An Introduction To Signals
Why Do You Need To Know About Signals?
What Are Signals?
Signal Representation - Sinusoidal Signals
     Amplitude
     Frequency
     Phase
What If Things Are A Little Different?
Problems
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Why Do You Need To Know About Signals?

        Electricity has been with us for a while now, and we use it in many ways that were never anticipated when we began to use electricity.

        AC systems prevailed, despite Edison's protestations.  AC systems were based on a time varying voltage.  (AC stands for Alternating Current.)  As people learned to deal with these time varying voltages they realized that time varying voltages could be used to transmit information.  With the invention of radio it was realized that information could be transmitted by controlling the parameters of a sinusoidally varying voltage.

        Today, we have vast industries that use electricity to distritute information.  Those industries are the TV networks (over the air, cable and satellite), phone systems, the internet and other radio communciation systems, for example.

        If you want to understand the basics of how the information distribution industry works you will need to know about the various forms of electrical signals.


Goals For This Lesson

        Goals for this lesson are simple.

   Given a signal,
   Be able to describe the signal when possible.
   Be able to use time-varying currents and voltages in KCL and KVL.

Electrical Signals

        There are many different kinds of electrical signals.  If we look at how signals are generated, we find that there are many different kinds of electrical signal sources.  Here are some.

        What could possibly all of these signals have in common?  Well, they are all electrically generated signals, and electrical engineers have to deal with them.         What is common through all of this is the need to be able to manipulate signals in different ways.  That might include doing one or more of the following. When you start to operate on signals in this way you are entering the realm of signal processing.  Today, signal processing is often done after digitizing a signal - making a digital version of the signal - and the processing that is done there is referred to as digital signal processing, or simply DSP.

        If you will be dealing with signals, then you will need to have some sort of model for the signals you work with.  Usually that will be some sort of mathematical representation.

        The simplest representation for a signal is to represent the signal as a function of time.  For example, the voltage that appears across the terminals of a microphone will vary in time when the microphone "picks up" a sound.  Then, we might say something like:

Microphone voltage = Vmike(t)

 Representing a signal as a time function is so common that there are many instruments and data gathering devices that give you a picture of a voltage time function.  The most common instrument that gives a picture of a voltage time function is the oscilloscope.

        At this point, we can consider some specific kinds of signals.  We'll start with periodic signals, and in particular we'll start with sinusoidal signals.


Periodic Signals

        Periodic signals are signals that repeat in time with a certain period.  The most fundamental periodic signal is the sinusoidal signal.  Any other periodic signal can be thought of as a combination of sinusoidal signals added together.  That approach is based on the Fourier Series and you can go to that topic when you know enough about sinusoidal signals.

        In this section you will begin to learn about sinusoidal signals.  Sinusoidal signals, based on sine and cosine functions, are the most important signals you will deal with.  They are important because virtually every other signal can be thought of as being composed of many different sine and cosine signals.  They form the basis for many other things you will do in signal processing and information transmission.  Eventually you will deal with signals as different as voice signals, radar signals, measurement signals and entertainment signals like those found in television and radio.  Sinusoidal signals are the starting point for almost all work in signal processing and information transmission.

        In this part of this lesson you will learn:


Representation of Sinusoidal Signals

        If you put a voltage signal into an oscilloscope you can get a picture of how the signal varies in time.  Sinusoidal signals are often voltages which vary sinusoidally in time.  (Sinusoidal signals could be, however, other physical variables like current, pressure, or virtually any other physical variable.)  Here's a simulator that will let you put various kinds of signals on a simulated oscilloscope.  Click the Start button to see a typical sinusoidal signal.

You can write a mathematical expression for the voltage signal as a function of time.  Call that mathematical expression V(t).  V(t) will have this general form.

V(t) = Vmaxsin(wt + f)

This signal has three parameters, the maximum voltage, or amplitude, denoted by Vmax, the angular frequency denoted by w and the phase angle denoted by f.  We'll examine these separately.



Amplitude of Sinusoidal Signals

        The amplitude of a sinusoidal signal is the largest value it takes (when the sine function has a value of +1 or -1).  Amplitude has whatever units the physical quantity has, so if it is a voltage signal, like the one below, it might have an amplitude of 10 volts.  On the other hand if the shock absorbers on your car are bad, your car can run down the road and vibrate up and down with an amplitude of three inches.  For the signal we saw above - repeated here - the amplitude is 100 volts.  However, you can change the amplitude of the sinusoidal signal here by typing in a different value.  Do that now.


Problem

P1.   Here is a sinusoidal signal.  Willy Nilly has measured this signal, and acquired it in a computer file and plotted it for you.  Determine the amplitude of this signal.

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


Frequency of Sinusoidal Signals

        Frequency is a parameter that determines how often the sinusoidal signal goes through a cycle.  It is usually represented with the symbol f, and it has the units hertz (sec-1 ).  Here is the simulator you saw above.

This signal repeats every 4 seconds.  We say that this signal has a period of 4 seconds, and we usually represent the period of the signal as "T".  Here we have T = 4 sec.  The frequency of the signal is the reciprocal of the period.  That's why the frequency is indicated as 0.25 in the simulator.  When "f" is the frequency, we have:

f = 1/T
f is in Hertz (Hz)
T is in seconds (sec)

For the signal above, the frequency is 0.25 Hz.  We get that from:

f = 1/T = 1/2 = 0.25 Hz.

Now, you can also change the frequency in the simulator above.  Change the frequency and observe what happens.  Try a frequency of 0.5 Hz, 1 Hz, etc.


Problem

P2.   Here is Willy Nilly's signal again.  Determine the frequency of this signal.  Hint:  First determine the period of the signal.

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


        A sinusoidal signal (sine or cosine) can be represented mathematically.  If we attempt to use the information we have on the sinusoidal signal we've been looking at to write a mathematical expression for the signal, we would write (Note the 2p factor in the expression!):

V(t) = A cos(2pft)

In this expression:

        Frequency of a sinusoid is something that you can perceive.  Frequency of a sinusoid determines the pitch you hear in the sound it make when a speaker is driven with a sinusoid.  Here are three signals you can listen to by clicking the hotwords.  They are chosen to be at frequencies in ratios of 2:1.

        Notice how these sounds are what a musician would call an octave apart.


Phase of Sinusoidal Signals

        Sinusoidal signals don't need to start at zero at t = 0.  There are other possibilities.  Here are two sinusoidal signals.

These two signals have the same amplitude and frequency, but they are not the same.  The difference in the two signals is in their phase.  Phase is another parameter of sinusoids.  Consider how we might write mathematical expressions for the signals in the plot above.

Note the following points about sinusoidal signals. Here is the simulator again.  This time you can vary the phase.  We have set things up so that you can input the phase in degrees rather than radians, and we have done the conversion internally.  That's the way EEs normally think of things anyway.


Problems

P3.   Here are two signals.  Both signals have the same amplitude.  Determine the amplitude of the two signals.

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:

P4.   Next, determine the frequency of the two signals.

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:

P5.   Now, determine the phase of the "blue" signal assuming that the "red" signal is the reference.  Give your answer in radians.

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


What If The Signal Isn't Sinusoidal?

        Sinusoidal signals aren't always the most interesting kind of signal.  They keep doing the same thing over and over.  However, other signals which contain information can often be thought of as combinations of sinusoidal signals.  That includes periodic signals - which repeat in time but not sinusoidally - and even non-periodic signals.  Even totally random signals are often viewed as having frequency components and that concept is borrowed from concepts that first arise when you consider sinusoidal signals.

        So, even if you don't ever see a sinusoidal signal again, you may well be trying to deal with sinusoidal components.  As you get into the study of signals you'll deal with numerical algorithms - like the FFT - that decompose signals into sinusoidal components.  You'll find many uses for whatever you learn about sinusoids.


What About Other Signals?

        There are many other kinds of signals besides sinusoidal signals.

        Here is an example of a periodic, non-sinusoidal signal.

Now, difficult as it is to imagine, this signal can be represented as a sum of sinusoidal signals.  While that is a subject for another lesson, you should be motivated to learn everything you can about sinusoidal signals.  The things you don't learn can prevent you from continuing when you get to concepts like representing this signal with a sum of sinusoidal signals.


Other Signals

        Information carrying signals will vary in time.  A constant signal really doesn't convey any information.  However, when you are dealing with time-varying signals things that you know about constant signals may still hold true.  In particular.

We'll ask you to use those in some of the problems below.

Problems
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