Signals and Systems (Baraniuk et al.)

Signal Classifications and Properties

Introduction

This module will begin our study of signals and systems by laying out some of the fundamentals of signal classification. It is essentially an introduction to the important definitions and properties that are fundamental to the discussion of signals and systems, with a brief discussion of each.

Classifications of Signals

Continuous-Time vs. Discrete-Time

As the names suggest, this classification is determined by whether or not the time axis is discrete (countable) or continuous (Figure ). A continuous-time signal will contain a value for all real numbers along the time axis.

In contrast to this, a discrete time signal, often created by sampling a continuous signal, will only have values at equally spaced intervals along the time axis. Figure

Analog vs. Digital

The difference between analog and digital is similar to the difference between continuous-time and discrete-time. However, in this case the difference involves the values of the function. Analog corresponds to a continuous set of possible function values, while digital corresponds to a discrete set of possible function values. An common example of a digital signal is a binary sequence, where the values of the function can only be one or zero.

The notion of signal size or energy is formally addressed through the mathematical concept of norms. There are many types of norms that can be defined for signals, some of the most important of which have been discussed here. For each type norm and each type of signal domain (continuous or discrete, and finite or infinite) there are vector spaces defined for signals of finite norm. Finally, while nonzero periodic signals have infinite energy, they have finite power if their single period units have finite energy.

Introduction

This module will look at two signal operations affecting the time parameter of the signal, time shifting and time scaling. These operations are very common components to real-world systems and, as such, should be understood thoroughly when learning about signals and systems.

Manipulating the Time Parameter

Time Shifting Time shifting is, as the name suggests, the shifting of a signal in time. This is done by adding or subtracting a quantity of the shift to the time variable in the function. Subtracting a fixed positive quantity from the time variable will shift the signal to the right (delay) by the subtracted quantity, while adding a fixed positive amount to the time variable will shift the signal to the left (advance) by the added quantity.