Assuming both symmetric and asymmetric exponentially distributed network link delays, this chapter is focused on finding the BLUE-OS and the MVUE for the clock offset between two nodes and evaluates their performance in terms of the MSE, which is chosen as the performance criterion throughout this book. The timing exchange mechanism between the two nodes is the same classical two-way message exchange mechanism adopted in protocols such as TPSN, NTP, etc.

The main topics in this chapter are as follows. First, BLUE-OS, an estimation scheme that has gone largely unnoticed in engineering literature, is investigated in the context of clock offset and the relevant clock offset estimators are derived. Second, the Rao–Blackwell–Lehmann–Scheffé theorem is used to derive the MVUE and it is shown that the MVUE coincides with the BLUE-OS. Therefore, in the class of unbiased estimators, BLUE-OS is the optimal solution and no other estimator can be found with less MSE (or variance, which is the same as MSE in the unbiased case) than MVUE. For the sake of completeness, the clock offset estimators are also derived for two scenarios, namely when the mean of the exponential link delays is known for each direction and when it is unknown for each direction. Third, a short commentary on whether the MVUE is the best possible solution as compared to the other estimators, such as the MLE, is presented.

We now turn our attention to a more accurate model defining the relationship between two clocks by the addition of clock skew. In practice, the time synchronization problem in WSNs generally involves two steps: synchronizing the nodes in the network to one common absolute time by adjusting clock phase offset (clock offset) among the nodes, and correcting the clock frequency offset (clock skew) relative to a certain standard frequency. The second step is required because the imperfections in quartz crystals and environmental conditions induce different clocks to run at slightly different frequencies. Actually, the effect of clock skew is the main reason why clock offset keeps drifting apart. Hence, adjusting clock skew guarantees long-term reliability of synchronization, and therefore reduces network-wide energy consumption in synchronization procedures. Indeed, developing long-term and network-wide time synchronization protocols that are energy-efficient represents one of the key strategies for the successful deployment of long-lived WSNs.

The main topics in this chapter are as follows. First, the MLE and the corresponding CRLB for the conventional clock offset model in a general sender–receiver protocol assuming a Gaussian model for the noise are derived. Second, the joint MLE and corresponding CRLB using a more realistic linear clock offset and skew model assuming Gaussian random delays are obtained. Third, the CRLB for the clock offset for the exponential delay model is derived as a performance threshold. Fourth, the joint MLE for the clock offset and skew under the exponential delay model is obtained and the corresponding algorithms to find these estimators are described in detail.

Much attention has been paid to WSNs due to their capability of serving a variety of purposes. Time synchronization is a significant factor in WSNs, and a number of fundamental operations, such as data fusion, power management, and transmission scheduling, require accurate time synchronization. Since the conventional time synchronization protocol for the Internet cannot be directly applied to WSNs, a number of synchronization protocols have been proposed to meet the unique requirements of sensor network applications.

The importance of time synchronization also comes from the evolution of WSNs which has been driven by technological advances in diverse areas. For instance, unlike the currently deployed WSNs, the next generation of sensor networks may consist of dynamic mobile sensors or a mixture of static and dynamic sensors. In this scenario, far more sophisticated time synchronization protocols that efficiently deal with the mobility of sensors will be required. Indeed, as the sensor network becomes more complicated, the role of time synchronization will become much more important.

In this book, the basic features and theoretical background of the time synchronization problem in WSNs were introduced and then the basic approaches were analyzed and compared to reveal the general ideas and features of time synchronization protocols in WSNs. In addition, a survey of existing time synchronization protocols in the literature was provided including the most recent results.

As a main feature of this book, the problem of time synchronization was studied from a statistical signal processing point of view. This book targeted the clock synchronization problem in a general sender–receiver and receiver–receiver timing packet exchange scenario.

What Is Modulation?

Data bits are mathematical entities that have no physical attributes. To send them over a channel, one needs to first map them into some physical signal, which is then “fed” into a channel to produce a physical signal at the channel's output. For example, when we send data over a telephone line, the data bits are first converted to an electrical signal, which then influences the voltage measured at the other end of the line. (We use the term “influences” because the signal measured at the other end of the line is usually not identical to the channel input: it is typically attenuated and also corrupted by thermal noise and other distortions introduced by various conversions in the telephone exchange system.) Similarly, in a wireless system, the data bits are mapped to an electromagnetic wave that then influences the electromagnetic field measured at the receiver antenna. In magnetic recording, data bits are written onto a magnetic medium by a mapping that maps them to a magnetization pattern, which is then measured (with some distortion and some noise) by the magnetic head at some later time when the data are read.

In the first example the bits are mapped to continuous-time waveforms corresponding to the voltage across an impedance, whereas in the last example the bits are mapped to a spatial waveform corresponding to different magnetizations at different locations across the magnetic medium. While some of the theory we shall develop holds for both cases, we shall focus here mainly on channels of the former type, where the channel input signal is some function of time rather than space.

Claude Shannon, the father of Information Theory, described the fundamental problem of point-to-point communications in his classic 1948 paper as “that of reproducing at one point either exactly or approximately a message selected at another point.” How engineers solve this problem is the subject of this book. But unlike Shannon's general problem, where the message can be an image, a sound clip, or a movie, here we restrict ourselves to bits. We thus envision that the original message is either a binary sequence to start with, or else that it was described using bits by a device outside our control and that our job is to reproduce the describing bits with high reliability. The issue of how images or text files are converted efficiently into bits is the subject of lossy and lossless data compression and is addressed in texts on information theory and on quantization.

The engineering solutions to the point-to-point communication problem greatly depend on the available resources and on the channel between the points. They typically bring together beautiful techniques from Fourier Analysis, Hilbert Spaces, Probability Theory, and Decision Theory. The purpose of this book is to introduce the reader to these techniques and to their interplay.

The book is intended for advanced undergraduates and beginning graduate students. The key prerequisites are basic courses in Calculus, Linear Algebra, and Probability Theory. A course in Linear Systems is a plus but not a must, because all the results from Linear Systems that are needed for this book are summarized in Chapters 5 and 6.