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.

Approximations to continuous univariate CDFs of MLEs in curved exponential and transformation families have been derived in Barndorff-Nielsen (1986, 1990, 1991) and are often referred to as r * approximations. These approximations, along with their equivalent approximations of the Lugannani and Rice/Skovgaard form, are presented in the next two chapters. Section 8.2 considers the conditional CDF for the MLE of a scalar parameter given appropriate ancillaries. The more complex situation that encompasses a vector nuisance parameter is the subject of chapter 9.

Other approaches to this distribution theory, aimed more toward p-value computation, are also presented in section 8.5. Fraser and Reid (1993, 1995, 2001) and Fraser et al. (1999a) have suggested an approach based on geometrical considerations of the inference problem. In this approach, explicit ancillary expressions are not needed which helps to simplify the computational effort. Along these same lines, Skovgaard (1996) also offers methods forCDF approximation that are quite simple computationally. Specification of ancillaries is again not necessary and these methods are direct approximations to the procedures suggested by Barndorff-Nielsen above.

Expressions for these approximate CDFs involve partial derivatives of the likelihood with respect the parameter but also with respect to the MLE and other quantities holding the approximate ancillary fixed. The latter partial derivatives are called sample space derivatives and can be difficult to compute. An introduction to these derivatives is given in the next section and approximations to such derivatives, as suggested in Skovgaard (1996), are presented in appropriate sections.

The ratio R = U/V of two random variables U and V, perhaps dependent, admits to saddlepoint approximation through the joint MGF of (U, V). If V > 0 with probability one, then the Lugannani and Rice approximation may be easily applied to approximate the associated CDF. Saddlepoint density approximation based on the joint MGF uses the Geary (1944) representation for its density. This approach was first noted in Daniels (1954, –9) and is discussed in section 12.1 below.

The ratio R is the root of the estimating equation U − RV = 0 and the distribution theory for ratios can be generalized to consider distributions for roots of general estimating equations. The results of section 12.1 are subsumed into the more general discussion of section 12.2 that provides approximate distributions for roots of general estimating equations. Saddlepoint approximations for these roots began in the robustness literature where M-estimates are the roots of certain estimating equations and the interestwas in determining their distributions when sample sizes are small. Hampel (1973), Field and Hampel (1982), and Field (1982) were instrumental in developing this general approach.

Saddlepoint approximation for a vector of ratios, such as for example (R1, R2, R3) = {U1/V, U2/V, U3/V}, is presented in section 12.3 and generalizes the results of Geary (1944). An important class of such examples to be considered includes vector ratios of quadratic forms in normal variables. A particularly prominent example in times series which is treated in detail concerns approximation to the joint distribution for the sequence of lag correlations comprising the serial autocorrelation function.

In engineering reliability and multistate survival analysis, the machine or patient is viewed as a stochastic system or process which passes from one state to another over time. In many practical settings, the state space of this system is finite and the dynamics of the process are modelled as either a Markov process or alternatively as a semi-Markov process if aspects of the Markov assumption are unreasonable or too restrictive.

This chapter gives the CGFs connected with first passage times in general semi-Markov models with a finite number of states as developed in Butler (1997, 2000, 2001). Different formulae apply to different types of passage times, however all of these CGF formulae have one common feature: they are all represented as explicit matrix expressions that are ratios of matrix determinants. When inverting these CGFs using saddlepoint methods, the required first and second derivatives are also explicit so that the whole saddlepoint inversion becomes a simple explicit computation. These ingredients when used with the parametric plot routine in Maple, lead to explicit plots for the first passage density or CDF that completely avoid the burden of solving the saddlepoint equation.

Distributions for first passage or return times to a single absorbing state are considered in section 13.2 along with many examples derived from various stochastic models that arise in engineering reliability and queueing theory. Distributions for first passage to a subset of states require a different CGF formula which is developed in section 13.3. Passage times for birth and death processes can be approached from a slightly different perspective due to the movement restriction to neighboring states.

Up to now, all of the saddlepoint formulas have involved the univariate normal density function ø(z) and its CDF Φ(z). These expressions are called normal-based saddlepoint approximations and, for the most part, they serve the majority of needs in a wide range of applications. In some specialized settings however, greater accuracy may be achieved by using saddlepoint approximations developed around the idea of using a different distributional base than the standard normal.

This chapter presents saddlepoint approximations that are based on distributions other than the standard normal distribution. Suppose this base distribution has density function λ(z) and CDF Λ(z) and define the saddlepoint approximations that use this distribution as (λ, Λ)-based saddlepoint approximations. Derivations and properties of such approximations are presented in section 16.1 along with some simple examples. Most of the development below is based on Wood et al. (1993).

Most prominent among the base distributions is the inverse Gaussian distribution. The importance of this base is that it provides very accurate probability approximations for certain heavy-tailed distributions in settings for which the usual normal-based saddlepoint approximations are not accurate. These distributions include various first passage times in random walks and queues in which the system is either unstable, so the first passage distribution may be defective, or stable and close to the border of stability. Examples include the first return distribution to state 0 in a random walk that is null persistent or close to being so in both discrete and continuous time. A second example considers a predictive Bayesian analysis for a Markov queue with infinite buffer capacity.

Saddlepoint approximations for conditional densities and mass functions are presented that make use of two saddlepoint approximations, one for the joint density and another for the marginal. In addition, approximations for univariate conditional distributions are developed. These conditional probability approximations are particularly important because alternative methods of computation, perhaps based upon simulation, are likely to be either very difficult to implement or not practically feasible. For the roles of conditioning in statistical inference, see Reid (1995).

Conditional saddlepoint density and mass functions

Let (X, Y) be a random vector having a nondegenerate distribution in ℜm with dim(X) = mx, dim(Y) = my, and mx + my = m.With all components continuous, suppose there is a joint density f (x, y) with support (x, y) ε X ε ⊆ ℜm. For lattice components on Im, assume there is a joint mass function p (j, k) for (j, k) ε X ε ⊆ Im. All of the saddlepoint procedures discussed in this chapter allowboth X and Y to mix components of the continuous and lattice type.

Approximations are presented below in the continuous setting by using symbols f, x, and y. Their discrete analogues simply amount to replacing these symbols with p, j, and k. We shall henceforth concentrate on the continuous notation but also describe the methods as if they were to be used in both the continuous and lattice settings. Any discrepancies that arise for the lattice case are noted.