BME Signals : Signals

A signal can represent any quantity that can be measured or modeled. Signals can represent phyiscal events such as electrical potentials in a beating heart or the arrival times of a bus.

In this course, we consider one-dimensional signals. This means that the signal level is a function of only one parameter. For example x(t) is a one-dimensional signal, while x(s,t) is not. In practice, signals can be a function of any number of parameter, but will first focus on understanding how to handle the more one-dimensional case. For convenience, we refer to the free parameter as "time." Often, this variable may actually represent time, but this is not typically relevant to the analysis.

In this course, signals are considered in two separate contexts: continuous and discrete.

A continuous signal represets a process that has a value at every value of its parameter. This value could be zero or undefined, but it makes sense to ask what the value it. We represent such signals with rounded parentheses: x(t). The variable "t" is typically used to indicate a continuous time function.

Here is an example of a continuous time signal. Sometimes, a vertical axis will be drawn in to specify the height of each of these amplitudes.

Other signals represent discontinuous processes, such as a list of the daily close of stock prices or digital signals represented on a computer. Only the series of numbers a limited set of points is available, and one cannot evaluate the function between sets of points. We represent these signals with square brackets: x[n]. The variable "n" is typically used to indicate a discrete time function.

Here is an example of a discrete time signal. Note that the samples are drawn a "lollipops" and that no lines connect adjacent signals.

Property
Periodic/Aperiodic	for any integer k and at least one period (interval) T.	A periodic signal is a signal that consists of a set of repeating sequences. Aperiodic signals are signals that are not periodic.
Bounded		Bounded signals are less than a finite value for all time. For example, sine and cosine are bounded, but exp(t) and exp(-t) are not bounded: Exp(t) goes to infinity as t goes to infinity, while exp(-t) goes to infinity as t goes to negative infinity.
Even Symmetric		Even signals are symmetric about the origin.
Odd Symmetric		Odd signals are anti-symmetric about the origin.

Every signal can be decomposed (broken down into) an even part and odd part. Consider an arbitrary signal :

Let:

Observe that can be recovered:

is the "even part" of , while is the "odd part" of .

These properties will be important to consider as we progress through this semester.