What is a signal?

The word 'signal' has been used in different contexts in the English language and it has several different meanings . In this class, we will use the term signal to mean a function of an independent variable that carries some information or describes some physical phenomenon. Often (not always) the independent variable will be time, and the signals will describe phenomena that change with time. Such a signal can be denoted by ${x}(t)$, where $t$ is the independent variable and ${x}(t)$ denotes the function of $t$. Notice that this is slightly in contrast to the notation that you may have been used to from your calculus courses. There, you may have used $y=f(x)$ to denote a function of $x$, where $x$ is the dependent variable and $y$ is the independent variable. In this course, since signals will be referred to as ${x}(t)$, ${x}$ refers to the dependent variable typically. Here are two examples of such signals.

This notation, even though fairly standard in the literature, is potentially confusing since $x(t)$ is used to refer to two related but different things. Consider the sentence "A recording of John's speech will be denoted by $x(t)$ and a recording of Adele's music will be denoted by $y(t)$". Here $x(t)$ and $y(t)$ refer to the entire signals, i.e., the audio waveforms. However, if you consider the sentence "find all values of $t$ for which $x(t) < 2$", here $x(t)$ refers to the value taken by the signal at time $t$. To elaborate further, it is the function $x$ evaluated at time $t$. In the Example 2 above $x(\pi)= \pi \cos\pi = -\pi$. This terminology is fairly standard in all text books, but in my opinion this leads to confusion. Therefore, we will use underlined variables to denote signals and variables without underlines will refer to values of the signals. With this notation, the signal will be denoted by $\underline{x}(t)$ and the value taken by this signal at time $t$ will be denoted by $x(t)$.

Continuous-time (CT) and Discrete-time (DT) signals

We will encounter two classes of signals in this course. The first class of signals are those for which the independent variable changes in a continuous manner or, equivalently, the signal $\underline{x}(t)$ is defined for every real value (or a continuum of values) of $t$ in the range $(a,b)$ ($a$ can be $-\infty$ and $b$ can be $\infty$). The two examples considered above are examples of CT signals. In contrast, we will also be interested in signals which are defined only for integer values of the independent variable. These signals are called discrete-time (DT) signals and will be denoted by $\underline{x}[n]$. Such signals arise in two situations - (i) the phenomenon that is being modeled is naturally one for which the independent variable takes only integer values or (ii) We can obtain a DT signal from a CT signal by ‘sampling’ the CT signal. For e.g., we can choose to keep only values of the signal $\underline{x}(t)$ at time instants $nT_s, \forall n$ for a fixed sampling interval $T_s$. From the sampled values we can construct a DT signal $\underline{x}[n]$ by assigning $x[n] = x(nT_s)$. The following two examples elaborate on these two methods.

How to specify or describe signals

There are two ways in which we will specify or describe signals in this course. The first way is to provide an explicit mathematical description of the signals such as $x(t) = \sin(200\pi t)$ or $x(t) = e^{-t}$. Sometimes, these signals may have to be described piecewise. Often, it will be easier to describe signals by sketching the function described by the signals or "drawing a picture of the signal". One of the skills that a student should develop from this part of the course is to be able to write a mathematical description for a signal defined pictorially and vice versa. The following examples illustrates these ideas.

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