CHAPTER 1

Basic Descriptions and Properties

This first chapter gives basic descriptions and properties of deterministic data and random data to provide a physical understanding for later material in this book. Simple classification ideas are used to explain differences between stationary random data, ergodic random data, and nonstationary random data. Fundamental statistical functions are defined bywords alone for analyzing the amplitude, time, and frequency domain properties of single stationary random records and pairs of stationary random records. An introduction is presented on various types of input/output linear system problems solved in this book, as well as necessary error analysis criteria to design experiments and evaluate measurements.

**1.1 DETERMINISTIC VERSUS RANDOM DATA**

Any observed data representing a physical phenomenon can be broadly classified as being either deterministic or nondeterministic. Deterministic data are those that can be described by an explicit mathematical relationship. For example, consider a rigid body that is suspended from a fixed foundation by a linear spring, as shown in Figure 1.1. Let *m* be the mass of the body (assumed to be inelastic) and *k* be the spring constant of the spring (assumed to be massless). Suppose the body is displaced from its position of equilibrium by a distance *X* and released at time *t* = 0. From either basic laws of mechanics or repeated observations, it can be established that the following relationship will apply: