Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

1.1 INTRODUCTION

A physical process, either natural or synthetic (human-made), can be viewed as a signal with time-varying features and properties that can be modeled as a random process. The term signal implies that some type of information is represented by the physical process, and random means that future outcomes of the process are unpredictable to some degree. Examples of natural random signals include: ultraviolet radiation impinging on a tree, a planet revolving around a star, and a tornado moving across an open field. Synthetic examples of random signals include: a microwave signal transmitted from a cell phone to a base station, an automobile traveling from Los Angeles to San Francisco, and a baseball thrown by a pitcher to a catcher. These examples could be modeled using a function that describes a trajectory in the three spatial dimensions as it varies over continuous time t. Obviously, these “signals” have different physical mechanisms, different levels of predictability, and contain different amounts of information. Figure 1.1 shows possible trajectories (called realizations) of a tornado in the x-direction as a function of time. In this single spatial dimension, we see that even the trajectories of a complicated natural process resembles our notion of a signal.

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