies, it is possible to scale depolarization with frequency [13]. The hydrometeor axes, however, are not equi‐aligned, and the dispersion of these orientations generally reduces the XPD. This aspect and the co‐presence of water and ice particles along the path makes the depolarization phenomenon complex and difficult to describe. Ice XPD is the critical issue for low margin systems at high frequencies (above Ka band) since, during rainy periods, the system is probably already in outage. On the other hand, the XPD due to rain can have an impact when attenuation is close to the fade margin of gateway systems with high availabilities.

Finally, turbulence effect must be mentioned. Turbulent eddies in the troposphere cause small‐scale inhomogeneity of the atmospheric refractive index along the propagation path, which produces signal scintillation, that is, rapid fluctuations of the received signal ampli‐ tude, above about 10 GHz. Scintillation, when expressed in decibels, is characterized by a Gaussian distribution with a stationary standard deviation over few minutes. In turn, scintil‐ lation standard deviation depends on the turbulence structure index; it increases with the frequency (but much less than rain attenuation) [14] and with the path length, and it decreases with the increasing antenna size, due to the aperture averaging effect. Scintillation can also be due to the variation of the arrival angle of the main ray; in this case, it increases with the decreasing beamwidth as a result of the effect of the antenna radiation pattern [14]. The impact of scintillation, typically quite smaller than that of attenuation, can be important for low elevation angles and low margin systems. Moreover, scintillation can interfere with track‐ ing systems or fade mitigation techniques. Scintillation can also occur during rain events and, in this case, the signal variations are mixed with the fast fluctuations induced by rain itself.

## **2. Fade mitigation in the Q/V feeder link channel**

The actual generation of fixed satellite services (FSS) for TLC applications (e.g. Eutelsat KA‐ SAT and Hughes EchoStar XVII) achieve throughputs of the order of 100 Gbit/s by exploit‐ ing full Ka‐band operation, multibeam architectures with frequency reuse on the user link side, and a number of high‐capacity gateway (GW) links on the feeder link side. A further increase in the capacity of future HTS by an order of magnitude up to the Terabit/s is now under investigation as it would be supported by the growing market demand for broad‐ band access [15]. However, it involves several technological challenges as well. Designing the general architecture of an HTS is a complex task and it is beyond the scope of this chap‐ ter. Here the focus is on the impact of the propagation channel on the design of the feeder link, that is, the connection between the GW station and the satellite. To this aim, a reference scenario for a future HTS is sketched and the most suitable FMTs to counteract atmospheric attenuation are reviewed.

## **2.1. Number of beams**

same effect in presence or absence of rain. The physical parameters that cause XPD are the anisotropy (i.e. the polarization dependent forward scattering of the particles, both for hydro‐ meteors and ice crystals) and the angle of the symmetry axes of the particles with respect to the polarization plane, named canting angle. Like rain attenuation, also XPD increases dra‐ matically with the frequency; by scaling anisotropies, it is possible to scale depolarization with frequency [13]. The hydrometeor axes, however, are not equi‐aligned, and the dispersion of these orientations generally reduces the XPD. This aspect and the co‐presence of water and ice particles along the path makes the depolarization phenomenon complex and difficult to describe. Ice XPD is the critical issue for low margin systems at high frequencies (above Ka band) since, during rainy periods, the system is probably already in outage. On the other hand, the XPD due to rain can have an impact when attenuation is close to the fade margin of

**Figure 4.** CCDF of rain attenuation (predicted by the SC EXCELL model [11]) for a satellite slant path (elevation angle 35°) from a ground station located in Milano (lat. 45.5°N, long. 9.2°E, alt. 120 m a.m.s.l.) at 20 GHz (black solid line), 50

GHz (blue dashed line) and 80 GHz (red dotted line) with vertical polarization.

44 Wave Propagation Concepts for Near-Future Telecommunication Systems

Finally, turbulence effect must be mentioned. Turbulent eddies in the troposphere cause small‐scale inhomogeneity of the atmospheric refractive index along the propagation path, which produces signal scintillation, that is, rapid fluctuations of the received signal ampli‐ tude, above about 10 GHz. Scintillation, when expressed in decibels, is characterized by a Gaussian distribution with a stationary standard deviation over few minutes. In turn, scintil‐ lation standard deviation depends on the turbulence structure index; it increases with the frequency (but much less than rain attenuation) [14] and with the path length, and it decreases with the increasing antenna size, due to the aperture averaging effect. Scintillation can also

gateway systems with high availabilities.

A simple calculation of the required number of GW beams and user beams to achieve Terabit/s capacity can be carried out once the available bandwidth, the frequency reuse scheme and the modulation scheme are given. An important assumption is that the traffic is uniform, that is, the system capacity is shared in equal parts among the user beams as well as among the GW beams. If the same MODCOD (MODulation and CODing) is transmitted, the relationship between total satellite capacity *C*tot (one‐way) and number of user beams *N*UB is given as:

$$\mathbf{C}\_{\text{tot}} = N\_{\text{tun}} \frac{N\_p}{F} \mathbf{B} \times \mathbf{S} E \tag{2}$$

where *N*p is the number of polarizations, *F* is the frequency reuse factor, *B* is the bandwidth per beam, and *SE* is the spectral efficiency of the MODCOD. For instance, assuming *B* = 0.5 GHz/ beam, that is, the entire exclusive bandwidth available for FSS across the European Community [16], *SE* = 3.433 (corresponding to the 32 APSK 5/6), *N*p = 2 and *F* = 4, we obtain

$$N\_{\rm tB} = C\_{\rm tot} \frac{F}{N\_p} \frac{1}{B \times SE} = 500 \frac{4}{2} \frac{1}{0.5 \times 3.433} = 583\tag{3}$$

being the 1 Terabit/s shared between uplink and downlink. Furthermore, the capacity on the user link side must equal the one of the feeder link, that is

$$C\_{\rm GW} N\_{\rm GW} = C\_{\rm UV} N\_{\rm UV} \tag{4}$$

In the Q‐V band, a much larger bandwidth (4–5 GHz) would be available even though in Europe there are no exclusive bandwidths for FSS [16]. Assuming that each GW has the entire Q‐V bandwidth at its disposal, that is, *B* = 4 GHz, and *SE* = 3.433, *N*p = 1 and *F* = 1, the required number of GW beams would approximately be equal to

$$N\_{\rm CW} = 500 \frac{1}{4 \times 3.433} = 36\tag{5}$$

Hence, each GW would serve 4×2/0.5 = 16 user beams.

In the design practice, other factors contribute to find the optimum number of spots, such as available on‐board power, complexity of the on‐board antenna system, interference issues, etc.

#### **2.2. Space diversity**

The drawback of higher frequencies, specifically Q/V‐band, is the increased vulnerability to the propagation impairments. Rain fades at Q/V‐band are much larger than the dynamic range of receiver systems with typical feeder link availability targets (99.9%), even when uplink power control is featured. To this aim, a well‐known FMT is single site diversity (SSD), where a GW is backed up by an idle GW located within the same feeder beam (**Figure 5**).

**Figure 5.** Single site diversity across the feeder link of an FSS.

When GW1 is faded, the traffic is switched to GW2 which transmits the same set of carriers as GW1 . The payload is therefore transparent to the process of allocating users from the GW in outage to the idle GW (switchover). The higher the distance between GW1 and GW2 the lower the probability that both are faded.

In the Q‐V band, a much larger bandwidth (4–5 GHz) would be available even though in Europe there are no exclusive bandwidths for FSS [16]. Assuming that each GW has the entire Q‐V bandwidth at its disposal, that is, *B* = 4 GHz, and *SE* = 3.433, *N*p = 1 and *F* = 1, the required

In the design practice, other factors contribute to find the optimum number of spots, such as available on‐board power, complexity of the on‐board antenna system, interference issues, etc.

The drawback of higher frequencies, specifically Q/V‐band, is the increased vulnerability to the propagation impairments. Rain fades at Q/V‐band are much larger than the dynamic range of receiver systems with typical feeder link availability targets (99.9%), even when uplink power control is featured. To this aim, a well‐known FMT is single site diversity (SSD), where a GW is backed up by an idle GW located within the same feeder beam (**Figure 5**).

4 × 3.433 = 36 (5)

number of GW beams would approximately be equal to *N*GW = 500 \_\_\_\_\_\_\_ 1

46 Wave Propagation Concepts for Near-Future Telecommunication Systems

Hence, each GW would serve 4×2/0.5 = 16 user beams.

**Figure 5.** Single site diversity across the feeder link of an FSS.

**2.2. Space diversity**

Conventional SSD, relying on the availability of an idle GW to back up a faded GW, is not a cost‐effective solution when many GWs are required to convey the user beams traffic, as it basically doubles the cost of the ground segment. On the other side, the diversity scheme in **Figure 6** achieves the required availability target while minimizing the number of redundant GWs with respect to SSD. There are *N* operational GWs and *P* < *N* redundant GWs (*N* = 2 and *P* = 1 in the figure, for the purpose of illustration), each located within its own feeder beam. The redundant capacity provided by the *P* back‐up GWs is shared among the user beams: when one of the operational GWs (GW1 or GW2 ) is faded, it is replaced by GW3 . The above "*N* + *P*" scheme tolerates up to *P* GWs in outage with no loss of capacity. As a drawback, *N* + *P* implies a higher degree of complexity both on the ground segment and on‐board. In fact, the GWs must be interconnected by high‐speed fibre optic links to form a smart‐gateway network (SGN). For instance, in **Figure 6**, the GWs nodes form a star topology where the central node, namely the

**Figure 6.** *N + P* diversity scheme across the feeder link of an FSS.

network operation centre (NOC), carries out resource management and controls traffic swi‐ tchover. Moreover, the payload should host *N* + *P* (instead of *N*) receiver‐transmitter front‐end chains and should provide rerouting capability from every user beam to any of the back‐up GWs. In general, a trade‐off between flexibility in managing system resources and complexity (both ground segment and payload) must be addressed.

With SSD and N + P diversity, a user beam is served by a single GW at a time. More complex diversity schemes have been proposed as well, where a single user‐beam is connected to sev‐ eral GWs at a time through frequency division multiple access (FDMA) [17]. When a GW is in outage, the user beam traffic conveyed by its carriers is switched to a different GW serving the same set of beams as the faded GW. If the system does have some redundancy (i.e. unused carriers in ideal conditions), an outage event does not imply a loss of capacity. These solutions are more challenging than *N* + *P* as they envisage a number of complex tasks including evalu‐ ation and distribution of the available resources, signalling and synchronization to the new carrier and assignment of the user terminals among the new set of carriers.

## **2.3.** *N* **+** *P* **performance**

Let us now assess the effectiveness of *N* + *P* in mitigating the impact of the atmospheric chan‐ nel on the Q‐V band feeder link for the reference scenario described above. The following figures of merit are considered:

The propagation margin can be calculated from the CCDF of rain attenuation at each GW site and basic statistical theory, provided that the rain process is assumed to occur independently over different GW links. On the other side, a full characterization of the propagation channel can be carried out only by models able to generate realistic time series of attenuation in every GW site, possibly taking into account the correlation of the rain process in time and space. Here, we have used the multisite time series synthesizer (MTS) proposed in Ref. [18] to simu‐ late the rainy atmospheric channel for a V‐band (50 GHz) feeder link (we consider here uplink only). The MTS generates synthetic time series of rain attenuation across an arbitrary ensem‐ ble of slant paths up to the V‐band by combining series of measured data carried out during the ITALSAT propagation campaign. First‐order statistics are reproduced and the spatial cor‐ relation of rain is duly taken into account. The contribution of non‐rainy components, that is, clear‐sky attenuation, is obviously non‐negligible at V‐band, as shown in the first part of this chapter. However, clear‐sky attenuation is on the order of few decibels and though time vari‐ ant, its fluctuations can be managed by FMTs such as uplink power control. On the other side, the peaked and fast fades produced by rain prompt for an approach based on space diversity.

Without loss of generality, let us consider a system of 10 GWs operating in a 9 + 1 diver‐ sity scheme. In fact, the 36‐GW system necessary to achieve the Terabit/s capacity could be divided into four independent clusters of 9 + 1 GWs each. Indeed, this architecture would result in a decrease in the number of high‐speed connections among the GWs, with respect to, for example, a 36 + 4 diversity scheme. The 10 selected GW sites are reported in **Table 1**, assuming a GEO satellite providing continental coverage across Europe, alongside the clima‐ tologic probability of rain and the slant‐path rain attenuation exceeded for 0.1 and 0.01% of time, respectively. The distance between the stations that should be maximized in the design process ranges from less than 500 km to about 3200 km.

network operation centre (NOC), carries out resource management and controls traffic swi‐ tchover. Moreover, the payload should host *N* + *P* (instead of *N*) receiver‐transmitter front‐end chains and should provide rerouting capability from every user beam to any of the back‐up GWs. In general, a trade‐off between flexibility in managing system resources and complexity

With SSD and N + P diversity, a user beam is served by a single GW at a time. More complex diversity schemes have been proposed as well, where a single user‐beam is connected to sev‐ eral GWs at a time through frequency division multiple access (FDMA) [17]. When a GW is in outage, the user beam traffic conveyed by its carriers is switched to a different GW serving the same set of beams as the faded GW. If the system does have some redundancy (i.e. unused carriers in ideal conditions), an outage event does not imply a loss of capacity. These solutions are more challenging than *N* + *P* as they envisage a number of complex tasks including evalu‐ ation and distribution of the available resources, signalling and synchronization to the new

Let us now assess the effectiveness of *N* + *P* in mitigating the impact of the atmospheric chan‐ nel on the Q‐V band feeder link for the reference scenario described above. The following

• The propagation margin, that is, the extra decibel margin to be included in the link budget

• The duration and the rate of fades, which, in turn, have an impact on the algorithms con‐