Introduction

Introduction

Energy and Power

Basic Operations

Practice Problems

Transformation of signals defined piecewise

Periodic signals

Even and Odd Signals

Commonly encountered signals

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. More detailed examples can be found Example 1 - Audio signal Example 1: One of the examples that we will use often in this class is that of audio signals. When a person speaks (or sings) into a microphone, the voltage at the output of the microphone changes with time. Such a signal is shown in the figure below. Example 2 Example2: Consider the signal $x(t)$ given by (1) \begin{align} x(t)=t \cos t \nonumber \end{align} We will also encounter signals that describe some phenomena that change with frequency. Such signals will be denoted by $X(\omega)$ where $\omega$ is the independent variable and the dependent variable $X$ changes with frequency. Here is an example. Example 3 - a signal with frequency being the independent variable Example3: Consider the signal $X(\omega)$ given by (2) \begin{align} X(\omega)= \frac{j\omega}{1+j \omega} \nonumber \end{align} For every $\omega$, $X(\omega)$ takes a complex value and hence it is a complex function of a real variable or equivalently a complex signal.

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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 $x(\pi)= \pi \cos\pi = -\pi$.

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) A CT signal is ‘sampled’ periodically by keeping their value only for discrete time instants typically spaced uniformly. 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.

Example 1 for DT signal 

Let $\underline{x}[n]$ denote the maximum temperature in College Station during the $n$th day in 2016. This definition makes sense only if $n$ takes integer values and an example of such an $\underline{x}[n]$ is shown below

Example 2 for DT signal - sampled signals

Sampling: We will often obtain DT Signals by sampling CT signals as shown in the figure below.

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.

Examples for mathematical descriptions of signals

$x(t) = \sin(200\pi t)$

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$\textit{x(t)}= e^{-t}$

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$x[n]= n^2$

Example of a signal defined piecewise

(3) \begin{align} x(t)= \begin{cases} 10 & \text{if}\ t<2 \\ 10e^{-t/RC} & \text{otherwise} \end{cases} \end{align}

Example of a signal defined pictorially

Consider the signal $x(t)$ shown in the picture below

The signal $x(t)$ can be mathematically expressed as,

(4)

\begin{align} x(t)= \begin{cases} 1-|t| & \text{if}\ -1<t<1 \\ 0 & \text{otherwise} \end{cases} \end{align}

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Practical Examples

Signals are everywhere in modern life. Here are a few examples.

Audio signals in cellular phones and CDs

Audio signals in cellular phones

A good example of where we encounter signals and systems is cellular phones. When we talk on a cellular phone, our voice is first converted to an electrical signal by a microphone in the phone. The output of the microphone is a CT signal $x(t)$. Starting from 2G cellular phones, digital communications has been used to transmit our voice signals. In such a system, the signal $x(t)$ is first sampled at the rate of 8000 samples/s and we obtain the DT signal $x[n] = x(nT_s)$, where $T_s = 1/8000$.

Audio signals in CDs

The same audio signal when recorded on a CD is sampled by taking 44,100 samples/s or, equivalently, $x[n] = x(nT_s), T_s = 1/44100$.

ECG signal

ECG signals

In an electrocardiogram, the activity of the heart is sensed by observing electrical changes which are sensed by electrodes placed under the skin. The electrodes essentially records a voltage as a function of time which corresponds to the activity of the heart. For more details see the [| ECG wiki page]. A typical ECG waveform is given below.

Images

Images

A digital image is a 2-D array of pixel and the amount of Red, Green and Blue light at pixel is typically recorded as three different arrays in order to capture a color image. Each one of the components R,G and B is a 2-D signal where the $X$ and $Y$ axes represent spatial dimension and the signal is a function of $x$ and $y$ coordinates $i$ and $j$, respectively. Thus, the image can be thought of as a 2-D signal $A(i,j)$ where the independent variables $i$ and $j$ represent spatial coordinates and do not represent time. A photo taken using a film camera would have produced a continuous-time version of the digital image, namely a continuous-space image.

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MATLAB Exercises

Exercise 1 - Create your own audio file that is at least 4 seconds long (the exact time duration is not really important, but do not make the file too long). Use the wavrecord command in MATLAB and a sampling frequency of 10000 Hz. You can also try to find a wav file online. Here is one that I like samplewavfile . Use the sound command in MATLAB to play the sound. Make sure the recording is fine.

Click here for Solutions

MATLAB code for Exercise 1

fs = 10000;
recObj = audiorecorder(fs, 8, 1);
recordblocking(recObj, t);
y = getaudiodata(recObj);
sound(y, fs);

Exercise 2 - Using the wavread command, read the signal into a vector called x. Also read the sampling frequency in to a variable called Fs. Make sure you understand what this sampling frequency means. Plot the received signal as a function of time. Your time axis must have units in seconds.

Click here for Solutions

MATLAB code for Exercise 2

[x, fs] = audioread('anykey.wav');
t = 1/fs:1/fs:length(x)/fs;
plot(t, x);
xlabel('time (s)');
ylabel('V');

Exercise 3 - Using the stem command in MATLAB plot the signal. What is the difference between this plot and the plot in Example 2

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MATLAB code for Exercise 2

[x, fs] = audioread('anykey.wav');
t = 1/fs:1/fs:length(x)/fs;
stem(t, x);
xlabel('time (s)');
ylabel('V');

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