Predicting the strength of a radio signal in the shadow of an obstacle is a vital function for propagation engineers. The mechanism by which a wave enters into the shadow of an obstacle is known as diffraction. Even the simplest of practical obstacles pose severe mathematical challenges. More easily solved approximations are adopted in order to estimate the strength of diffracted signals. The starting point for diffraction problems is the case where a receiver is in the shadow of a perfectly absorbing ‘knife-edge’ obstacle. This is then extended to encompass the situation where there are several such obstacles on the path. Many approximate multiple-knife-edge prediction methods exist and the most commonly used are analysed and compared. More accurate ‘near-exact’ methods are discussed. Although these methods usually make better predictions of the signal strength in the shadow of obstacles, they require significantly more computing time as well as being significantly more complicated to implement. Once an understanding of the properties of a diffracted signal has been obtained, it is possible to derive clearance requirements for a point-to-point path so that diffraction effects may be safely ignored. The insights gained by investigating the mechanism of diffraction into the shadow of an obstacle can be used to analyse two related phenomena: reflection from a finite surface and the formation of the radiation pattern of an aperture antenna.

Knife-edge diffraction

Diffraction is the name given to the mechanism by which waves enter into the shadow of an obstacle.

Point-to-area transmission is the generic name given to the way in which broadcasting transmitters or base stations for mobile communications provide coverage to a given area. A general overview is provided in order to deliver the most significant information as quickly as possible. Following that, the concept of electric field strength as an alternative to power density, prediction methods and the effect of frequency are explained in more detail. Digital mobile radio is selected for further study as a specific example of point-to-area communication. Path-loss-prediction methods specific to digital mobile radio are examined and various types of antenna that are used for the base stations of these systems are described. Finally, the effect that interference can have on the coverage range of a base station is explained.

Overview

The simplest form of antenna used is an ‘omni-directional’ antenna. These radiate equally in all directions in the horizontal plane. They often have a narrower beam (perhaps less than 20 degrees) in the vertical plane. Such antennas typically have a gain of 10 dBi. Two collinear wire elements that are fed at their junction with a signal form what is known as a dipole antenna. The basic omni-directional antenna is a dipole that is half a wavelength in height (a ‘half-wave dipole’ with each of the wire elements being a quarter of a wavelength in length). This has a wide vertical beam and has a gain of 2.1 dBi.

A further benefit from using the decibel scale becomes apparent when dealing with what is known as the link budget. Using decibels, a power budget on a radio link becomes as straightforward as a simple financial budget. Transmit power and gains can be thought of as equivalent to income, with losses and required margins being equivalent to expenditure.

For example, suppose that we transmit with a power of 30 dBm (1 watt). There are feeder losses, antenna gains, free-space loss, absorption loss, fading margin etc. The link budget is really a method of organising these parameters so as to make the calculation of the received signal level (under conditions of maximum fade, if the fade margin is considered) as straightforward as possible. The received signal level should be sufficient to deliver an acceptably low bit error ratio. Table A4.1 gives an example of such a link budget.

An attempt has been made to provide an insight into the way in which signal strength can be predicted for a variety of situations. Information has been presented with the intention of stimulating an intuitive understanding of radio wave propagation together with essential formulas that will allow rapid estimates of signal strength to be made. It is the sort of information that experienced radio-propagation engineers will carry around in their heads (with the exception of the more complicated equations). Further, detailed information will be gained from consulting more detailed books such as those recommended as further reading and the ITU recommendations (also listed). Further, a radio-propagation engineer will often have software modules available that implement the ITU recommendations and other methods for propagation prediction such as the Okumura–Hata method.

Although radio wave propagation is really a single subject, all the diverse factors that affect the strength of a received signal make a comprehensive calculation of signal strength almost impossible. As a result, radio-propagation engineers concentrate on the factors that have the most significant effect for the circumstances in hand. It is seen that the task of predicting the signal received when propagation is in free space is relatively straightforward and depends upon antenna gains, path length and frequency. The concept of antennas possessing gain, although they are passive devices, is explained: the ‘gain’ is associated with the ability of an antenna to direct the transmitted energy in the required direction and prevent the energy spreading as it travels.

The objective of this book is to allow the reader to predict the received signal power produced by a particular radio transmitter. The first two chapters examine propagation in free space for point-to-point and point-to-area transmission, respectively. This is combined with a discussion regarding the characteristics of antennas for various purposes. In chapter 3, the effect of obstacles, whether buildings or mountains, is discussed and analytical methods, whereby the strength of a signal is the shadow of an obstacle can be predicted, are presented. The following chapter investigates the nature of reflections and the effect that reflections have on the nature of a received signal. Chapter 5 shows how the level of a received signal can be predicted considering all propagation mechanisms. The many effects on a radio wave that are caused by precipitation and the structure of the atmosphere are explained in chapter 6. Chapter 7 demonstrates how knowledge gained can be used to design point-to-point radio links, broadcast systems, Earth–space systems and in-building systems. In chapter 8, the value of software tools in the planning of various radio networks is explained.

A radio wave can often travel from a transmitter to a particular point by a number of routes: directly, by diffraction, by reflection, by penetration. At any point, the power received by a receiving antenna will be a combination of all these propagation mechanisms. Because the combined signal is a phasor sum of all the individual contributions an accurate prediction of the electric field strength is very difficult to obtain: it would need knowledge of the distance travelled for each propagation mechanism to within about a tenth of a wavelength plus details of the electrical properties of any materials involved in the paths involving reflection or penetration. Usually, all that is practical is to estimate the strength of the signal that would be achieved by each propagation path in isolation. The total received power is then estimated as the sum of these individual contributions. This gives an estimate of what is called the ‘local-mean’ level. That means that the actual power received would vary about this level by an amount that depends upon the relative strengths and directions of the individual contributions. If the angular separation of the individual contributions, when viewed from the receiver, is small then the signal will not vary very quickly with distance. Further, if one of the contributions is dominant and provides the majority of the signal power on its own then the variation will not be as great as if all the different propagation paths contributed nearly equal amounts of power.

The test of the usefulness of the knowledge gained can best be determined by undertaking some practical exercises that give an insight into problems encountered by radio system designers in the ‘real world’. Firstly, the value of propagation studies in helping to identify the most appropriate frequency for various services is discussed. The system design of microwave links at 10 GHz (at which multipath fading will dominate) and at 23 GHz (at which rain fading will dominate) is explained in some detail. The fact that many thousands of microwave links will be required in an industrialised country leads to a need for interference management, so this topic is introduced. Next, attention is turned to the design of broadcasting systems with a view to obtaining maximum coverage whilst investigating methods of limiting interference at great distances. Additionally, an example of designing a link to a geostationary satellite is presented. Finally, special methods needed for providing and predicting the signal strength for in-building systems are presented.

Determining the most appropriate frequencies for specific services

We have seen that the frequency of operation affects the way in which a radio wave is affected both by obstacles and by the atmosphere and rain. At first sight, it seems that the lower the frequency, the better. Obstacles cause lower levels of diffraction loss at longer wavelengths. Also, the effect of rain and atmospheric absorption is almost negligible below about 5 GHz. Further, the penetration of materials such as concrete is better at lower frequencies.

This chapter introduces the basic concepts of radio signals travelling from one antenna to another. The aperture antenna is used initially to illustrate this, being the easiest concept to understand. The vital equations that underpin the day-to-day lives of propagation engineers are introduced. Although this chapter is introductory in nature, practical examples are covered. The approach adopted is to deliver the material, together with the most significant equations, in a simplified manner in the first two subsections before providing more detail. Following this, the focus is on developing methods of predicting the received signal power on point-to-point links given vital information such as path length, frequency, antenna sizes and transmit power.

Propagation in free space: simplified explanation

Radio waves travel from a source into the surrounding space at the ‘speed of light’ (approximately 3.0 × 108 metres per second) when in ‘free space’. Literally, ‘free space’ should mean a vacuum, but clear air is a good approximation to this. We are interested in the power that can be transmitted from one antenna to another. Because there are lots of different antennas, it is necessary to define a reference with which others can be compared. The isotropic antenna in which the transmitted power is radiated equally in all directions is commonly used as a reference. It is possible to determine the ratio between the power received and that transmitted in linear units, but it is more common to quote it in decibels (dB).

Chapter 1 introduced expressions which define the various saddlepoint approximations along with enough supplementary information to allow the reader to begin making computations. This chapter develops some elementary properties of the approximations which leads to further understanding of the methods. Heuristic derivations for many of the approximations are presented.

Simple properties of the approximations

Some important properties possessed by saddlepoint density/mass functions and CDFs are developed below. Unless noted otherwise, the distributions involved throughout are assumed to have MGFs that are convergent on open neighborhoods of 0.

The first few properties concern a linear transformation of the random variable X to Y = σX + μ with σ ≠ 0. When X is discrete with integer support, then Y has support on a subset of the σ-lattice {μ,μ ± σ, μ ± 2σ, …}. The resulting variable Y has a saddlepoint mass and CDF approximation that has not been defined and there are a couple of ways in which to proceed. The more intriguing approach would be based on the inversion theory of the probability masses, however, the difficulty of this approach places it beyond the scope of this text. A more expedient and simpler alternative approach is taken here which adopts the following convention and which leads to the same approximations.

Lattice convention. The saddlepoint mass function and CDF approximation for lattice variable Y, with support in {μ, μ ± σ,μ ± 2σ, …} for σ > 0, are specified in terms of their equivalents based on X = (Y − μ) /σ with support on the integer lattice.