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Signal Classifications and Properties

By Melissa Selik, Richard Baraniuk, Michael Haag

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 1). 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 1

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.

Figure 2

Periodic vs. Aperiodic

Periodic signals repeat with some period TT, while aperiodic, or nonperiodic, signals do not (Figure 3). We can define a periodic function through the following mathematical expression, where t t can be any number and T T is a positive constant:

ft=fT+t f t f T t
1
The fundamental period of our function, ft f t , is the smallest value of T T that the still allows Equation 1 to be true.

(a) A periodic signal with period T 0 T 0
(b) An aperiodic signal
Figure 3

Finite vs. Infinite Length

As the name implies, signals can be characterized as to whether they have a finite or infinite length set of values. Most finite length signals are used when dealing with discrete-time signals or a given sequence of values. Mathematically speaking, ft f t is a finite-length signal if it is nonzero over a finite interval t 1 <ft< t 2 t 1 f t t 2 where t 1 > t 1 and t 2 < t 2 . An example can be seen in Figure 4. Similarly, an infinite-length signal, ft f t , is defined as nonzero over all real numbers: ft f t

Figure 4: Finite-Length Signal. Note that it only has nonzero values on a set, finite interval.

Causal vs. Anticausal vs. Noncausal

Causal signals are signals that are zero for all negative time, while anticausal are signals that are zero for all positive time. Noncausal signals are signals that have nonzero values in both positive and negative time (Figure 5).

(a) A causal signal
(b) An anticausal signal
(c) A noncausal signal
Figure 5

Even vs. Odd

An even signal is any signal ff such that ft=ft f t f t . Even signals can be easily spotted as they are symmetric around the vertical axis. An odd signal, on the other hand, is a signal ff such that ft=ft f t f t (Figure 6).

(a) An even signal
(b) An odd signal
Figure 6

Using the definitions of even and odd signals, we can show that any signal can be written as a combination of an even and odd signal. That is, every signal has an odd-even decomposition. To demonstrate this, we have to look no further than a single equation.

ft=12(ft+ft)+12(ftft) f t 1 2 f t f t 1 2 f t f t
2
By multiplying and adding this expression out, it can be shown to be true. Also, it can be shown that ft+ft f t f t fulfills the requirement of an even function, while ftft f t f t fulfills the requirement of an odd function (Figure 7).

Example 1
(a) The signal we will decompose using odd-even decomposition
(b) Even part: et=12(ft+ft) e t 1 2 f t f t
(c) Odd part: ot=12(ftft) o t 1 2 f t f t
(d) Check: et+ot=ft e t o t f t
Figure 7

Deterministic vs. Random

A deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table. Because of this the future values of the signal can be calculated from past values with complete confidence. On the other hand, a random signal has a lot of uncertainty about its behavior. The future values of a random signal cannot be accurately predicted and can usually only be guessed based on the averages of sets of signals (Figure 8).

(a) Deterministic Signal
(b) Random Signal
Figure 8
Example 2

Consider the signal defined for all real tt described by

f ( t ) = sin ( 2 π t ) / t t 1 0 t < 1 f ( t ) = sin ( 2 π t ) / t t 1 0 t < 1
3

This signal is continuous time, analog, aperiodic, infinite length, causal, neither even nor odd, and, by definition, deterministic.

Signal Classifications Summary

This module describes just some of the many ways in which signals can be classified. They can be continuous time or discrete time, analog or digital, periodic or aperiodic, finite or infinite, and deterministic or random. We can also divide them based on their causality and symmetry properties. There are other ways to classify signals, such as boundedness, handedness, and continuity, that are not discussed here but will be described in subsequent modules.